volume 49 number 1 june 2019 - …

ISSN 0304-9892 (Print) ISSN 2455-7463 (Online)

(HALF YEARLY JOURNAL OF MATHEMATICS)

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VOLUME 49

NUMBER 1

JUNE 2019

Published by : The Vijñāna Parishad of India

[ Society for Applications of Mathematics ] DAYANAND VEDIC POSTGRADUATE COLLEGE

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Jnanabha, Vol. 49(1) (2019), 1-10

TIME SERIES ANALYSIS OF HEAT STROKEBy

Rashmi Bhardwaj(1)∗ and Varsha Duhoon(2)

(1)∗University School of Basic and Applied Sciences, Non-linear Dynamics Research Lab,Guru Gobind Singh Indraprastha University, Sec-16C, Dwarka, New Delhi-110078, India

Email:[email protected](2)University School of Basic and Applied Sciences, Non-linear Dynamics Research Lab,Guru Gobind Singh Indraprastha University, Sec-16C, Dwarka,New Delhi-110078, India

Email:[email protected]

(Received : January 24, 2018 ; Revised: March 26, 2019)

Abstract

India in terms of Travel and tourism has been ranked at the 40th globally as perthe list of World Economic Forum. Tourism is one of the main source of earning inour country, as it helps in Earning of Foreign Exchange from Tourism (PR) in termsof INR (1 crore=10 million), 135193 crores, Rate of Annual Growth 9.6%, in US$terms Billion US $ 21.07, Rate of Annual Growth 4.1%, which means a lot for thecountry and holds good share in GDP of India. The amount of carbon dioxide releasedby the public transports such as buses, cars and air ways are affecting the air in theatmosphere highly. The rapid increase in the temperature causing deaths due to heatstroke, thus, it is important to take preventive measures to reduce the emission ofcarbon dioxide to control global warming. The paper studies about the change in thepredictability of the temperature using time series analysis for the factors which mayaffect chaotic situation leading to the increase in death rates due to heat stroke alsostudying the regression analysis and descriptive statistics of the data of both tourismand temperature. Increase in temperature causes hazardous change in environment inform of Global Warming. Anti-persistence behavior is observed which is alarming dueto the chaotic nature thus government may take directives as emphasized by globalinitiative like Paris Climate Agreement 2015 in direction of controlling the carbonemission and reducing Global Warming.2010 Mathematics Subject Classification: 37C45Keywords and phrases: Time Series, Predictability Index, Global Warming, COemission, Chaos.

1 IntroductionIndia is a country of flora and fauna with different climate throughout the year as summer,winter, spring, autumn, monsoon. Industry of Tourism is increasing day by day in terms ofvolume and in terms of its importance economically. Many places have gained attention overpast few years as Holiday destination. Despite the fact that now and since past few yearsthe economy of the Holiday destination has become vulnerable as a result of Climate change.Temperature is increasing both globally and regionally as well. Climate change have leadincrease in risk of illness across the world which will result in discouragement of Tourism.

As per the suggestion of climate models a future warning of 0.2 0.3 oC per decade andit is expected a rise of 4-10 cm in the sea level per decade. Climate has important role

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to play in promotion of tourism for places such as beaches, Hill Station and water sportsand the continuous increase in hazardous gases affecting climate and weather is also gravelyaffecting the other economic activities. The aviation sector is going through tremendouspressure since the intergovernmental panel on climate change (IPCC) and environmentalcampaign groups have discussed and signed the reason of rise in the global greenhouse gasemissions due to air travels. It has been seen that world-wide, 16000 commercial air waysproduces above 600 million tons of CO2 every year, India produces 10 million tons of CO2from all sectors ranking 5th next to US, China, Russia, Japan. Increase in green-house gasconcentration can affect the all over rise in days of rain to an intensity of 1-4mm/day apartof areas in northwest India their rainfall has decreased intensity by 1mm/day. The paperstudies about the nature of temperature pattern of every five years from 1901 to 2015 andfurther the death of human beings due to heat stroke increasing every year. The paper willalso study about the increase in the tourism sector and further modes of transportation usedwhich contributes in the emission of CO2 causing global warming, hence increase in thetemperature.

Bhardwaj et al [2-8] studied various statistical characteristics, Wavelets and FractalMethods of various real-life phenomena to forecast weather, rainfall, environment data.Becken et al. [9] studied the impacts of weather on tourist Travel. Bodri [10] appliedRescaled Analysis in order to analyze the Annual mean surface Temperature of differentstations in Hungary. Hu. et al. [15] measured destination attractiveness: a contextualapproach. Maddison [16] studied the change in the flow of British tourism as an impact ofclimate change. Morgan et al. [17] calculated climate index for improved user-based beach.Nyaupane et al. [18] studied the impact of climate change on nature-based tourism in theNepalese Himalayas. Robinson et al. [19] discussed the effects on environment of rise inCarbon Di Oxide. Witt et al. [20] forecasted tourism demand. Wei et al. [23] gave conceptsof Time Series Analysis – Univariate and Multivariate Methods. None of the authors havestudied the time series, predictability analysis of heat stroke.

2 MethodologyThe paper studies different measures of analysis to study the temperature patterns.2.1 Analysis of Regression:Analysis of Regression is a method for studying relationship among the dependent andindependent variable. regression Analysis is used for the purpose of Forecasting and also tostudy how dependent and independent variables are related. Equation can be representedas , where Y is the variable which depends (implies ordinate hence Y axis), X is a variablewhich is independent (implies abscissa hence X axis), b is the slope; y-intercept is denotedas a.2.2 Hurst exponent:Hurst exponent refers to the index of dependence. Hurst exponent value ranges between 0and 1. H equals to 0.5 implies true random walk (that is time series which is Brownian).When H is in range of 0.5 to 1 persistence response (positive autocorrelation) exists. H isin range of 0 and 0.5 then anti-persistence response (non-positive autocorrelation) exists.A cyclic response is seen as Increase will be followed by decrease and vice-verse. MeanReversion term used for this behavior.Algorithm for the calculation on Rescaled Range:

2

1. Average of the Series is Calculated by:

y =1

nΣni=1yi. (2.1)

2. Generate a Series of Adjusted Averages:

Zt = Yt − y, t = 1, 2, 3, ..., n. (2.2)

3. Now, Generating Series of deviated Sum:

Xt = Σni=1Zt, t = 1, 2, 3, ..., n. (2.3)

4. Series of Range is Generated as R:

Rt = max(a1, a2, ..., an)−min(a1, a2, ..., an), t = 1, 2, 3, 4, ..., n. (2.4)

5. Generate Series of Standard Deviation S:

St =

√1

tΣnt=1(Xi − x(t))

2

, t = 1, 2, 3, 4, ..., n, (2.5)

where, x(t) is denoted as the average of Series ‘t’6. Obtaining the rescaled range series (R/S)

(R

S)t =

Rt

St, t = 1, 2, 3, 4, ..., n. (2.6)

2.3 Fractal Dimension:Fractal dimension is a method of studying how completely fractal fill in the space, asexpanded to superior to superior scale. Dimension of Fractal is the Haussdrof Dimension

DH = limφ−→∞

[ln[X(φ)]

(ln[φ])],

where [X(φ)] is denoted as number of radius of open balls to complete the whole set.

Now, as per the above calculations of Hurst exponent; further calculations can be done.2.4 Predictability Index:It analyzes the nature of series of time. Predictability Index (P.I.) is given by

P.I = 2|D − 1.5|.The value of P.I. rises resulting in D more or less than 1.5. If value comes near 0 then it

is called Brownian motion and hence non-predictable.

3 ResultsForeign Tourist arrival is the major source of income in the tourism sector as every yearwe have increasing number of tourists visiting India to the India monuments, Beaches, Hillstations. The main source of transportation used by the tourists is Airways whose fuel emitsCO2 and other hazardous gases which hence leads to increase in the temperate, Humidity,weather changes and further climate changes such as late monsoon, unexpected rainfall.Table 3.1 shows the number of FTAs per year in India. Global warming is affecting thetourism in other way, the number of FTAs is not even throughout the year the peak monthsare May, June, July [18].

May and June are the lean months as they are the hottest months of the year and asa result FTAs are lesser in these months. The months of highest attraction are January,March and December. The most opted mode of transportation by the tourists is air waysand then followed by the land and then sea.

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Table 3.1: Number of FTAs in India since 1981-2015

Year FTAs in India Year FTAs in India1981 1279210 1999 24819281982 1288162 2000 26493781983 1304976 2001 25377821984 1193752 2002 23843641985 1259384 2003 27262141986 1451076 2004 34574771987 1484290 2005 39186101988 1590661 2006 44471671989 1736093 2007 50815041990 1707158 2008 52826031991 1677508 2009 51676991992 1867651 2010 57756921993 1764830 2011 63092221994 1886433 2012 65777451995 2123683 2013 69676011996 2287860 2014 76790991997 2374094 2015 80271331998 2358629

Table 3.2: Descriptive statistics of Temperature and Tourism

STATISTICAL MEASURES TEMPERATURE TOURISMMINIMUM VALUE 29.050 1193752MAXIMUM VALUE 30.030 8027133QUARTILE 1 29.363 1714391.750QUARTILE 3 29.818 4922919.750MEDIAN 29.710 2379229MEAN 29.649 3259631.118VARIANCE (n-1) 0.116 4255831032363.020STANDARD DEVIATION (n-1) 0.340 2062966.561

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Now, The Statistical measurements made in the above table are necessary in order toanalyze central tendencies of data. Mean gives the expected average value about whichthe data is supposed to be symmetrically distributed while the standard deviation givesinformation about the deviation of data points from the mean. Variance shows the deviationin the data from the mean. Quartile is middle value such as Q1 is the middle number betweenthe smallest number and the median of the data set; Q3 is the middle value between themedian and the highest value of the data set. Median is the value which is average of themiddle most value of the data set.

Figure 3.1: Graph 1: Box plot of Temperature and Tourism

Further, regression calculation has been done as follows; where X is temperature theindependent variable and Y implies tourism the dependent variable:

Table 3.3: Regression calculation of Temperature and Tourism

REGRESSION XYMULTIPLE R 0.529277R SQUARED 0.280134STANDARD ERROR 0.292363

As per study it can be seen that there are certain months which are peak months of FTA andfew are Lean month; such as U.S, U.K, Bangladesh, Canada, Malaysia, Australia, Russia,Singapore and Pakistan have peal months as December with a share of 14%, 9.4%, 12.2%,13.7%, 10.5%, 18.2%, 19.5%, 12%, 11.9% respectively. March has FTAs with a share of10.7%, 12.6% from Sri Lanka and Germany. January shares 10.8%, 10% share of FTAs fromJapan and France and for October and November the FTAs arrival from Nepal and China is11.7% and 11.2%. With peak months there are some Lean months too; the months in whichthe FTA arrival is lesser such as 4.2%, 4.5%, 5.1%, 4.8%, 1.7%, 6.7% of Bangladesh, Canada,

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Australia, Germany, Russia, Pakistan and 6%, 4.8%, 6.3% share of May of the countries SriLanka, France, China. April shows 6.5% 6.7%, 6.7% for Malaysia, Nepal, Singapore. ForJanuary, July, September share is 6.5% U.K, 6.8% Japan, 4.9% U.S.

Figure 3.2: Pie-Chart of Mode of Travel of Foreign Tourist Arrivals in India, 2015

The below data shows the most preferred mode of transportation is air ways by the FTAs,and air ways have high contribution to the emission of CO2 which is further causing globalwarming and hence increase in temperature. The change in the color of the marble of TajMahal to slightly yellowish, increase in the death rate due to heat stroke, drought, meltingof ice, increase in sea water level.

0

500

1000

1500

2000

2500

3000

1985 1990 1995 2000 2005 2010 2015 2020

deaths

Figure 3.3: Time series of yearly Deaths due to heat stroke from 1991-2015 [National Disaster Management Authority]

The increasing death rate of the human beings can be seen in the graph which hasincreased substantially past 10 years (fig 2)

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Table 3.4: Regression Equation , Slope , Hurst Exponent (H), Fractal Dimension (D), Predictability Index (PI) and natureof maximum temperature 1901-2015

Year Regression equation b H D PI Nature1901-1905 y = 0.695X+0.5334 0.695 0.694976 1.305024 0.389952 P1906-1910 y = 0.6823X+0.5293 0.6823 0.682337 1.317663 0.364674 P1911-1915 y = 0.534X+0.7427 0.534 0.53401 1.46599 0.06802 P1916-1920 y = 0.632X+0.6228 0.632 0.631982 1.368018 0.263964 P1921-1925 y = 0.517X+0.7461 0.517 0.517023 1.482977 0.034046 P1926-1930 y = 0.4852X+0.7879 0.4852 0.485244 1.514756 0.029512 AP1931-1935 y = 0.5567X+0.6915 0.5567 0.556715 1.443285 0.11343 P1936-1940 y= 0.4698X+0.8145 0.4698 0.469757 1.530243 0.060486 AP1941-1945 y = 0.5623X+0.685 0.5623 0.563183 1.436817 0.126366 P1946-1950 y = 0.5442X+0.7154 0.5442 0.544243 1.455757 0.088486 P1951-1955 y = 0.5952X+0.6447 0.5952 0.595218 1.404782 0.190436 P1956-1960 y = 0.6067X+0.6421 0.6067 0.606662 1.393338 0.213324 P1961-1965 y = 0.4772X+0.7886 0.4772 0.477165 1.522835 0.04567 AP1966-1970 y = 0.5808X+0.6676 0.5808 0.580774 1.419226 0.161548 P1971-1975 y = 0.5776X+0.6676 0.5776 0.577592 1.422408 0.155184 P1976-1980 y = 0.5301X+0.7531 0.5301 0.530065 1.469935 0.06013 P1981-1985 y = 0.5708X+0.6868 0.5708 0.570755 1.429245 0.14151 P1986-1990 y = 0.5896X+0.6513 0.5896 0.589614 1.410386 0.179228 P1991-1995 y = 0.5721X+0.684 0.5721 0.572117 1.42789 0.14422 P1996-2000 y = 0.6379X+0.5993 0.6379 0.637852 1.362148 0.275704 P2001-2005 y = 0.5452X+0.7053 0.5452 0.545243 1.454757 0.090486 P2006-2010 y = 0.6052X+0.6335 0.6052 0.60527 1.3948 0.2104 P2011-2015 y = 0.4729X+0.7905 0.4729 0.4729 1.5271 0.0542 AP

* P- Persistent nature; AP - Anti persistent behavior

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Table 4 shows the calculation of Regression Equation, Slope, Hurst Exponent, FractalDimension, Predictability Index and nature. If the Fractal Dimension is less than 0.5 then itis considered to be Anti-persistent hence showing chaotic behavior and if Fractal Dimensionis greater than 0.5 then it is persistent hence predictable. The table shows at some point thevalue of Fractal Dimension going below 0.5 which means Anti-Persistence behavior which isresult of sudden increase in maximum temperature further followed by intense heat strokewhich resulted in increased death. The anti-persistence behavior of temperature is observedfor past decade. It is to bring to the notice of the concern authorities to take preventivemeasures in order to control the global warming hence leading to temperature change whichis demoralizing tourism which will gradually affect the GDP especially in terms of tertiarysector of country and many low-level income groups. The focus should be to generate fuelswhich are environment friendly and emit lesser hazardous gases. Global warming is the mostfaced challenge for the tourism industry in mountain areas.

4 ConclusionWith India moving a step ahead every day, it is also important for us to look at whathappening in our environment too. The day to day modes of transportation are providingus comfort but at the same time release hazardous gases which have directly or indirectlydifferent impact on the living and non-living at the same time. The chaotic behavior oftemperature is a matter of concern. Increase in temperature is causing hazardous changein environment in form of Global Warming. The present period of 2005-2015 show Anti-persistence behavior which is alarming due to the chaotic nature. This emphasizes thenecessity for global initiative like Paris Climate Agreement 2015 in direction of controllingthe carbon emission and reducing Global Warming.Acknowledgements. For providing research facilities and financial support authors arethankful to Guru Gobind Singh Indraprastha University, Delhi, India. Authors are alsothankful to the referee for his suggestions to bring the paper in its present form.

References[1] J. Aylen, K. Albertson and G. Cavan, The impact of weather and climate on tourist

demand: the case of Chester Zoo, Climatic Change (2014), 127-183.[2] R. Bhardwaj, Wavelets and Fractal Methods with environmental applications. Math-

ematical Models, Methods and Applications, Eds: Siddiqi, A.H., Manchanda, P.,Bhardwaj, R.; (2016), 173-195..

[3] R. Bhardwaj, D. Pruthi, Predictability and Wavelet Analysis of Air Pollutants forCommercial and Industrial Regions in Delhi, Indian Journal of Industrial and AppliedMathematics, 7(2) (2016), 165-174.

[4] R. Bhardwaj, A. Bangia, Complex Dynamics of Meditating Body, Indian Journal ofIndustrial and Applied Mathematics, 7(2) (2016), 106-116.

[5] R. Bhardwaj, Wavelet and Correlation analysis of weather data, International Journalof Current Engineering & Technology, 2(1) (2012), 178-183.

[6] R. Bhardwaj, Wavelet & Correlation analysis of air pollution parameters using Haarwavelet (Level 3), International Journal of Thermal Technologies, 2(2) (2012), 160-164.

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[7] R. Bhardwaj, A. Kumar, P. Maini, S.C. Kar and L.S. Rathore, Bias free rainfallforecast and temperature trend-based temperature forecast based upon T-170 Modelduring monsoon season, Meteorological Applications, 14(4) (2007), 351-360.

[8] R. Bhardwaj and K. Srivastava, Real time Nowcast of a Cloudburst and a Thunder-storm event with assimilation of Doppler Weather Radar data, Natural Hazards, 70(2)(2014), 1357-1383.

[9] S. Becken and J. Wilson, The impacts of weather on tourist Travel, TourismGeographies, 15(4) (2013), 620-639.

[10] C. Bodri, Fractal analysis of climatic data: Mean Annual temperature records inHungary, Theoretical and applied climatology, 49 (1993), 53-57.

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[12] H. Ghanim, Z. Bargaoui and C. Mallet, Investigation of the fractal dimension of rainfalloccurrence in a semi-arid Mediterranean climate, Hydrological Sciences Journal, 58(3)(2013), 483-497.

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[14] J. Fletcher, Input-Output analysis and tourism impact studies, Annals of TourismResearch ,16 (1989), 514529.

[15] Y. Hu and J. Ritchie, Measuring destination attractiveness: a contextual approach,Journal of Travel Research, 32(2) (1993), 2534.

[16] D. Maddison, In search of warmer climates? The impact of climate change on flows ofBritish tourists, Climatic Change, 49 (2001), 193208.

[17] R. Morgan, E. Gatell, R. Junyent, A. Micallef, E. Ozhan and A.T. Williams, Animproved user-based beach climate index, Journal of Coastal Conservation, 6(1)(2000), 41-50.

[18] G. Nyaupane, and N. Chhetri, Vulnerability to climate change of nature-based tourismin the Nepalese Himalayas, Tourism Geographies, 11(1) (2009), 95-119.

[19] A.B. Robinson, N.E. Robinson and W. Soon, Environmental effects of increased at-mospheric carbon dioxide,Journal of American physicians and surgeons,12(3) (2007),79-90.

[20] Witt, F. Stephen and A. Christine, (1995) Forecasting tourism demand: A review ofempirical research, International Journal of Forecasting, 11(3) (1995), 447-475.

[21] W.W.S. Wei, Time Series Analysis Univariate and Multivariate Methods,Pearson/Addison-Wesley,2, (2006).

[22] E. Wilkins, S. De Urioste-Stone, A. Weiskittel and T. Gabe, Weather sensitivity andclimate change perceptions of tourists: A segmentation analysis. Tourism Geographies,Taylor & Francis (2015), 1-17.

[23] E. Wilkins, Effects of Weather Conditions on Tourism Spending: Implications forFuture Trends under Climate Change, Journal of Travel Research, 57(8) (2018), 1042-1053.

[24] L. Wang, B. Fang and R. Law, Effect of air quality in the place of origin on outboundtourism demand: Disposable income as a moderator, Tourism Management, 68 (2018),152-161.

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[25] F. Stephen, Witt, and A. Christine, Forecasting tourism demand: A review of empiricalresearch, International Journal of Forecasting, 11(3) (1995), 447-475.

[26] B. Susann and W. Jude, The impacts of weather on tourist travel. Tourism Geogra-phies, 15(4) (2013), 620-639.

[27] R. Steiger, B. Abegg and L. Janicke, Rain, Rain, Go Away, Come Again Another Day.Weather Preferences of Summer Tourists in Mountain Environments. Atmosphere. 7(5)(2016).

[28] http://tourism.gov.in. Government of India, Ministry of Tourism, Market ResearchDivision, Indian Tourism Statistics 2015.

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Jnanabha, Vol. 49(1) (2019), 11-25

STUDY OF TRAFFIC CONGESTION FLOW USING QUEUEING MODELBy

Jitendra Kumar and Vikas ShindeDepartment of Applied Mathematics

Madhav Institute of Technology & Science, Gwalior - 474005, Madhya Pradesh, IndiaEmail:[email protected], v p [email protected]

(Received : August 06, 2018 ; Revised: December 05, 2018)

Abstract

In this paper, we discuss continuous flow intersection (CFI) using queueing model.Here, A multi dimensional traffic flow is analyze under CFI intersection design whichreducing the delay and increasing traffic capacity. We address such issue using queueingmodel along with its performance measures as (Average number of vehicle in the systemand queue, Average time spent in the system, Average waiting time in queue andtraffic intensity etc.). We evaluate CFI under heterogamous traffic flow conditionsusing MATLAB. The analysis of results shows that the CFI design is certainly moreefficient in comparison to conventional traffic design.2010 Mathematics Subject Classifications: 60K25Keywords and phrases: Continuous flow intersection, Queueing Model, Trafficintensity, CFI T-design & Two-leg design.

1 IntroductionThe road traffic of Indian cities and other country cities are growing rapidly and are becomingeven more heterogeneous as varied vehicles are introduced in the traffic. The developmentof urban transport infrastructure is not possible because of inadequate space there is animmediate need to find alternative solutions of the traffic problem. Optimal utilization ofavailable infrastructure (road space) is redesign and reconstruction in such a way that thecarrying capacity of the roadway is enhanced. This may be achieved by the Continuous FlowIntersection (CFI) concept is an innovative traffic signal intersection. The concept of the CFIhas been considered since quite recently as an alternative intersection design to traditionalat grade and grade-separated intersections. It enables one or more conflicting movements totake place away from the main intersection at a new crossover intersection, which reduces thenumber of conflicts at the central node and also increases the capacity without demandingmore space. This study is concerned with evaluation of CFI under heterogeneous trafficflow conditions. Traffic flow problems have been addressed by many researchers. Vedagiriand Daydar [16] studied continuous flow intersection under heterogeneous conditions usingcomputer simulation. Jagannatha and Bared [7] discussed the design methodologies forproviding pedestrian access and related pedestrian signal timings for CFI with base signaltimings optimized for vehicular traffic performance. Reid and Hummer [15] conductedsimulation experiments using turning movement data from seven existing intersections ofvarying sizes to compare the travel time of conventional and unconventional designs fairly.Vedagiri and Daydar [17] considered the displaced right-turn intersection with a sub levelbetween grade separated and at grade intersections. Wang et al. [18] described the vehiclequeue to reduce the vehicle queue system waiting time and explores methods to solve the

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problem of vehicle queues at intersections. Nico et al [13] discussed traffic flow using queueingmodel. Iglehart and Witt [6] studied heavy traffic congestion flow using multiple channelqueueing models. Abane [1], Analyzed the urban transportation problem in cities of thethird world in the problem of traffic congestion. Jabari [9], described the model traffic flownetworks and deterministic models often, consider traffic flow either from the perspective offluid dynamics. Jabari and Lin [10] studied of microscopic traffic models the behaviour ofindividual road users can be randomized. Woensel [19] Studied of different queueing modelsfor traffic on road used to adequately model uninterrupted traffic flows. Utilize of this tool tofacilitate the optimal positioning of the country points on a highway. Parmar [14] Analyzedof any traffic system is the analysis of delay. Delay is a more different to understand concept.Developed to analyse issues common to highway toll plazas and parking entry/ exit plazas.Also analyze different queueing models in traffic problem through spruced sheet. Modaresand Fakhe [12] discussed the most important issues of material flow and inter-plant trafficwith queueing model ad also, traffic analysis of single queue model (M/M/1) is analyzed ina Carmakes Company (IKCO) through the analysis, the production stop rate and relevantcost are estimated.

In the present investigation, we study a queueing model for continuous flow intersection(CFI). The remaining paper is arranged as follows: In section 2, description of continuous flowintersection and its types. In the section 3 and 4 described queueing model and performancemeasures for single and multiple servers. Sections 5 present CFI design for T and Two-legintersections. In section 6, we have carried out numerical illustration with different session(as Morning, Afternoon and Evening) and the influence of the parameters for the trafficsystem corresponding to performance. Finally, section 7 summarizes the study done anddraws the conclusions.

2 Description of Continuous Flow Intersection (CFI) & Its TypeThe continuous flow intersection (CFI) has been implemented in several locations across theIndia and other countries. The main feature of CFI is to eliminate the conflict betweenleft turn and opposing through traffic by relocating the left turn way several hundred feetupstream of the primary intersection, where they can cross the opposing through traffic.Such a control strategy has the advantage of allowing the through and left turn traffic to runconcurrently at the primary intersection, reducing the number of signal phases. However,it creates four additional signalized crossover intersections to facilitate a left turn crossoveralong each leg of the intersection. These unique geometric features give CFI a larger footprintthan a conventional intersection design.

In the following four figures, we discuss the different CFI design, which is capable toprovide traffic flow by reducing the number of conflicting point.

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Figure 2.1: Graphical illustration of CFI two-leg design

Figure 2.2: Graphical illustration of a CFI-T design

Figure 2.3: Conventional Intersection (16 conflicting points)

Figure 2.4: CFI Intersection (12 conflicting points)

13

3 Description of Queueing ModelQueueing is quite common in many areas, for examples in telephone exchange, in asupermarket, at computer system and traffic etc. We discuss the traffic problems by usingqueueing theory. Two queueing models are employed to analyze the traffic of vehicles on theroad. Different types of intersection points are determined to corresponding traffic flow onthe road with vehicles.

4 Performance Measures of Queueing ModelsTo characterize a queueing system, we have to identify the probabilistic properties of theincoming flow at request, service times and service disciplines. The arrival process can becharacterized by the distribution of the inter-arrival times of the vehicles.

Description of Task Queueing Models

M/M/1 M/M/S

Traffic Intensity: (ρ)

Average number of Vehicles

waiting in Queue:

(Lq) Where


in the system:

(Ls)


waiting time in queue:

(Wq)


time spent in the system:

(Ws)

5 Design of CFI using Queueing ModelWe consider two types design for CFI traffic (i) T-intersection (ii) Two-leg intersection andalso study the different cases corresponding to various way using queueing models.(i) CFI for T Intersection We consider CFI-T intersection form in which left turn crossoveris one approach. (This CFI design comprises two intersections, referred as the major andminor intersections, based on the number of intersection conflict points). We discuss eightdifferent routes of CFI T- intersection where queues may occur either to a traffic signal

14

control or to merging maneuvers of traffic flows. The different formations of queues aregiven below:

Q7 Q4 Q2

Q5 Q6 Q1

Q3

Q8

Figure 5.1: Classification of queue locations in a CFI-T design.

Q1 : Northbound right-turn queue at the major intersection;Q2 : Eastbound through queue at the minor intersection;Q3 : Northbound left-turn queue at the minor intersection;Q4 : Westbound left-turn queue at the major intersection;Q5 : Northbound left-turn queue at the major intersection;Q6 : Southbound left-turn queue at the minor intersection;Q7 : Northbound left-turn queue before the merging point;Q8 : Eastbound right-turn queue before the merging point.These all possible routes or ways are assumed into four combinations. Case I: present at

the signal stop line, is caused by the through or right-turning traffic volume (Q1, Q2). CaseII: queue is caused by left turning vehicles (Q3, Q4). Those vehicles filtered by the upstreamsignals and thus forming the queue at the stop line of the downstream signals are denotedCase III: queue (Q5, Q6). Case IV: queue is commonly observed in a merging area to predictthe queue length in a merging area. The merging flow tends to merge into the mainline duringacceptable gaps. Such a relationship can be described with a classical M/M/1and M/M/3queueing models. The service time is equivalent to the merge time which is assumed to bean exponential distribution. We have carried out the average number of vehicles waitingin queue (Lq), average number of vehicles in the system (Ls), average number of vehicleswaiting time in queue (Wq) and average number of vehicles time spent in the system (Ws)with traffic intensity in (Q7, Q8). The following models:

15

• Case I: It presents signal stop line which is caused by through and right turning trafficvolume (Q1, Q2),• Case II: It is formed by left-turning vehicles (Q3, Q4),• Case III: It is generated by case II which is filtered by upstream signal and forming the

queue at the stop line of the downstream signal (Q1,Q2). This type of traffic queuesusually occur when left turning flow from a CFI leg must consecutively pass two signalsto reach their destination.• Case IV: such queue is observed in merging area where approaching flow exceed the

merging capacity (Q7, Q8).

(ii) CFI for Two-leg IntersectionTwo different designs exist for two-leg CFIs: a symmetrical partial CFI, with its two CFI

legs running in opposite directions; and an asymmetrical partial CFI, with its two CFI legsrunning along with two adjacent directions. Two designs have quite similar structures, thesame set of queue formulations can be used to evaluate their performance. All possible pathsof two-leg CFI are described in four pair of queues mentioned in figure 3.

Q1: Northbound through queue at the major intersection;Q2: Northbound left-turn queue at the south crossover;Q3: Northbound right-turn queue at the major intersection;Q4: Northbound left-turn queue at the major intersection;Q5: Southbound through queue at the south crossover;Q6: Southbound through queue at the major intersection;Q7: Southbound right-turn queue at the north crossover;Q8: Southbound left-turn queue at the north crossover;Q9: Southbound left-turn queue at the major intersection;Q10: Eastbound through queue at the major intersection;Q11: Eastbound left-turn queue at the major intersection;Q12: Westbound through queue at the major intersection; andQ13: Westbound left-turn queue at the major intersection.Based on these paths we create the following pair of combinations.

• Case I: combinationof queues as (Q1, Q6, Q10, Q12),• Case II: combination of queues as (Q2, Q8, Q11, Q13),• Case III: combination of queues as (Q4, Q9),• Case IV: combination of queues as (Q5, Q10).

6 Numerical IllustrationWe consider the following activities for specific process and task. Every task is described byspecific object in different ways. These activities have been described in earlier section. Weconstruct the four tables for M/M/1 model and two tables for M/M/C model with differenttraffic flows. We have also obtained various performance measures using the above queueingmodels, i.e. the average number of vehicle in the system and queue, average time spent inthe system, average waiting time in queue for both models. Further, we plot the graph forstudying between the various performance measures.(i) M/M/1

16

Q8

Q7 Q10 Q9

Q6

Q11 Q10

Q12 Q13

Q1 Q4 Q5

Q3

Q2

Figure 3 Classification of queue locations in a CFI Two-leg

Q1: Northbound through queue at the major intersection;Q2: Northbound left-turn queue at the south crossover; Q3: Northbound right-turn queue at the major intersection;Q4: Northbound left-turn queue at the major intersection;Q5: Southbound through queue at the south crossover; Q6: Southbound through queue at the major intersection;Q7: Southbound right-turn queue at the north crossover; Q8: Southbound left-turn queue at the north crossover; Q9: Southbound left-turn queue at the major intersection;Q10: Eastbound through queue at the major intersection; Q11: Eastbound left-turn queue at the major intersection;

6

Figure 5.2: Classification of queue locations in a CFI Two-leg

We consider the following pattern of vehicles arrival in different time slot (i.e. Morning,Afternoon and Evening) in table 6.1 & 6.3. We evaluated the arrival and service rate CFIfor T and Two-leg intersection respectively. Different performance measures are evaluatedfor T and Two-leg intersection and shown in table 6.2 & 6.4. Similarly we develop table6.5 & 6.6 for multi channels with same input of arrival and service rate and also obtainedvarious performance indices.

17

Table 6.1: CFI T -Intersection

7

Based on these paths we create the following pair of combinations.

Case I: combination of queues as ,

Case II: combination of queues as ,

Case III: combination of queues as ,

Case IV: combination of queues as .

6. Numerical Illustration We consider the following activities for specific process and task. Every task is described by specific object in different

ways. These activities have been described in earlier section. We construct the four tables for M/M/1 model and two

tables for M/M/C model with different traffic flows. We have also obtained various performance measures using the

above queueing models, i.e. the average number of vehicle in the system and queue, average time spent in the system,

average waiting time in queue for both models. Further, we plot the graph for studying between the various performance

measures.

(i) M/M/1

Location

Session/

Timing Arrival Service Rate Arrival Rate

(λ) Arrival Rate (λ)

Service Rate ( μ) Vehicle Min. Vehicle Min.

Type 1 Queue

Model (Q1, Q2)

Morning 35 1.19 42 1.01 29 41 35

Afternoon 32 1.55 27 1.07 21 25 32

Evening 41 18.1 39 1.03 23 38 41

Type 2 Queue

Model (Q3, Q4)

Morning 36 1.32 47 1.44 27 33 36

Afternoon 31 1.2 34 1.01 26 34 31

Evening 29 1.58 39 1.21 18 26 29

Type 3 Queue

Model (Q5, Q6)

Morning 27 1.05 35 1.04 26 34 27

Afternoon 39 1.59 38 1.06 25 36 39

Evening 61 2.09 52 1.45 29 35 61

Type 4 Queue

Model (Q7, Q8)

Morning 38 1.25 45 1.02 30 44 38

Afternoon 39 2.3 59 1.19 17 50 39

Evening 65 2.55 75 2.55 25 29 65

Table 2 Values of Ls, Lq, Ws and Wq for the CFI T – Intersection

Location Session Traffic

Intensity

(ρ)

Average

number

of

Vehicles

in the

system

Average

number

of

Vehicles

waiting in

Queue

Average

number of

Vehicles

time spent

in the

system

Average

number of

Vehicles

waiting

time in

Queue

Type 1 Queue

Model

(Q1, Q2)

Morning 0.707 2 2 0.08 0.06

Afternoon 0.84 5 4 0.25 0.21

Evening 0.605 2 1 0.06 0.04

Type 2 Queue

Model

(Q3, Q4)

Morning 0.82 5 4 0.17 0.14

Afternoon 0.765 3 2 0.13 0.1

Evening 0.694 2 2 0.13 0.09

Type 3 Queue

Model

(Q5, Q6)

Morning 0.764 3 2 0.13 0.09

Afternoon 0.694 2 2 0.09 0.06

Evening 0.829 5 4 0.17 0.14

Type 4 Queue

Model

(Q7, Q8)

Morning 0.682 2 2 0.07 0.06

Afternoon 0.34 5 0.18 0.03 0.01

Evening 0.862 6 5 0.25 0.22

Table 6.2: Values of Ls, Lq, Ws and Wq for the CFI T -Intersection

7

Based on these paths we create the following pair of combinations.

Case I: combination of queues as ,

Case II: combination of queues as ,

Case III: combination of queues as ,

Case IV: combination of queues as .

6. Numerical IllustrationWe consider the following activities for specific process and task. Every task is described by specific object in different

ways. These activities have been described in earlier section. We construct the four tables for M/M/1 model and two

tables for M/M/C model with different traffic flows. We have also obtained various performance measures using the

above queueing models, i.e. the average number of vehicle in the system and queue, average time spent in the system,

average waiting time in queue for both models. Further, we plot the graph for studying between the various performance

measures.

(i) M/M/1

Table 1 CFI T – Intersection

Location

Session/

Timing Arrival Service Rate Arrival Rate

(λ) Arrival Rate (λ)

Service Rate ( μ) Vehicle Min. Vehicle Min.

Type 1 Queue

Model (Q1, Q2)

Morning 35 1.19 42 1.01 29 41 35

Afternoon 32 1.55 27 1.07 21 25 32

Evening 41 18.1 39 1.03 23 38 41

Type 2 Queue

Model (Q3, Q4)

Morning 36 1.32 47 1.44 27 33 36

Afternoon 31 1.2 34 1.01 26 34 31

Evening 29 1.58 39 1.21 18 26 29

Type 3 Queue

Model (Q5, Q6)

Morning 27 1.05 35 1.04 26 34 27

Afternoon 39 1.59 38 1.06 25 36 39

Evening 61 2.09 52 1.45 29 35 61

Type 4 Queue

Model (Q7, Q8)

Morning 38 1.25 45 1.02 30 44 38

Afternoon 39 2.3 59 1.19 17 50 39

Evening 65 2.55 75 2.55 25 29 65


Intensity

(ρ)

Average

number

of

Vehicles

in the

system

Average

number

of

Vehicles

waiting in

Queue

Average

number of

Vehicles

time spent

in the

system

Average

number of

Vehicles

waiting

time in

Queue

Type 1 Queue

Model

(Q1, Q2)

Morning 0.707 2 2 0.08 0.06

Afternoon 0.84 5 4 0.25 0.21

Evening 0.605 2 1 0.06 0.04

Type 2 Queue

Model

(Q3, Q4)

Morning 0.82 5 4 0.17 0.14

Afternoon 0.765 3 2 0.13 0.1

Evening 0.694 2 2 0.13 0.09

Type 3 Queue

Model

(Q5, Q6)

Morning 0.764 3 2 0.13 0.09

Afternoon 0.694 2 2 0.09 0.06

Evening 0.829 5 4 0.17 0.14

Type 4 Queue

Model

(Q7, Q8)

Morning 0.682 2 2 0.07 0.06

Afternoon 0.34 5 0.18 0.03 0.01

Evening 0.862 6 5 0.25 0.22

18

Table 6.3: Two-leg CFI - Intersection

8

Location

Session/

Arrival

Service Rate Arrival Rate (λ) Arrival Rate (λ)

Service Rate ( μ)

Timing Vehicle Min. Vehicle Min.

Type 1 Queue

Model

(Q1, Q2)

Morning 67 1.19 71 1.01 56 70 35

Afternoon 58 2.55 62 1.07 23 58 32

Evening 82 1.16 92 1.1 71 84 41 Type 2 Queue

Model

(Q3, Q4)

Morning 72 2.32 79 1.44 31 55 36

Afternoon 88 1.33 92 1.25 66 74 31


Model

(Q5, Q6)

Morning 78 2.02 84 1.04 39 80 27

Afternoon 71 1.59 75 1.06 45 71 39


Model

(Q7, Q8)

Morning 122 2.25 135 2.02 54 67 38

Afternoon 104 2.3 128 1.75 45 73 39

Evening 131 2.55 152 2.45 51 62 65


Intensity

(ρ)

Average

number of

Vehicles in

the system

Average number

of Vehicles

waiting in Queue

Average number

of Vehicles time

spent in the

system

Average number

of Vehicles

waiting time in

Queue

Type 1 Queue

Model (Q1,

Q2)

Morning 0.707 2 2 0.08 0.06

Afternoon 0.84 5 4 0.25 0.21

Evening 0.605 2 1 0.06 0.04

Type 2 Queue

Model (Q3,

Q4)

Morning 0.82 5 4 0.17 0.14

Afternoon 0.765 3 2 0.13 0.1

Evening 0.694 2 2 0.13 0.09

Type 3 Queue

Model (Q5,

Q6)

Morning 0.764 3 2 0.13 0.09

Afternoon 0.694 2 2 0.09 0.06

Evening 0.829 5 4 0.17 0.14

Type 4 Queue

Model (Q7,

Q8)

Morning 0.682 2 2 0.07 0.06

Afternoon 0.34 5 0.18 0.03 0.01

Evening 0.862 6 5 0.25 0.22

(ii) M/M/C


Intensity

(ρ)

Average

number of

Vehicles

in the

system

Average

number of

Vehicles waiting

in Queue

Average number

of Vehicles time

spent in the system

Average number

of Vehicles

waiting time in

Queue

Type 1 Queue

Model (Q1, Q2)

Morning 0.24 0.72 0.012 0.025 0.0004

Afternoon 0.28 0.86 0.023 0.041 0.0011

Evening 0.2 0.61 0.006 0.027 0.0003

Type 2 Queue

Model (Q3, Q4)

Morning 0.27 0.84 0.021 0.031 0.0008

Afternoon 0.26 0.78 0.016 0.03 0.0006

Evening 0.23 0.7 0.011 0.039 0.0006

Type 3 Queue

Model (Q5, Q6)

Morning 0.25 0.78 0.016 0.03 0.0006

Afternoon 0.23 0.71 0.011 0.028 0.0004

Evening 0.28 0.85 0.022 0.029 0.0007

Type 4 Queue

Model (Q7, Q8)

Morning 0.23 0.69 0.01 0.023 0.0003

Afternoon 0.11 0.34 0 0.02 0

Evening 0.29 0.89 0.025 0.036 0.001

Table 6.4: Values of Ls, Lq, Ws and Wq for Two-leg CFI - Intersection

8

Location

Session/

Arrival


Service Rate ( μ)


Type 1 Queue

Model

(Q1, Q2)

Morning 67 1.19 71 1.01 56 70 35

Afternoon 58 2.55 62 1.07 23 58 32


Model

(Q3, Q4)

Morning 72 2.32 79 1.44 31 55 36

Afternoon 88 1.33 92 1.25 66 74 31


Model

(Q5, Q6)

Morning 78 2.02 84 1.04 39 80 27

Afternoon 71 1.59 75 1.06 45 71 39


Model

(Q7, Q8)

Morning 122 2.25 135 2.02 54 67 38

Afternoon 104 2.3 128 1.75 45 73 39

Evening 131 2.55 152 2.45 51 62 65


Intensity

(ρ)

Average

number of

Vehicles in

the system

Average number

of Vehicles

waiting in Queue

Average number

of Vehicles time

spent in the

system

Average number

of Vehicles

waiting time in

Queue

Type 1 Queue

Model (Q1,

Q2)

Morning 0.707 2 2 0.08 0.06

Afternoon 0.84 5 4 0.25 0.21

Evening 0.605 2 1 0.06 0.04

Type 2 Queue

Model (Q3,

Q4)

Morning 0.82 5 4 0.17 0.14

Afternoon 0.765 3 2 0.13 0.1

Evening 0.694 2 2 0.13 0.09

Type 3 Queue

Model (Q5,

Q6)

Morning 0.764 3 2 0.13 0.09

Afternoon 0.694 2 2 0.09 0.06

Evening 0.829 5 4 0.17 0.14

Type 4 Queue

Model (Q7,

Q8)

Morning 0.682 2 2 0.07 0.06

Afternoon 0.34 5 0.18 0.03 0.01

Evening 0.862 6 5 0.25 0.22

(ii) M/M/C


Intensity

(ρ)

Average

number of

Vehicles

in the

system

Average

number of

Vehicles waiting

in Queue

Average number

of Vehicles time

spent in the system

Average number

of Vehicles

waiting time in

Queue

Type 1 Queue

Model (Q1, Q2)

Morning 0.24 0.72 0.012 0.025 0.0004

Afternoon 0.28 0.86 0.023 0.041 0.0011

Evening 0.2 0.61 0.006 0.027 0.0003

Type 2 Queue

Model (Q3, Q4)

Morning 0.27 0.84 0.021 0.031 0.0008

Afternoon 0.26 0.78 0.016 0.03 0.0006

Evening 0.23 0.7 0.011 0.039 0.0006

Type 3 Queue

Model (Q5, Q6)

Morning 0.25 0.78 0.016 0.03 0.0006

Afternoon 0.23 0.71 0.011 0.028 0.0004

Evening 0.28 0.85 0.022 0.029 0.0007

Type 4 Queue

Model (Q7, Q8)

Morning 0.23 0.69 0.01 0.023 0.0003

Afternoon 0.11 0.34 0 0.02 0

Evening 0.29 0.89 0.025 0.036 0.001

19

Table 6.5: Values of Ls, Lq, Ws and Wq for the CFI T -Intersection

8

Location

Session/

Arrival


Service Rate ( μ)


Type 1 Queue

Model

(Q1, Q2)

Morning 67 1.19 71 1.01 56 70 35

Afternoon 58 2.55 62 1.07 23 58 32


Model

(Q3, Q4)

Morning 72 2.32 79 1.44 31 55 36

Afternoon 88 1.33 92 1.25 66 74 31


Model

(Q5, Q6)

Morning 78 2.02 84 1.04 39 80 27

Afternoon 71 1.59 75 1.06 45 71 39


Model

(Q7, Q8)

Morning 122 2.25 135 2.02 54 67 38

Afternoon 104 2.3 128 1.75 45 73 39

Evening 131 2.55 152 2.45 51 62 65


Intensity

(ρ)

Average

number of

Vehicles in

the system

Average number

of Vehicles

waiting in Queue

Average number

of Vehicles time

spent in the

system

Average number

of Vehicles

waiting time in

Queue

Type 1 Queue

Model (Q1,

Q2)

Morning 0.707 2 2 0.08 0.06

Afternoon 0.84 5 4 0.25 0.21

Evening 0.605 2 1 0.06 0.04

Type 2 Queue

Model (Q3,

Q4)

Morning 0.82 5 4 0.17 0.14

Afternoon 0.765 3 2 0.13 0.1

Evening 0.694 2 2 0.13 0.09

Type 3 Queue

Model (Q5,

Q6)

Morning 0.764 3 2 0.13 0.09

Afternoon 0.694 2 2 0.09 0.06

Evening 0.829 5 4 0.17 0.14

Type 4 Queue

Model (Q7,

Q8)

Morning 0.682 2 2 0.07 0.06

Afternoon 0.34 5 0.18 0.03 0.01

Evening 0.862 6 5 0.25 0.22

(ii) M/M/C


Intensity

(ρ)

Average

number of

Vehicles

in the

system

Average

number of

Vehicles waiting

in Queue

Average number

of Vehicles time

spent in the system

Average number

of Vehicles

waiting time in

Queue

Type 1 Queue

Model (Q1, Q2)

Morning 0.24 0.72 0.012 0.025 0.0004

Afternoon 0.28 0.86 0.023 0.041 0.0011

Evening 0.2 0.61 0.006 0.027 0.0003

Type 2 Queue

Model (Q3, Q4)

Morning 0.27 0.84 0.021 0.031 0.0008

Afternoon 0.26 0.78 0.016 0.03 0.0006

Evening 0.23 0.7 0.011 0.039 0.0006

Type 3 Queue

Model (Q5, Q6)

Morning 0.25 0.78 0.016 0.03 0.0006

Afternoon 0.23 0.71 0.011 0.028 0.0004

Evening 0.28 0.85 0.022 0.029 0.0007

Type 4 Queue

Model (Q7, Q8)

Morning 0.23 0.69 0.01 0.023 0.0003

Afternoon 0.11 0.34 0 0.02 0

Evening 0.29 0.89 0.025 0.036 0.001

Table 6.6: Values of Ls, Lq, Ws and Wq for Two-leg CFI - Intersection

9


Intensity

(ρ)

Average

number of

Vehicles in the

system

Average

number of

Vehicles

waiting in

Queue

Average

number of

Vehicles time

spent in the

system

Average

number of

Vehicles

waiting time in

Queue

Type 1 Queue

Model (Q1, Q2)

Morning 0.27 0.82 0.019 0.015 0.338

Afternoon 0.13 0.4 0.001 0.017 0.053

Evening 0.28 0.87 0.024 0.012 0.331

Type 2 Queue

Model (Q3, Q4)

Morning 0.19 0.57 0.005 0.018 0.156

Afternoon 0.3 0.92 0.029 0.014 0.439

Evening 0.31 0.96 0.033 0.016 0.553

Type 3 Queue

Model (Q5, Q6)

Morning 0.16 0.49 0.003 0.013 0.07

Afternoon 0.21 0.64 0.008 0.014 0.17

Evening 0.25 0.76 0.014 0.026 0.49

Type 4 Queue

Model (Q7, Q8)

Morning 0.27 0.83 0.02 0.015 0.361

Afternoon 0.21 0.62 0.007 0.013 0.152

Evening 0.27 0.84 0.021 0.017 0.414

We consider the following pattern of vehicles arrival in different time slot (i.e. Morning, Afternoon and Evening) in table 1 & 3. We evaluated the arrival and service rate CFI for T and Two-leg intersection respectively. Different performance measures are evaluated for T and Two-leg intersection and shown in table 2 & 4. Similarly we develop table 5 & 6 for multi channels with same input of arrival and service rate and also obtained various performance indices.

1 2 3 4 5 6 7 8 9 10 11 120.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Position of Session (Morning, Afternoon & Evening )

Vule

s of T

raffic

Intensity

Traffic Intensity for CFI T design

Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1

Position of Session (Morning, Afternoon & Evening)

Valu

es of T

raffic

Intensity

Traffic Intensity of CFI Two-leg design

Traffic Intensity

Figure 4 Figure 5

Traffic Intensity for CFI T-design Traffic Intensity for CFI Two-leg design

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of Lq, Ls, W

q &

W

s

Traffic Intensity for CFI T-design (M/M/3)

Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of T

raffic

Intensity

Traffic Intensity for CFI Two-leg design (Multi-Server)

Traffic Intensity

Figure 6 Figure 7

Traffic Intensity for CFI T-design (Multi-Server) Traffic Intensity for CFI Two-leg design (Multi-Server)

20

9

Table 6 Values of Ls, Lq, Ws and Wq for Two-leg CFI – Intersection Location Session Traffic

Intensity

(ρ)

Average

number of

Vehicles in the

system

Average

number of

Vehicles

waiting in

Queue

Average

number of

Vehicles time

spent in the

system

Average

number of

Vehicles

waiting time in

Queue

Type 1 Queue

Model (Q1, Q2)

Morning 0.27 0.82 0.019 0.015 0.338

Afternoon 0.13 0.4 0.001 0.017 0.053

Evening 0.28 0.87 0.024 0.012 0.331

Type 2 Queue

Model (Q3, Q4)

Morning 0.19 0.57 0.005 0.018 0.156

Afternoon 0.3 0.92 0.029 0.014 0.439

Evening 0.31 0.96 0.033 0.016 0.553

Type 3 Queue

Model (Q5, Q6)

Morning 0.16 0.49 0.003 0.013 0.07

Afternoon 0.21 0.64 0.008 0.014 0.17

Evening 0.25 0.76 0.014 0.026 0.49

Type 4 Queue

Model (Q7, Q8)

Morning 0.27 0.83 0.02 0.015 0.361

Afternoon 0.21 0.62 0.007 0.013 0.152

Evening 0.27 0.84 0.021 0.017 0.414

1 2 3 4 5 6 7 8 9 10 11 120.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Vule

s of T

raffic

Intensity


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of T

raffic

Intensity


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of Lq, Ls, W

q &

W

s


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of T

raffic

Intensity


Traffic Intensity

Figure 6.1: Traffic Intensity for CFI T-design

9


Intensity

(ρ)

Average

number of

Vehicles in the

system

Average

number of

Vehicles

waiting in

Queue

Average

number of

Vehicles time

spent in the

system

Average

number of

Vehicles

waiting time in

Queue

Type 1 Queue

Model (Q1, Q2)

Morning 0.27 0.82 0.019 0.015 0.338

Afternoon 0.13 0.4 0.001 0.017 0.053

Evening 0.28 0.87 0.024 0.012 0.331

Type 2 Queue

Model (Q3, Q4)

Morning 0.19 0.57 0.005 0.018 0.156

Afternoon 0.3 0.92 0.029 0.014 0.439

Evening 0.31 0.96 0.033 0.016 0.553

Type 3 Queue

Model (Q5, Q6)

Morning 0.16 0.49 0.003 0.013 0.07

Afternoon 0.21 0.64 0.008 0.014 0.17

Evening 0.25 0.76 0.014 0.026 0.49

Type 4 Queue

Model (Q7, Q8)

Morning 0.27 0.83 0.02 0.015 0.361

Afternoon 0.21 0.62 0.007 0.013 0.152

Evening 0.27 0.84 0.021 0.017 0.414

1 2 3 4 5 6 7 8 9 10 11 120.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Vule

s of T

raffic

Intensity


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of T

raffic

Intensity


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of Lq, Ls, W

q &

W

s


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of T

raffic

Intensity


Traffic Intensity

Figure 6.2: Traffic Intensity for CFI Two-leg design

9


Intensity

(ρ)

Average

number of

Vehicles in the

system

Average

number of

Vehicles

waiting in

Queue

Average

number of

Vehicles time

spent in the

system

Average

number of

Vehicles

waiting time in

Queue

Type 1 Queue

Model (Q1, Q2)

Morning 0.27 0.82 0.019 0.015 0.338

Afternoon 0.13 0.4 0.001 0.017 0.053

Evening 0.28 0.87 0.024 0.012 0.331

Type 2 Queue

Model (Q3, Q4)

Morning 0.19 0.57 0.005 0.018 0.156

Afternoon 0.3 0.92 0.029 0.014 0.439

Evening 0.31 0.96 0.033 0.016 0.553

Type 3 Queue

Model (Q5, Q6)

Morning 0.16 0.49 0.003 0.013 0.07

Afternoon 0.21 0.64 0.008 0.014 0.17

Evening 0.25 0.76 0.014 0.026 0.49

Type 4 Queue

Model (Q7, Q8)

Morning 0.27 0.83 0.02 0.015 0.361

Afternoon 0.21 0.62 0.007 0.013 0.152

Evening 0.27 0.84 0.021 0.017 0.414

1 2 3 4 5 6 7 8 9 10 11 120.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Vule

s of T

raffic

Intensity


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1

Position of Session (Morning, Afternoon & Evening)V

alu

es of T

raffic

Intensity


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of Lq, Ls, W

q &

W

s


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of T

raffic

Intensity


Traffic Intensity

Figure 6.3: Traffic Intensity for CFI T-design(Multi-Server)

9


Intensity

(ρ)

Average

number of

Vehicles in the

system

Average

number of

Vehicles

waiting in

Queue

Average

number of

Vehicles time

spent in the

system

Average

number of

Vehicles

waiting time in

Queue

Type 1 Queue

Model (Q1, Q2)

Morning 0.27 0.82 0.019 0.015 0.338

Afternoon 0.13 0.4 0.001 0.017 0.053

Evening 0.28 0.87 0.024 0.012 0.331

Type 2 Queue

Model (Q3, Q4)

Morning 0.19 0.57 0.005 0.018 0.156

Afternoon 0.3 0.92 0.029 0.014 0.439

Evening 0.31 0.96 0.033 0.016 0.553

Type 3 Queue

Model (Q5, Q6)

Morning 0.16 0.49 0.003 0.013 0.07

Afternoon 0.21 0.64 0.008 0.014 0.17

Evening 0.25 0.76 0.014 0.026 0.49

Type 4 Queue

Model (Q7, Q8)

Morning 0.27 0.83 0.02 0.015 0.361

Afternoon 0.21 0.62 0.007 0.013 0.152

Evening 0.27 0.84 0.021 0.017 0.414

1 2 3 4 5 6 7 8 9 10 11 120.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Vule

s of T

raffic

Intensity


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of T

raffic

Intensity


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of Lq, Ls, W

q &

W

s


Traffic Intensity

1 2 3 4 5 6 7 8 9 10 11 12

0.4

0.5

0.6

0.7

0.8

0.9

1


Valu

es of T

raffic

Intensity


Traffic Intensity

Figure 6.4: Traffic Intensity for CFI Two-leg design(Multi-Server)

21

Va

lues

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

S

Valu

es

of

Lq,L

s, W

q &

Ws

V

alu

es

of

Lq,

Ls,

Wq

& W

s V

alue

s of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Valu

es

of

Lq,

Ls,

Wq

& W

s

Va

lues

of

Lq,

Ls,

Wq

& W

s

Type -1 for CFI T-design

6

5

Lq Ls Wq Ws

4.5

4

3.5

Type 2 for CFI T-design

Lq Ls Wq Ws

4 3

2.5 3

2

2 1.5

1 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


Figure 4(a) Type1 for CFI T-design Figure 4(b) Type2 for CFI T-design

5

4.5 Lq

Type -3 for CFI Tdesign

7

Lq

Type-4 for CFI T-design

4

3.5

Ls Wq

Ws

6 Ls Wq Ws

5

3 4

2.5

2 3

1.5 2

1

1 0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Position of Session (morning, Afternoon & Evening)

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


Figure 4(c) Type 3 for CFI T-design Figure 4(d) Type 4 for CFI T-design

6

Lq Ls

Type-1for CFI Two-leg design

12

Lq Ls

Type 2 for CFI Two-leg design

5 Wq Ws

10 Wq Ws

4 8

3 6

2 4

1 2

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Position of Session ( Morning, Afternoon & Evening)

Figure 5(a) Type 1 for CFI Two-leg design Figure 5(b) Type 2 for CFI Two-leg design

Type-3 for CFI Two-leg design 3

Lq


Lq

2.5 Ls Lq

Ws

2

4.5

4

3.5

Ls

Wq Ws

1.5

3

2.5

2 1

1.5

0.5 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


Figure 5(c) Type 3 for CFI Two-leg design Figure 5(d) Type 4 for CFI Two-leg design

0.9

0.8

0.7

Type 1 for CFI T-design (Multi-Server)

Ws Ls Wq Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


Figure 6(a) Type 1 for CFI T-design (Multi-Server) Figure 6(b) Type 2 for CFI T-design (Multi-Server)

10

Figure 6.1a: Type 1 for CFI T-design

Va

lues

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

S

Valu

es

of

Lq,L

s, W

q &

Ws

V

alu

es

of

Lq,

Ls,

Wq

& W

s V

alue

s of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Valu

es

of

Lq,

Ls,

Wq

& W

s

Va

lues

of

Lq,

Ls,

Wq

& W

s


6

5

Lq Ls Wq Ws

4.5

4

3.5


Lq Ls Wq Ws

4 3

2.5 3

2

2 1.5

1 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



5

4.5 Lq


7

Lq


4

3.5

Ls Wq

Ws

6 Ls Wq Ws

5

3 4

2.5

2 3

1.5 2

1

1 0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



6

Lq Ls


12

Lq Ls


5 Wq Ws

10 Wq Ws

4 8

3 6

2 4

1 2

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




Lq


Lq

2.5 Ls Lq

Ws

2

4.5

4

3.5

Ls

Wq Ws

1.5

3

2.5

2 1

1.5

0.5 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.9

0.8

0.7


Ws Ls Wq Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



10

Figure 6.1b: Type 2 for CFI T-design

Va

lues

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

S

Valu

es

of

Lq,L

s, W

q &

Ws

V

alu

es

of

Lq,

Ls,

Wq

& W

s V

alue

s of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Valu

es

of

Lq,

Ls,

Wq

& W

s

Va

lues

of

Lq,

Ls,

Wq

& W

s


6

5

Lq Ls Wq Ws

4.5

4

3.5


Lq Ls Wq Ws

4 3

2.5 3

2

2 1.5

1 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



5

4.5 Lq


7

Lq


4

3.5

Ls Wq

Ws

6 Ls Wq Ws

5

3 4

2.5

2 3

1.5 2

1

1 0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



6

Lq Ls


12

Lq Ls


5 Wq Ws

10 Wq Ws

4 8

3 6

2 4

1 2

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




Lq


Lq

2.5 Ls Lq

Ws

2

4.5

4

3.5

Ls

Wq Ws

1.5

3

2.5

2 1

1.5

0.5 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.9

0.8

0.7


Ws Ls Wq Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



10

Figure 6.1c: Type 3 for CFI T-design

Va

lues

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

S

Valu

es

of

Lq,L

s, W

q &

Ws

V

alu

es

of

Lq,

Ls,

Wq

& W

s V

alue

s of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Va

lues

of

Lq,

Ls,

Wq

& W

s


6

5

Lq Ls Wq Ws

4.5

4

3.5


Lq Ls Wq Ws

4 3

2.5 3

2

2 1.5

1 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



5

4.5 Lq


7

Lq


4

3.5

Ls Wq

Ws

6 Ls Wq Ws

5

3 4

2.5

2 3

1.5 2

1

1 0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



6

Lq Ls


12

Lq Ls


5 Wq Ws

10 Wq Ws

4 8

3 6

2 4

1 2

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




Lq


Lq

2.5 Ls Lq

Ws

2

4.5

4

3.5

Ls

Wq Ws

1.5

3

2.5

2 1

1.5

0.5 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.9

0.8

0.7


Ws Ls Wq Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



10

Figure 6.1d: Type 4 for CFI T-design

Va

lues

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

S

Valu

es

of

Lq,L

s, W

q &

Ws

V

alu

es

of

Lq,

Ls,

Wq

& W

s V

alue

s of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Va

lues

of

Lq,

Ls,

Wq

& W

s


6

5

Lq Ls Wq Ws

4.5

4

3.5


Lq Ls Wq Ws

4 3

2.5 3

2

2 1.5

1 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



5

4.5 Lq


7

Lq


4

3.5

Ls Wq

Ws

6 Ls Wq Ws

5

3 4

2.5

2 3

1.5 2

1

1 0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



6

Lq Ls


12

Lq Ls


5 Wq Ws

10 Wq Ws

4 8

3 6

2 4

1 2

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




Lq


Lq

2.5 Ls Lq

Ws

2

4.5

4

3.5

Ls

Wq Ws

1.5

3

2.5

2 1

1.5

0.5 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.9

0.8

0.7


Ws Ls Wq Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



10

Figure 6.2a: Type 1 for CFI Two-leg design

Va

lues

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

S

Valu

es

of

Lq,L

s, W

q &

Ws

V

alu

es

of

Lq,

Ls,

Wq

& W

s V

alue

s of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Va

lues

of

Lq,

Ls,

Wq

& W

s


6

5

Lq Ls Wq Ws

4.5

4

3.5


Lq Ls Wq Ws

4 3

2.5 3

2

2 1.5

1 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



5

4.5 Lq


7

Lq


4

3.5

Ls Wq

Ws

6 Ls Wq Ws

5

3 4

2.5

2 3

1.5 2

1

1 0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



6

Lq Ls


12

Lq Ls


5 Wq Ws

10 Wq Ws

4 8

3 6

2 4

1 2

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




Lq


Lq

2.5 Ls Lq

Ws

2

4.5

4

3.5

Ls

Wq Ws

1.5

3

2.5

2 1

1.5

0.5 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.9

0.8

0.7


Ws Ls Wq Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



10

Figure 6.2b: Type 2 for CFI Two-leg design

Va

lues

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

S

Valu

es

of

Lq,L

s, W

q &

Ws

V

alu

es

of

Lq,

Ls,

Wq

& W

s V

alue

s of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Va

lues

of

Lq,

Ls,

Wq

& W

s


6

5

Lq Ls Wq Ws

4.5

4

3.5


Lq Ls Wq Ws

4 3

2.5 3

2

2 1.5

1 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



5

4.5 Lq


7

Lq


4

3.5

Ls Wq

Ws

6 Ls Wq Ws

5

3 4

2.5

2 3

1.5 2

1

1 0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



6

Lq Ls


12

Lq Ls


5 Wq Ws

10 Wq Ws

4 8

3 6

2 4

1 2

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




Lq


Lq

2.5 Ls Lq

Ws

2

4.5

4

3.5

Ls

Wq Ws

1.5

3

2.5

2 1

1.5

0.5 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.9

0.8

0.7


Ws Ls Wq Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



10

Figure 6.2c: Type 3 for CFI Two-leg design

Va

lues

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

S

Valu

es

of

Lq,L

s, W

q &

Ws

V

alu

es

of

Lq,

Ls,

Wq

& W

s V

alue

s of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Va

lues

of

Lq,

Ls,

Wq

& W

s


6

5

Lq Ls Wq Ws

4.5

4

3.5


Lq Ls Wq Ws

4 3

2.5 3

2

2 1.5

1 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



5

4.5 Lq


7

Lq


4

3.5

Ls Wq

Ws

6 Ls Wq Ws

5

3 4

2.5

2 3

1.5 2

1

1 0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



6

Lq Ls


12

Lq Ls


5 Wq Ws

10 Wq Ws

4 8

3 6

2 4

1 2

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




Lq


Lq

2.5 Ls Lq

Ws

2

4.5

4

3.5

Ls

Wq Ws

1.5

3

2.5

2 1

1.5

0.5 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.9

0.8

0.7


Ws Ls Wq Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



10

Figure 6.2d: Type 4 for CFI Two-leg design

22

Va

lues

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

S

Valu

es

of

Lq,L

s, W

q &

Ws

V

alu

es

of

Lq,

Ls,

Wq

& W

s V

alue

s of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Valu

es

of

Lq

, L

s,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Va

lues

of

Lq,

Ls,

Wq

& W

s


6

5

Lq Ls Wq Ws

4.5

4

3.5


Lq Ls Wq Ws

4 3

2.5 3

2

2 1.5

1 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



5

4.5 Lq


7

Lq


4

3.5

Ls Wq

Ws

6 Ls Wq Ws

5

3 4

2.5

2 3

1.5 2

1

1 0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



6

Lq Ls


12

Lq Ls


5 Wq Ws

10 Wq Ws

4 8

3 6

2 4

1 2

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




Lq


Lq

2.5 Ls Lq

Ws

2

4.5

4

3.5

Ls

Wq Ws

1.5

3

2.5

2 1

1.5

0.5 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.9

0.8

0.7


Ws Ls Wq Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



10

Figure 6.3a: Type 1 for CFI T-design (Multi-Server)

Va

lues

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

S

Valu

es

of

Lq,L

s, W

q &

Ws

V

alu

es

of

Lq,

Ls,

Wq

& W

s V

alue

s of

Lq,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Val

ues

of L

q, L

s, W

q &

Ws

Va

lue

s o

f L

q,

Ls,

Wq

& W

s

Va

lues

of

Lq,

Ls,

Wq

& W

s


6

5

Lq Ls Wq Ws

4.5

4

3.5


Lq Ls Wq Ws

4 3

2.5 3

2

2 1.5

1 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



5

4.5 Lq


7

Lq


4

3.5

Ls Wq

Ws

6 Ls Wq Ws

5

3 4

2.5

2 3

1.5 2

1

1 0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



6

Lq Ls


12

Lq Ls


5 Wq Ws

10 Wq Ws

4 8

3 6

2 4

1 2

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




Lq


Lq

2.5 Ls Lq

Ws

2

4.5

4

3.5

Ls

Wq Ws

1.5

3

2.5

2 1

1.5

0.5 1

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.9

0.8

0.7


Ws Ls Wq Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



10

Figure 6.3b: Type 2 for CFI T-design (Multi-Server)

Valu

es

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s V

alu

es

of

Lq,

Ls,

Wq

& W

s

0.9

0.8

0.7


Ws Ls Wq

Lq

0.9

0.8

0.7

Type 4 for CFI T-deign (Multi-Server)

Ws Ls Wq Lq

0.6

0.6

0.5

0.4

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


Figure 6(c) Figure 6(d) Type 3 for CFI T-design (Multi-Server) Type 4 for CFI T-design (Multi-Server)

0.9 Type 1 for CFI Two-leg design (Multi-Server)

1

Lq Lq

Type 2 for CFI Two-leg design (Multi-Server)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Ls Wq

Ws

0.9 Ls

0.8 Wq Ws

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


Figure 7(a). Figure 7(b). Type 1 for CFI Two-leg design (Multi-Server) Type 2 for CFI Two-leg design (Multi-Server)

0.8

0.7

0.6

Lq Ls

Wq Ws


0.9

0.8

0.7


Lq Ls Wq Ws

0.5

0.6

0.4

0.3

0.5

0.4

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Position ofSession (Morning, Afternoon & Evening)

Figure 7(c). Figure 7(d). Type 3 for CFI Two-leg design (Multi-Server) Type 4 for CFI Two-leg design (Multi-Server)

In figures 4 to 7, we plot the graph between time and traffic intensity for CFI T and Two-leg design for single and multiple channels respectively. We observed that the pattern of the graph is zigzag. In figure 4(a) and (b), we have taken type 1 & 2 as slot (Q1, Q2) & (Q3, Q4) in which we draw the graph between time for different session vs Lq, Ls, Wq and Ws respectively. Where Wq and Ws are parallel to timely session and Lq and Ls are going up and there after its decode very fast in figure 4(a) however in figure 4(b) all Lq, Ls, Wq and Ws are going down gradually. In figure 4(a), 4(d), 5(a), 5(d) and 7(a), we have obtained graph between time session vs Lq, Ls, Wq and Ws particularly for CFI T and Two-leg design. Wq and Ws are appeared in parallel fashion however Ls and Lq are coming down in middle slot of timing and their after it’s going increasing with time in both figures. In figure 5 (b) and 5(c), we have seen that Wq and Ws are appearing parallel to time scale and Lq and Ls both are increasing after (Q3,Q4) and (Q5,Q6). In figure 6(a), Lq, Wq and Ws are parallel to time and Ls is getting peak value corresponding to the time session and after a specific period of time and it is decreasing. In figures 6(b), 6(c) and 6(d) the values of Lq, Wq and Ws are similar to the pervious graph but Ls is slightly decreasing in figure 6(b) and however Ls is increasing in figure 6(c) and 6(d) after a specific period of time. In figures 7(a), 7(b), 7(c) &7 (d), it has been observed that the Wq, Ws and Lq are parallel to the time. In figure

11

Figure 6.3c: Type 3 for CFI T-design (Multi-Server)

Valu

es

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s V

alu

es

of

Lq,

Ls,

Wq

& W

s

0.9

0.8

0.7


Ws Ls Wq

Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.4

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




1

Lq Lq


0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Ls Wq

Ws

0.9 Ls

0.8 Wq Ws

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.8

0.7

0.6

Lq Ls

Wq Ws


0.9

0.8

0.7


Lq Ls Wq Ws

0.5

0.6

0.4

0.3

0.5

0.4

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




11

Figure 6.3d: Type 4 for CFI T-design (Multi-Server)

Valu

es

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s V

alu

es

of

Lq,

Ls,

Wq

& W

s

0.9

0.8

0.7


Ws Ls Wq

Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.4

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




1

Lq Lq


0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Ls Wq

Ws

0.9 Ls

0.8 Wq Ws

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.8

0.7

0.6

Lq Ls

Wq Ws


0.9

0.8

0.7


Lq Ls Wq Ws

0.5

0.6

0.4

0.3

0.5

0.4

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




11

Figure 6.4a: Type 1 for CFI Two-leg design (Multi-Server)

Valu

es

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s V

alu

es

of

Lq,

Ls,

Wq

& W

s

0.9

0.8

0.7


Ws Ls Wq

Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.4

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




1

Lq Lq


0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Ls Wq

Ws

0.9 Ls

0.8 Wq Ws

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.8

0.7

0.6

Lq Ls

Wq Ws


0.9

0.8

0.7


Lq Ls Wq Ws

0.5

0.6

0.4

0.3

0.5

0.4

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




11

Figure 6.4b: Type 2 for CFI Two-leg design (Multi-Server)

Valu

es

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s V

alu

es

of

Lq,

Ls,

Wq

& W

s

0.9

0.8

0.7


Ws Ls Wq

Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.4

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




1

Lq Lq


0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Ls Wq

Ws

0.9 Ls

0.8 Wq Ws

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.8

0.7

0.6

Lq Ls

Wq Ws


0.9

0.8

0.7


Lq Ls Wq Ws

0.5

0.6

0.4

0.3

0.5

0.4

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




11

Figure 6.4c: Type 3 for CFI Two-leg design (Multi-Server)

Valu

es

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s

Valu

es

of

Lq,

Ls,

Wq

& W

s

Val

ues

of

Lq,

Ls,

Wq

& W

s V

alu

es

of

Lq,

Ls,

Wq

& W

s

0.9

0.8

0.7


Ws Ls Wq

Lq

0.9

0.8

0.7


Ws Ls Wq Lq

0.6

0.6

0.5

0.4

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




1

Lq Lq


0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Ls Wq

Ws

0.9 Ls

0.8 Wq Ws

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3



0.8

0.7

0.6

Lq Ls

Wq Ws


0.9

0.8

0.7


Lq Ls Wq Ws

0.5

0.6

0.4

0.3

0.5

0.4

0.3

0.2

0.2

0.1

0.1

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3




11

Figure 6.4d: Type 4 for CFI Two-leg design (Multi-Server)

In figures 6.1 to 6.4, we plot the graph between time and traffic intensity for CFI T andTwo-leg design for single and multiple channels respectively. We observed that the patternof the graph is zigzag. In figure 6.1a and 6.1b, we have taken type 1 & 2 as slot (Q1, Q2)

23

& (Q3, Q4) in which we draw the graph between time for different session vs Lq, Ls, Wqand Ws respectively, where Wq and Ws are parallel to timely session and Lq and Ls aregoing up and there after its decode very fast in figure 6.1a however in figure 6.1b all Lq, Ls,Wq and Ws are going down gradually. In figure 6.1a, 6.1d, 6.2a, 6.2d and 6.4a, we haveobtained graph between time session vs Lq, Ls, Wq and Ws particularly for CFI T andTwo-leg design. Wq and Ws are appeared in parallel fashion however Ls and Lq are comingdown in middle slot of timing and their after it’s going increasing with time in both figures.In figure 6.2b and 6.2c, we have seen that Wq and Ws are appearing parallel to time scaleand Lq and Ls both are increasing after (Q3,Q4) and (Q5,Q6). In figure 6.3a, Lq, Wq andWs are parallel to time and Ls is getting peak value corresponding to the time session andafter a specific period of time and it is decreasing. In figures 6.3b, 6.3c and 6.4d the valuesof Lq, Wq and Ws are similar to the pervious graph but Ls is slightly decreasing in figure6.3b and however Ls is increasing in figure 6.3c and 6.3d after a specific period of time. Infigures 6.4a, 6.4b, 6.4c & 6.4d, it has been observed that the Wq, Ws and Lq are parallel tothe time. In figure 6.4a & 6.4d Ls is decreasing after certain period of time thereafter, it isgoing up. In figure 6.4b, the value of Ls is going up and after a period of time it is sustainand in figure 6.4c, the value of Ls is appearing in increasing way.

7 ConclusionIn this investigation, we have analyzed two continuous flow intersection (CFI) designs whichare helpful to control heavy traffic volume in lesser time. Performance measure of M/M/1and M/M/C queueing models have been carried out to examine the traffic flow in differenttime slot. The proposed study is useful to traffic engineers for planning the new CFI designby which they can handle the increasing traffic flow without congestion and reduce thedelays at the intersection. Such results will definitely provide new direction or dimension fordesigners of the forthcoming smart cities.Acknowledgement. The Authors are thankful to the referee for his valuable suggestions.

References[1] A.M. Abane, Taking Traffic Congestion in Accra, Ghana: A Road users Perspective,

Journal of Advanced Transportation, 27 (1993), 193-206.[2] C.F. Daganzo, Fundamentals of Transportation and Traffic Operations, Elsevier

Science Ltd. Oxford, 1987.[3] M.P. Gupta and R.B. Khanna, Quantitative Techniques for Decision Making, Prentice

Hall, New Delhi, 2007.[4] D. Gross and C. Harris, Fundamental of Queueing Theory, 3rd Edition, John Wiley,

Chichester, 1998.[5] R.L. Gordon, R.A. Reiss, H. Haenal, E.R. Case and R.L. French, A. Mohaddes and R.

Wolcott, Traffic Control Systems HandbookRevised Edition 1996. Report FHWA-SA-95-032, FHWA, U.S. Department of Transportation, (1996).

[6] D.L. Iglehart and W. Witt, Multiple channel queues in heavy traffic, II: Sequences,networks and batches, Advance International APPL. Probability, 2 (1970), 355-369.

[7] R. Jagannathan and J.G. Bared, Design and Operational Performance of CrossoverDisplaced Left-Turn Intersections. In Transportation Research Record: Journal ofthe Transportation Research Board, No. 1881, Transportation Research Board of theNational Academies, Washington D.C., (2004), 1-10.

24

[8] E.H. Joseph, Unconventional Left-Turn Alternatives for Urban and SuburbanArterials-update, Transportation Research e-Circular, (2000).

[9] S.E. Jabari, Node Modeling for Congested Unban Road networks, TransportationResearch part B Methodological, 46(10) (2016), 1639-1656.

[10] S.E. Jabari and H.V. Lin, A stochastic model of traffic flow: Theoretical Foundation,Transportation Research part B Methodological, 46(1) (2012), 156-174.

[11] L. Kleinrock, Queieng Systems, John Wiliy & Sons, New York, 1976.[12] M. Modares and H.B. Fakhe, Traffic flow analysis based on queueing models, Journal

of Industrial Engineering, 2 (2009), 19-23.[13] V. Nico, T.V. Woensel and A.A. Verbruggen, Queuieng Based Traffic Flow Model,

Transportation Research, 5 (2000), 212-135.[14] D. Parmar, Analysis of different queueing model in traffic flow problem, International

conference on Engineering Trends in Scientific Research, C.U. Shah, University, 2015.[15] J.D. Reid and E.H. Joseph, Travel Time Comparisons between Seven Unconventional

Arterial Intersection Designs. In Transportation Research Record: Journal of theTransportation Research Board, No. 1751, TRB, National Research Council, Wash-ington, D.C. (2001), 56-66.

[16] P. Vedagiri and S. Daydar, Evaluation of Displaced Right-Turn using ComputerSimulation, Sustainable Urban and Transportation Planning Issues and ManagementStrategies (SUTRIMS-11), NIT Surat, (2010).

[17] P. Vedagiri and S. Daydar, Performance Analysis of Continuous Flow Intersection inMixed Traffic Condition, ACEE International Journal on Transportation and UrbanDevelopment, 2(1) (2012), 20-25.

[18] F. Wong, Y. Chuning, Y. Zhang and L. Yan, Simulation Analysis and Improvement ofthe Vehicle Queueing System on Intersection Based on Matlab, The open Cyberneticsand Systemic Journal, 8 (2014), 217-223.

[19] T.V. Woensel, Modeling traffic flows with queueing models: A review, Asia PacificJournal of Operation Research, 24(4) (2007), 435-461.

25

Jnanabha, Vol. 49(1) (2019), 26-39

NUMERICAL SOLUTION OF THE CONVECTION DIFFUSIONEQUATION BY THE LEGENDRE WAVELET METHOD

ByDevendra Chouhan

Dept. of Mathematics, IES, IPS Academy, Indore, IndiaEmail:[email protected]

R.S. ChandelGovt. Geetanjali Girls College, Bhopal, India

Email:rs [email protected]

(Received : January 08, 2019 ; Revised: June 03, 2019)

Abstract

In this paper, a numerical method is proposed for solving Convection Diffusionequations. The method is based upon the Legendre wavelet expansion. The Legendrewavelets operational matrix of integration is derived and utilized to transform theequation to a system of algebraic equations by combining collocation method. Theproposed method is very convenient for solving such problems since conditions aretaken into account automatically. Illustrative examples are included to demonstratethe validity and applicability of the proposed Legendre wavelets method.2010 Mathematics Subject Classifications: 42C40, 35A08, 65L60.Keywords and phrases: Two - dimensional Legendre wavelets, Operational matrixof integration, Partial differential equations, Convection diffusion equation, Collocationmethod.

1 IntroductionConvection diffusion equations are widely used for modeling and simulations of various com-plex phenomena in science and engineering, such as dispersion of chemicals in reactors, smokeplume in atmosphere, tracer dispersion in a porous medium, migration of contaminants in astream etc. Since it is impossible to solve Convection diffusion equations analytically for mostapplication problems, efficient numerical algorithms are becoming increasingly importantto numerical simulations involving Convection diffusion equations. For this model, someauthors have studied the numerical techniques such as the Crank - Nicholson method [19],ADI method [12], the Bessel collocation method [20], the Wavelet - Galerkin method [3, 8],the finite difference method [1, 5, 18], the finite element method [6, 7] and the Piecewise -analytical method [16].

Among these methods, the Wavelets method is more attractive. Wavelet theory isrelatively new and an emerging area in mathematical research. It has been applied in awide range of engineering disciplines. Wavelets are used in system analysis, signal analysisfor wave - form representation and segmentations, optimal - control, numerical analysis, time- frequency analysis and fast algorithms for easy implementation [17]. However wavelets arejust another basis set which offers considerable advantages over alternative basis sets andallows us to attack problems not accessible with conventional numerical methods. Theirmain advantages are given in [9].

26

In the last two decades wavelets methods have been applied for solving partial differentialequations [2, 4, 10, 14, 15]. It is worth mentioning that Legendre wavelets have both ofspectral accuracy, orthogonality and other properties of wavelets.

In this work, a numerical method based on the two - dimensional Legendre wavelets isproposed for solving Convection diffusion equation with Dirichlet initial boundary conditionsgiven as follows

∂u

∂t+ a(x)

∂u

∂x+ b(x)

∂2u

∂x2= g(x, t), 0 ≤ x ≤ 1, 0 ≤ t ≤ 1 (1.1)

with the conditionsu(x, 0) = f0(x), u(x, 1) = f1(x), 0 ≤ x ≤ 1 (1.2)

andu(0, t) = g0(t), u(1, t) = g1(t), 0 ≤ t ≤ 1 (1.3)

where a(x) and b(x)(6= 0) are continuous functions.In the proposed method, both of the operational matrices of integration and derivative are

mutually employed to obtain numerical solutions of the mentioned problem. The proposedmethod is very convenient for solving such problems since the given conditions are takeninto account automatically. Numerical results demonstrate the efficiency of this Legendrewavelets method in solving convection diffusion equation.

2 Basic Definitions, Mathematical Preliminaries and NotationsIn this section some necessary definitions and mathematical preliminaries of Wavelet theoryand Legendre wavelets are given which will be used further in this paper.2.1 Wavelets and Legendre WaveletsWavelets is a family of functions constructed from dilation parameter ’a’ and translationparameter ’b’ of a single function called the ’mother wavelet’ ψ(t). They are defined by

ψa,b(t) =1√|a|ψ(t− ba

), a, b ∈ R, a 6= 0.

Now for the discrete values of a and b, a = a−k0 , b = nb0a−k0 , a0 > 1, b0 > 0, where n

and k are positive integers. We have the following family of discrete wavelets:

ψk,n(t) = |a|−12ψ(ak0t− nb0)

where ψk,n(t) forms a basis of L2(R).Legendre Wavelets: Legendre wavelet ψnm(t) = ψ(k, n,m, t) have four arguments n =2n−1, n = 1, 2, 3, ..., 2k−1, k can be any positive integer, m is order for Legendre polynomialsand t is the normalized time [11]. They are defined on the interval [0, 1) by

ψnm(t) =

2k2

√m+

1

2Pm(2kt− n)for

n− 1

2k≤ t ≤ n+ 1

2k

0 otherwise.

The coefficient√m+ 1

2is for orthonormality, the dilation parameter is 2−k and the

translation parameter is n2−k. Here Pm(t) are the well known Legendre polynomials of

27

order m which are orthogonal with respect to the weight function w(t) = 1 on the interval[−1, 1] and satisfy the following formulae,

P0(t) = 1, P1(t) = t

and

Pm+1(t) = (2m+ 1

m+ 1)tPm(t)− (

m

m+ 1)Pm−1(t),m = 1, 2, 3, ...

2.2 Function Approximation:A function f(t) defined over [0, 1] may be expanded by Legendre wavelets as

f(t) =∞∑n=1

∞∑m=0

cnmψnm(t), (2.1)

where cnm = (f(t), ψnm(t)) in which, denotes the inner product.If the infinite series in equation (2.1) is truncated, then it can be rewritten as

f(t) ≈2k−1∑n=1

M−1∑m=0

cnmψnm(t) = CTΨ(t), (2.2)

where T indicates transposition and C and Ψ(t) are m = 2k−1M column vectors.For simplicity eq. (2.2) can be written as

f(t) ≈m∑i=1

ciψi(t) = CTΨ(t), (2.3)

where ci = cnm, ψi(t) = ψnm(t).The index i, is determined by the relation i = M(n− 1) +m+ 1, thus we have

C∆= [c1, c2, ..., cm]T ,

Ψ(t) = [ψ1(t), ψ2(t), ..., ψm(t)]T . (2.4)

In the same way, an arbitrary function of two variables u(x, y) defined over [0 , 1)× [0 ,1) may be expanded into Legendre wavelets basis as

u(x, y) ≈m∑i=1

m∑j=1

uijψi(x)ψj(y) = ΨT (x)UΨ(y),

where U = [uij] and uij = (ψi(x), (u(x, y), ψj(y))).By taking the collocation points ti = 2i−1

2m, (i = 1, 2, ...,m) in the Ψ(t).

Now define Legendre wavelets matrix ϕm×m as

ϕm×m∆= [Ψ(

1

2m),Ψ(

3

2m), ...,Ψ(

2m− 1

2m)].

Here ϕm×m has a diagonal form.

28

3 Operational MatricesThe integration of integer order α of the vector Ψ(t), defined in (2.4) can be expressed as

(IαΨ)(t) ≈ PαΨ(t),

where Pα is the m×m Legendre wavelet operational matrix of integration of integer orderα. This matrix Pα can be approximated as

Pα ≈ ϕm×mPαBϕ−1m×m,

where PαB is the operational matrix of integration of integer order α of the Block - Pulse

functions (BPFs), which is given by [13].

PαB =

1

mα

1

(α + 1)!

1 ξ1 ξ2 · · · ξm−1

0 1 ξ1 · · · ξm−2

0 0 1 · · · ξm−3...

......

......

0 0 0 · · · 1

,where ξi = (i+ 1)α+1 − 2iα+1 + (i− 1)α+1.

We define a m - set of Block pulse functions (BPF) as:

bi(t) =

1,

i

m≤ t <

i+ 1

m0, otherwise

where i = 0, 1, 2, ...,m− 1The functions bi(t) are disjoint and orthogonal, that is

bi(t)bj(t) =

0, i 6= j

bi(t), i = j,

∫ 1

0

bi(t)bj(t)dt =

0, i 6= j

1

m, i = j

.

On taking the derivative of integer order α of the vector Ψ(t), defined in (2.4), we have

(DαΨ)(t) ≈ QαΨ(t),

where Qα is Legendre wavelet operational matrix of derivative of integer order α. Here Qαisinverse of matrix Pα, so it can be expressed as

Qα = ϕm×mP−αB ϕ−1

m×m,

where P−αB is the operational matrix of derivative of integer order α of the BPFs, which isgiven by

P−αB = mα(α + 1)!

1 d1 d2 · · · dm−1

0 1 d1 · · · dm−2

0 0 1 · · · dm−3...

......

......

0 0 0 · · · 1

,where di = −

∑ij=1 ξjdi−j for all i = 1, 2, ...,m− 1 and d0 = 1.

29

4 The Proposed MethodIn this section we apply the Legendre wavelets operational matrices of integer order forsolving convection diffusion equation with variable coefficients given by equation (1.1) alongwith the conditions given in (1.2) and (1.3).

For solving the given problem we approximate

∂3u

∂t∂x2= Ψ(t)TUΨ(x), (4.1)

where U = [ui,j]m×m is an unknown matrix which should be found and Ψ(.) is the vectorwhich is defined in (2.4).

By integration of order 2 of (4.1) with respect to x we have∂u

∂t≈ Ψ(t)TUP 2Ψ(x) + [

∂u

∂t+ x

∂

∂x(∂u

∂t)]x=0. (4.2)

On putting x = 1 and using (1.3), we have by (4.2)

[∂

∂x(∂u

∂t)]x=0 ≈

∂g1

∂t− ∂g0

∂t−Ψ(t)TUP 2Ψ(1). (4.3)

Now on substituting (4.3) into (4.2), we have

∂u

∂t≈ Ψ(t)TUP 2Ψ(x) + (1− x)

∂g0

∂t− xΨ(t)TUP 2Ψ(1) + x

∂g1

∂t. (4.4)

Moreover by integrating (4.1) with respect to t, we have∂2u

∂x2≈ (PΨ(t))TUΨ(x) + [

∂2u

∂x2+ t

∂

∂t(∂2u

∂x2)]t=0. (4.5)

By putting t = 1 in (4.5) and using (1.2), we have by (4.5)

[∂

∂t(∂2u

∂x2)]t=0 ≈

∂2f1

∂x2− ∂2f0

∂x2− (PΨ(t))TUΨ(x). (4.6)

On substituting (4.6) into (4.5), we have∂2u

∂x2≈ (PΨ(t))TUΨ(x) + (1− t)∂

2f0

∂x2+ t

∂2f1

∂x2− t(PΨ(1))TUΨ(x). (4.7)

Now by integration of (4.4) with respect to t and using conditions (1.2) and (1.3), we get

u(x, t) ≈ (PΨ(t))TUP 2Ψ(x)− x(PΨ(t))TUP 2Ψ(1)− t(PΨ(1))TUP 2Ψ(x)

+ xt(PΨ(1))TUP 2Ψ(1) +H(x, t), (4.8)where

H(x, t) = f0(x) + (1− x)(g0(t)− g0(0)− tg′0(0)) + x(g1(t)− g1(0)− tg′1(0))+t(f1(x)− f0(x))− t(1− x)g0(1)− g0(0)− g′0(0)− xt(g1(1)− g1(0)− g′1(0)).

On differentiating (4.8) with respect to x, we have∂u

∂x≈ (PΨ(t))TUPΨ(x)− (PΨ(t))TUP 2Ψ(1)− t(PΨ(1))TUPΨ(x)

+ t(PΨ(1))TUP 2Ψ(1) +∂

∂xH(x, t). (4.9)

Now by substituting (4.4), (4.7), (4.9) into (1.1) and taking collocation points

xi, ti =2i− 1

2m, i = 1, 2, ...,m,

into the obtained equation, we have a nonlinear system of algebraic equations. This nonlinearsystem can be solved by using an iterative method such as Newton iteration method. Bysolving this system and finding U , we obtain the numerical solution of the problem bysubstituting U into (4.8).

30

5 Numerical ExamplesIn this section, we will use the proposed method to solve the convection diffusion equationwith variable or constant coefficients. The following numerical examples are given to showthe efficiency and reliability of the proposed method and the results have been comparedwith the exact solution.

Example 5.1. Consider convection diffusion eq. (1.1) with a(x) = −0.1, b(x) = −0.01 andg(x, t) = 0.

The given conditions are

u(x, 0) = e−x, u(x, 1) = e−x−0.09

andu(0, t) = e−0.09t, u(1, t) = e−1−0.09t.

The exact solution of this problem is u(x, t) = e−x−0.09t.The space - time diagram of the numerical solution for M = 6, k = 2 is shown in figure

5.1.Absolute errors between the numerical and analytical solution are shown in figure 5.2.The graph of analytical and approximate solutions for some nodes on [0, 1) × [0, 1) is

presented in figure 5.3.Absolute errors between the numerical and analytical solutions at different times are

shown in figure 5.4.

Example 5.2. Consider convection diffusion eq. (1.1) with a(x) = −x6, b(x) = −x2

12and

g(x, t) = 0.The given conditions are

u(x, 0) = x3, u(x, 1) = x3e

andu(0, t) = 0, u(1, t) = et.

The exact solution of this problem is u(x, t) = x3et.The space - time diagram of the numerical solution for M = 6, k = 2 is shown in figure

5.5.Absolute errors between the numerical and analytical solution are shown in figure 5.6.The graph of analytical and approximate solutions for some nodes on [0, 1) × [0, 1) is



Example 5.3. Consider convection diffusion eq. (1.1) with a(x) = 2, b(x) = −1 andg(x, t) = −2et−x.

The given conditions are

u(x, 0) = e−x, u(x, 1) = e1−x

andu(0, t) = et, u(1, t) = et−1.

The exact solution of this problem is u(x, t) = et−x.

31

The space - time diagram of the numerical solution for M = 6, k = 2 is shown in figure5.9.

Absolute errors between the numerical and analytical solutions are shown in figure 5.10.The graph of analytical and approximate solutions for some nodes on [0, 1) × [0, 1) is



Figure 5.1: Approximate Solution of Example 5.1.

Figure 5.2: Absolute Errors of Example 5.1.

32

Figure 5.3: Numerical and Exact Solutions in different values of t for Example 5.1.

Figure 5.4: Absolute Errors in different values of t for Example 5.1.

33



34



35



36



37

6 ConclusionIn this paper, we have derived the two dimensional Legendre wavelets operational matricesof integration and differentiation and proposed a numerical method to approximate thesolution of convection diffusion equation with variable or constant coefficients. The methodis computationally efficient and the algorithm can be implemented easily on a computer. Theadvantage of the method is that only small size operational matrix is required to provide thesolution at high accuracy. Numerical examples are given to show that the proposed methodis applicable, efficient and accurate.Acknowledgement. We are very much thankful to the referee for his valuable suggestionsto bring the paper in its present form.

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[17] A. D. Singh, R. S. Chandel and D. Chouhan, A Numerical approach for Solvingboundary value problems for fractional differential equations using Shannon Wavelet,J. Math. Comput. Sci., 6(6) (2016), 1085 - 1099.

[18] A. D. Singh, R. S. Chandel and D. Chouhan, Numerical solution of fractional orderdifferential equations using Haar Wavelet operational matrix, Palestine Journal ofMathematics, 6(2) (2017), 515 - 523.

[19] W.Q. Wang, The alternating segment Crank-Nicolson method for solving convection-diffusion equation with variable coefficient, Appl. Math. Mech., 24 (2003), 32-42.

[20] S. Yuzbasi and N. Sahin, Numerical solutions of singularly perturbed one-dimensionalparabolic convection-diffusion problems by the bessel collocation method, Appl. Math.Comput., 220 (2013), 305-315.

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Jnanabha, Vol. 49(1) (2019), 40-49

SOME COUPLED FIXED POINT THEOREMS IN CONVEX METRICSPACES

ByMemudu Olaposi Olatinwo

Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria.1

Email:[email protected]/[email protected]/[email protected]

(Received : April 01, 2019 ; Revised: May 26, 2019)

Abstract

We establish some coupled fixed point theorems in convex cone metric spaces for(k, µ)−Lipschitzian and (k, µ, L)−Lipschitzian mappings. We assume that the conehas nonempty interior. Our results generalize and extend several known results in theexisting literature.2010 Mathematics Subject Classifications: 47H10, 47H09.Keywords and phrases: Coupled fixed point; convex cone metric spaces.

1 IntroductionThe term ‘coupled fixed point’ was coined from the seminal paper of Guo and Lakshmikan-tham [13] in which the concept was introduced in 1987. Several coupled fixed point theoremsestablished in cone metric space and partially ordered metric space are available in theliterature. In addition to [13], some of the various authors whose contributions are alsoof colossal value in the study of the notion of coupled fixed point are Abbas and Beg [1],Bhaskar and Lakshmikantham [4], Beg et al. [7], Chang and Ma [8], Chang et al. [9], Duaand Li [11], Guu [14] as well as a host of others in the literature.

Bhaskar and Lakshmikantham [4] established a coupled fixed point theorem in a metricspace endowed with partial order by employing a weak contractive type condition. Morerecently, the result of [4] was further generalized and extended by Lakshmikantham andCiric [16]. See also Ciric and Lakshmikantham [10].

We shall employ the following definitions in the sequel:

Definition 1.1. Let E be a real Banach space. A nonempty convex closed subset P ⊂ E iscalled a cone in E if it satisfies the following:(i) P is closed, nonempty and P 6= 0;(ii) a, b ∈ IR, a, b ≥ 0 and x, y ∈ P =⇒ ax+ by ∈ P ;(iii) x ∈ P and −x ∈ P =⇒ x = 0.

For a given cone P ⊂ E, the partial ordering with respect to P is defined by x y ifand only if y − x ∈ P. If y − x ∈ int P , we write x ≺≺ y (where int P denotes the interiorof P ). Also, we use x ≺ y if x y and x 6= y.

Definition 1.2. Let X be a nonempty set and let E be a real Banach space equipped with thepartial ordering with respect to the cone P ⊂ E. Suppose that the mapping d : X×X → Esatisfies the following conditions:(i) 0 d(x, y), ∀ x, y ∈ X and d(x, y) = 0 ⇐⇒ x = y;

1ORCID https: //orcid.org/0000-0002-1995-9225

40

(ii) d(x, y) = d(y, x), ∀ x, y ∈ X;(iii) d(x, y) d(x, z) + d(z, y), ∀ x, y, z ∈ X.Then, d is called a cone metric on X, and the pair (X, d) is called a cone metric space.

The notion of convex metric spaces was introduced by Takahashi [23] and he establishedthat all normed spaces and their convex subsets are convex metric spaces. In addition, healso gave several examples of convex metric spaces which are not imbedded in any normedspace. Several papers have been devoted to the study of fixed point theory in convex metricspaces in the literature. See Agarwal et al [2], Beg [5, 6] and Guay et al [12]. The aim of thispaper is to prove coupled fixed point theorems in convex metric spaces for (k, µ)−Lipschitzianand (k, µ, L)−Lipschitzian mappings.

The following definitions are also pertinent to the study:

Definition 1.3. [6, 23] Let (X, d) be a metric space. A mapping W : X ×X × [0, 1] → Xis said to be a convex structure on X if for each(x, y, λ) ∈ X ×X × [0, 1] and u ∈ X,

d(u,W (x, y, λ)) λd(u, x) + (1− λ)d(u, y). (1.1)

A metric space X having the convex structure W is called a convex metric space.Let (X, d,W ) be a convex metric space. A nonempty subset C of X is said to be convex

if W (x, y, λ) ∈ C whenever (x, y, λ) ∈ C × C × [0, 1].

Definition 1.4. [4, 16]Let (X, d) be a metric space. An element (x, y) ∈ X ×X is said tobe a coupled fixed point of the mapping T : X ×X → X if T (x, y) = x and T (y, x) = y.

Definition 1.5. [15, 21] Let (X, d) be a cone metric space. Let xn∞n=1 ⊆ X and x ∈ X.Then,(i) xn∞n=1 converges to x, that is, lim

n→∞xn = x, if for every c ∈ E with 0 ≺≺ c there exists

a natural number N such that d(xn, x) ≺≺ c for all n N ;(ii) xn∞n=1 is a Cauchy sequence if for every c ∈ E with 0 ≺≺ c, there exists a naturalnumber N such that d(xn, xm) ≺≺ c for all n, m N.

A cone metric space (X, d) is said to be complete if every Cauchy sequence in X convergesto a point x ∈ X.

Definition 1.6. Let C be a nonempty subset of a convex metric space (X, d,W ).(i) A mapping T : C × C → C is said to be a (k, µ)−Lipschitzian if and only if there existtwo constants k, µ ∈ [0,∞), such that

d(T (x, y), T (u, v)) kd(x, u) + µd(y, v), ∀ x, y, u, v ∈ C. (1.2)

(ii) A mapping T : C × C → C is said to be (k, µ, L)−Lipschitzian if and only if there existconstants k, µ ∈ [0,∞) and L 0, such that

d(T (x, y), T (u, v)) kd(x, u) + µd(y, v) + Ld(T (x, y), x), ∀ x, y, u, v ∈ C. (1.3)

Definition 1.7. [3] : (a) A function φ : IR+ → IR+ is called a comparison function if: (i)φ is monotone increasing; (ii) lim

n→∞φn(t) = 0, ∀ t ≥ 0.

(b) A comparison function satisfying∑∞

n=0 φn(t) converges for all t 0 is called a

(c)−comparison function.

41

In addition, we shall require the following iterative process in the sequel:For (x0, y0) ∈ C × C, define the sequence (xn, yn) ⊂ C × C in terms of a convex

structure by

xn+1 = W (xn, T (xn, yn); 1− αn), yn+1 = W (yn, T (yn, xn); 1− αn), (1.4)

n = 0, 1, 2, · · · , αn ∈ [0, 1]. The iterative process defined in (1.4) above is a Mann-typeiterative process.

Remark 1.1. If αn = 1 in (1.4), then we obtain the iterative process in Sabetghadam et al.[21] and some others in the reference section which is again stated below:

For (x0, y0) ∈ C × C, define the sequence (xn, yn) ⊂ C × C by

xn+1 = T (xn, yn), yn+1 = T (yn, xn). (1.4?)

2 Main resultsTheorem 2.1. Let (X, d,W ) be a complete convex cone metric space, C a nonempty closedconvex subset of X and T : C × C → C is a (k, µ, L)−Lipschitzian mapping. Suppose thatφ : IR+ → IR+ is a (c)−comparison function such that, for arbitrary x, y ∈ C, there existu, v ∈ C with(i) d(T (u, v), u) φ(d(T (x, y), x));(ii) d(u, x) bd(T (x, y), x), b 0.For (x0, y0) ∈ C × C, let (xn, yn) ⊂ C × C be defined by (4?).

Then, T has a coupled fixed point in C.

Proof. Let (x0, y0) ∈ C × C be an arbitrary point. We consider sequencesxn∞n=0, yn∞n=0 ⊂ C such that by conditions (i) and (ii) of the theorem above, we have

d(T (xn+1, yn+1), xn+1) φ(d(T (xn, yn), xn)), n = 0, 1, 2, · · · , (2.1)

andd(xn+1, xn) bd(T (xn, yn), xn), b 0, n = 0, 1, 2, · · · . (2.2)

We obtain by induction in (2.1) that

lld(T (xn+1, yn+1), xn+1) φ(d(T (xn, yn), xn))

φ2(d(T (xn−1, yn−1), xn−1)) · · · φn+1(d(T (x0, y0), x0)). (2.3)

Using (2.3) in (2.2) gives

d(xn+1, xn) bφn(d(T (x0, y0), x0)). (2.4)

In a similar manner, we obtain by conditions (i) and (ii) of the theorem that

d(T (yn+1, xn+1), yn+1) φ(d(T (yn, xn), yn)), n = 0, 1, 2, · · · , (2.5)

andd(yn+1, yn) bd(T (yn, xn), yn), b 0, n = 0, 1, 2, · · · . (2.6)

By induction in (2.5), we have that

d(T (yn+1, xn+1), yn+1) φ(d(T (yn, xn), yn)) φ2(d(T (yn−1, xn−1), yn−1)) · · · φn+1(d(T (y0, x0), y0)).

(2.7)

Using (2.7) in (2.6) yields

d(yn+1, yn) bφn(d(T (y0, x0), y0)). (2.8)

42

For each r ∈ IN, we obtain by using (2.4) and (2.8) in the repeated application of thetriangle inequality that

d(xn, xn+r) + d(yn, yn+r) [d(xn, xn+1) + d(xn+1, xn+2) + · · ·+ d(xn+r−1, xn+r)]+ [d(yn, yn+1) + d(yn+1, yn+2) + · · ·+ d(yn+r−1, yn+r)]

b[∑n+r−1

k=n φk(d(T (x0, y0), x0)) +∑n+r−1

k=n φk(d(T (y0, x0), y0))]

= b[∑n+r−1

k=0 φk(d(T (x0, y0), x0))−∑n−1

k=0 φk(d(T (x0, y0), x0))]

+ b[∑n+r−1

k=0 φk(d(T (y0, x0), y0))−∑n−1

k=0 φk(d(T (y0, x0), y0))]

→ 0 as n→∞,(2.9)

and noting that φ is a (c)−comparison function. It follows from (??) that forc ∈ E, 0 ≺≺ c and for large n, we haveb[∑n+r−1

k=n φk(d(T (x0, y0), x0)) +∑n+r−1

k=n φk(d(T (y0, x0), y0))] ≺≺ c,thus, leading to the fact that d(xn, xn+r) + d(yn, yn+r) ≺≺ c.

Therefore, it follows that the sequence (xn, yn) is a Cauchy sequence in C × C. SinceX is complete, we have that C is also a complete subspace of X. Suppose that there existx∗, y∗ ∈ C such that lim

n→∞xn = x∗ and lim

n→∞yn = y∗. That is, for c ∈ E, 0 ≺≺ c, there exists

q ∈ IN such that

d(xq, x∗) ≺≺ c

3(1 + k)and d(yq, y

∗) ≺≺ c

3(1 + µ), for all n ≥ q, (2.10)

with k ≺ 1 + µ, µ ≺ 1 + k.Using (1.3), (2.3), (2.10) as well as the triangle inequality give

d(T (x∗, y∗), x∗) d(T (x∗, y∗), T (xq, yq)) + d(T (xq, yq), x∗)

= d(T (xq, yq), T (x∗, y∗)) + d(xq+1, x∗)

kd(xq, x∗) + µd(yq, y

∗) + Ld(T (xq, yq), xq) + d(xq+1, x∗)

Lφq(d(T (x0, y0), x0)) + kd(xq, x∗) + µd(yq, y

∗) + d(xq+1, x∗),


∗) + d(xq+1, x∗),

(2.11)

since φq(d(T (x0, y0), x0))→ 0 as q →∞, noting that φ is a comparison function. Therefore,we have from (2.11) that

d(T (x∗, y∗), x∗) kd(xq, x∗) + µd(yq, y

∗) + d(xq+1, x∗)

≺≺ kc3(1+k)

+ µc3(1+µ)

+ c3(1+k)

≺ c3

+ c3

+ c3

= c,

from which it follows that d(T (x∗, y∗), x∗) ≺≺ c.Thus, d(T (x∗, y∗), x∗) = 0, that is, T (x∗, y∗) = x∗.Similarly, from (1.3), (2.7), (2.10) and the triangle inequality, we get

d(T (y∗, x∗), y∗) d(T (y∗, x∗), T (yq, xq)) + d(T (yq, xq), y∗)

= d(T (yq, xq), T (y∗, x∗)) + d(yq+1, y∗)

kd(yq, y∗) + µd(xq, x

∗) + Ld(T (yq, xq), yq) + d(yq+1, y∗)

Lφq(d(T (y0, x0), y0)) + kd(yq, y∗) + µd(xq, x

∗) + d(yq+1, y∗),


∗) + d(yq+1, y∗),

(2.12)

since φq(d(T (y0, x0), y0)) → 0 as q → ∞, and observing that φ is a comparison function.Now, since k < 1 + µ, µ < 1 + k, then, we obtain from (2.12) that

d(T (y∗, x∗), y∗) kd(yq, y∗) + µd(xq, x

∗) + d(yq+1, y∗)

≺≺ kc3(1+µ)

+ µc3(1+k)

+ c3(1+µ)

≺ (1+µ)c3(1+µ)

+ (1+k)c3(1+k)

+ c3(1+µ)

≺ c3

+ c3

+ c3

= c,

43

from which it follows again that d(T (y∗, x∗), y∗) ≺≺ c.Therefore, d(T (y∗, x∗), y∗) = 0, that is, T (y∗, x∗) = y∗.Hence, (x∗, y∗) is a coupled fixed point of T.

Theorem 2.1 can be extended also to the next result under the assumption of two metricsρ and d such that (X, ρ) is complete and T : C×C → C is a (k, µ, L)−Lipschitzian mappingwith respect to d.

Theorem 2.2. Let X be a nonempty set, C ⊂ X, ρ and d are two metrics on X andT : C × C → C is a mapping. Suppose that:(H1) there exists a (c)−comparison function φ : IR+ → IR+ such that, for arbitrary x, y ∈ C,there exist u, v ∈ C with(i) d(T (u, v), u) φ(d(T (x, y), x));(ii) d(u, x) bd(T (x, y), x), b 0;(H2) there exist real numbers A 0, 0 ≺ R ≺ 1 such that, for arbitrary x, y ∈ C, thereexists u ∈ C with(i) ρ(u, x) Ad(u, x);(ii) ρ(T (x, y), x) Rd(T (x, y), x);(H3) (X, ρ,W ) is a complete convex cone metric space and C is a closed convex subset of(X, ρ,W );(H4) T : (C, d)× (C, d)→ (C, d) is a (k, µ, L)−Lipschitzian.For (x0, y0) ∈ C × C, let (xn, yn) ⊂ C × C be defined by (1.4?).Then, T has a coupled fixed point in C.

Proof. Let x0 ∈ C be an arbitrary point. By using conditions (H1)(i) and (H1)(ii), we obtainas in Theorem 2.1 that for r ∈ IN,d(xn, xn+r) + d(yn, yn+r) b[

∑n+r−1k=n φk(d(T (x0, y0), x0)) +

∑n+r−1k=n φk(d(T (y0, x0), y0))]

= b[∑n+r−1

k=0 φk(d(T (x0, y0), x0))−∑n−1

k=0 φk(d(T (x0, y0), x0))]

+ b[∑n+r−1

k=0 φk(d(T (y0, x0), y0))−∑n−1

k=0 φk(d(T (y0, x0), y0))]

→ 0 as n→∞,(2.13)

from which it follows by the right-hand side expression in (2.13) that for c ∈ E, 0 ≺≺ c andfor large n, we have

b

[n+r−1∑k=n

φk(d(T (x0, y0), x0)) +n+r−1∑k=n

φk(d(T (y0, x0), y0))

]≺≺ c, (2.14)

thus, showing thatd(xn, xn+r) + d(yn, yn+r) ≺≺ c. (2.15)

It follows again that the sequence (xn, yn) is a Cauchy sequence in C×C with respectto d.

We now show that (xn, yn) is a Cauchy sequence in C × C with respect to ρ too:By Condition (H2)(i), we have that for r ∈ IN,

ρ(xn, xn+r) Ad(xn, xn+r) and ρ(yn, yn+r) Ad(yn, yn+r). (2.16)We obtain from (2.13) and (2.16) that

ρ(xn, xn+r) + ρ(yn, yn+r) A[d(xn, xn+r) + d(yn, yn+r)]

bA[∑n+r−1

k=n φk(d(T (x0, y0), x0)) +∑n+r−1

k=n φk(d(T (y0, x0), y0))]→ 0 as n→∞.

(2.17)

44

Using (2.17), we obtain as in (2.14) and (2.15) that for c ∈ E, 0 ≺≺ c and for large n,

ρ(xn, xn+r) + ρ(yn, yn+r) ≺≺ c. (2.18)

Thus, we obtain from (22) that (xn, yn) is a Cauchy sequence in C × C with respectto ρ too.

By (H3), (X, ρ,W ) is a complete convex cone metric space. Therefore, (C, ρ,W ) is acomplete subspace of complete convex cone metric space (X, ρ,W ). By this reason, thereexist x∗, y∗ ∈ C such that lim


n→∞xn = y∗. That is, for

c ∈ E, 0 ≺≺ c, there exists q ∈ IN such that the inequality Conditions (14) hold.Using (1.3), (2.3), Conditions (H2)(ii) and (H4) as well as the triangle inequality, we

have thatρ(T (x∗, y∗), x∗) Rd(T (x∗, y∗), x∗)

R[d(T (x∗, y∗), T (xq, yq)) + d(T (xq, yq), x∗)]

= R[d(T (xq, yq), T (x∗, y∗)) + d(xq+1, x∗)]

R[kd(xq, x∗) + µd(yq, y

∗) + Ld(T (xq, yq), xq) + d(xq+1, x∗)]

R[Lφq(d(T (x0, y0), x0)) + kd(xq, x∗) + µd(yq, y

∗) + d(xq+1, x∗)],

R[Lφq(d(T (x0, y0), x0)) + kd(xq, x∗) + µd(yq, y

∗) + d(xq+1, x∗)]

≺ kd(xq, x∗) + µd(yq, y

∗) + d(xq+1, x∗), 0 ≺ R ≺ 1,

(2.19)where φq(d(T (x0, y0), x0)) → 0 as q → ∞ (since φ is a comparison function again).Therefore, we have from (2.10) and (2.19) that

ρ(T (x∗, y∗), x∗) ≺ kd(xq, x∗) + µd(yq, y

∗) + d(xq+1, x∗)

≺≺ kc3(1+k)

+ µc3(1+µ)

+ c3(1+k)

≺ c3

+ c3

+ c3

= c,

from which it follows that ρ(T (x∗, y∗), x∗) ≺≺ c. Thus, ρ(T (x∗, y∗), x∗) = 0, that is,T (x∗, y∗) = x∗.Similarly, using (1.3), (11), Conditions (H2)(ii) and (H4) as well as the triangle inequality,we obtainρ(T (y∗, x∗), y∗) Rd(T (y∗, x∗), y∗)

R[d(T (y∗, x∗), T (yq, xq)) + d(T (yq, xq), y∗)]

= R[d(T (yq, xq), T (y∗, x∗)) + d(yq+1, y∗)]

R[kd(yq, y∗) + µd(xq, x

∗) + Ld(T (yq, xq), yq) + d(yq+1, y∗)]

R[Lφq(d(T (y0, x0), y0)) + kd(yq, y∗) + µd(xq, x

∗) + d(yq+1, y∗)],

≺ kd(yq, y∗) + µd(xq, x

∗) + d(yq+1, y∗), since 0 ≺ R ≺ 1,

(2.20)

where φq(d(T (y0, x0), y0)) → 0 as q → ∞ (since φ is a comparison function again). Sincek ≺ 1 + µ, µ ≺ 1 + k, then, we obtain from (2.10) and (2.20) that

d(T (y∗, x∗), y∗) ≺ kd(yq, y∗) + µd(xq, x

∗) + d(yq+1, y∗)

≺≺ kc3(1+µ)

+ µc3(1+k)

+ c3(1+µ)

≺ (1+µ)c3(1+µ)

+ (1+k)c3(1+k)

+ c3(1+µ)

≺ c3

+ c3

+ c3

= c,

from which it follows again that d(T (y∗, x∗), y∗) ≺≺ c. Therefore, d(T (y∗, x∗), y∗) = 0, thatis, T (y∗, x∗) = y∗.

Hence, (x∗, y∗) is a coupled fixed point of T.

Theorem 2.3. Let (X, d,W ) be a complete convex cone metric space, C a nonempty closedconvex subset of X and T : C × C → C is a (k, µ)−Lipschitzian mapping. Suppose thatφ : IR+ → IR+ is a (c)−comparison function such that, for arbitrary x, y ∈ C, there exist

45

u, v ∈ C with(i) d(T (u, v), u) φ(d(T (x, y), x));(ii) d(u, x) bd(T (x, y), x), b 0.For (x0, y0) ∈ C × C, let (xn, yn) ⊂ C × C be defined by (4?).Then, T has a coupled fixed point in C.

Proof. The proof is similar to that of Theorem 2.1, but with L = 0.

Theorem 2.4. Let (X, d,W ) be a complete convex metric space, C a nonempty closed convexsubset of X and T : C × C → C is a (k, µ, L)−Lipschitzian mapping. Suppose that forarbitrary x, y ∈ C, there exist u, v ∈ C such that(i) d(T (u, v), x) rd(T (x, y), x)), 0 r ≺ 1;

(ii) d(T (x, y), x) kd(x,u)+µd(y,v)k+µ+w

, w 0.

For (x0, y0) ∈ C × C, let (xn, yn) ⊂ C × C be defined by (4?), with αn α, ∀ n,α ∈ [0, 1]. Then, T has a coupled fixed point in C, if α ≺ 1−r

k+µ+w+L−r .

Proof. For any x, y ∈ C, let u = W (x, T (x, y); 1− αn). Then,d(u, x) = d(W (x, T (x, y); 1− αn), x)

(1− αn)d(x, x) + αnd(T (x, y), x) = αnd(T (x, y), x).(2.21)

Also,d(u, T (u, v)) = d(W (x, T (x, y); 1− αn), T (u, v))

(1− αn)d(x, T (u, v)) + αnd(T (x, y), T (u, v)) r(1− αn)d(T (x, y), x)

+ αn[ kd(x, u) + µd(y, v) + Ld(T (x, y), x) ] r(1− αn)d(T (x, y), x) + αn(k + µ+ w + L)d(T (x, y), x)= [r + (k + µ+ w + L− r)αn]d(T (x, y), x)= βd(T (x, y), x),

(2.22)

where β = r + (k + µ+ w + L− r)αn and 0 ≤ β < 1 since α ≺ 1−rk+µ+w+L−r .

From (??), we have by a similar process as in Theorem 2.1 thatd(T (xn+1, yn+1), xn+1) βd(T (xn, yn), xn)

≤ β2d(T (xn−1, yn−1), xn−1) · · · ≤ βn+1d(T (x0, y0), x0).(2.23)

Using (2.23) in (2.21) givesd(xn+1, xn) αnd(T (xn, yn), xn) αβnd(T (x0, y0), x0). (2.24)

Similarly, we have from (2.22) again thatd(T (yn+1, xn+1), yn+1) βd(T (yn, xn), yn)

β2d(T (yn−1, xn−1), yn−1) · · · βn+1d(T (y0, x0), y0),(2.25)

from which we get by using (2.25) in (2.21) thatd(yn+1, yn) αnd(T (yn, xn), yn) ≤ αβnd(T (y0, x0), y0). (2.26)

For each r ∈ IN, we obtain by using (2.24) and (2.26) in the repeated application of thetriangle inequality thatd(xn, xn+r) + d(yn, yn+r) [d(xn, xn+1) + d(xn+1, xn+2) + · · ·+ d(xn+r−1, xn+r)]

+ [d(yn, yn+1) + d(yn+1, yn+2) + · · ·+ d(yn+r−1, yn+r)] αβn[1 + β + β2 + · · ·+ βr−1][d(T (x0, y0), x0) + d(T (y0, x0), y0)]

=αβn(1− βr)

1− β[d(T (x0, y0), x0) + d(T (y0, x0), y0)] ,

(since αn ≤ α, ∀ n), → 0 as n→∞ since β ∈ [0, 1).(2.27)

46

It follows from (2.27) that for c ∈ E, 0 ≺≺ c and for large n, we have

αβn(1− βr)1− β

[d(T (x0, y0), x0) + d(T (y0, x0), y0)] ≺≺ c,

and thus, leads to the fact that d(xn, xn+r) + d(yn, yn+r) ≺≺ c.Hence, it follows that the sequence (xn, yn) is a Cauchy sequence in C×C. Since X is

complete, then C is also complete as a subspace of X. Suppose that there exist x∗, y∗ ∈ Csuch that lim


n→∞yn = y∗. That is, for c ∈ E, 0 ≺≺ c, there exists q ∈ IN

such that

d(xq, x∗) ≺≺ c

2(1 + k)and d(yq, y

∗) ≺≺ c

2(1 + µ), for all n q, (2.28)

with k ≺ µ and µ ≺ 1 + k.Using (1.3), (2.23), (2.28) as well as the triangle inequality give

d(T (x∗, y∗), x∗) d(T (x∗, y∗), T (xq, yq)) + d(T (xq, yq), xq) + d(xq, x∗)

= d(T (xq, yq), T (x∗, y∗)) + d(T (xq, yq), xq) + d(xq, x∗)


∗) + Ld(T (xq, yq), xq)+ d(T (xq, yq), xq) + d(xq, x

∗)= (1 + k)d(xq, x

∗) + µd(yq, y∗) + (1 + L)d(T (xq, yq), xq)

= βq(1 + L)d(T (x0, y0), x0) + (1 + k)d(xq, x∗) + µd(yq, y

∗) (1 + k)d(xq, x

∗) + µd(yq, y∗),

(2.29)

since βqd(T (x0, y0), x0))→ 0 as q →∞.It follows from (2.28) and (2.29) that

d(T (x∗, y∗), x∗) (1 + k)d(xq, x∗) + µd(yq, y

∗)≺≺ (1 + k) c

2(1+k)+ µc

2(1+µ)≺ c

2+ c

2= c,

from which it follows that d(T (x∗, y∗), y∗) ≺≺ c. Thus, d(T (x∗, y∗), x∗) = 0, that is,T (x∗, y∗) = x∗.

In a similar manner, from (1.3), (2.25), (2.28) and the triangle inequality, we get

d(T (y∗, x∗), y∗) d(T (y∗, x∗), T (yq, xq)) + d(T (yq, xq), yq) + d(yq, y∗)

= d(T (yq, xq), T (y∗, x∗)) + d(T (yq, xq), yq) + d(yq, y∗)


∗) + Ld(T (yq, xq), yq) + d(yq, y∗)

+ d(T (yq, xq), yq) + d(yq, y∗)

= (1 + L)d(T (yq, xq), yq) + (1 + k)d(yq, y∗) + µd(xq, x

∗) βq(1 + L)d(T (y0, x0), y0)) + (1 + k)d(yq, y

∗) + µd(xq, x∗)

(1 + k)d(yq, y∗) + µd(xq, x

∗),

(2.30)

since βq(d(T (y0, x0), y0))→ 0 as q →∞.Since k ≺ µ, µ ≺ 1 + k, then, we obtain from (2.30) that

d(T (y∗, x∗), y∗) kd(yq, y∗) + µd(xq, x

∗)

≺≺ (1 + k) c2(1+µ)

+ µc2(1+k)

≺ (1 + µ) c2(1+µ)

+ (1+k)c2(1+k)

= c2

+ c2

= c,

from which it follows again that d(T (y∗, x∗), y∗) ≺≺ c. Therefore, d(T (y∗, x∗), y∗) = 0, thatis, T (y∗, x∗) = y∗.

Hence, (x∗, y∗) is a coupled fixed point of T.

47

Theorem 2.5. Let (X, d,W ) be a complete convex metric space, C a nonempty closed convexsubset of X and T : C×C → C is a (k, µ)−Lipschitzian mapping. Suppose that for arbitraryx, y ∈ C, there exist u, v ∈ C such that(i) d(T (u, v), x) rd(T (x, y), x)), 0 r ≺ 1;

(ii) d(T (x, y), x) kd(x,u)+µd(y,v)k+µ+w

, w 0.

For (x0, y0) ∈ C × C, let (xn, yn) ⊂ C × C be defined by (1.4), with αn α, ∀ n,α ∈ [0, 1]. Then, T has a coupled fixed point in C, if α ≺ 1−r

k+µ+w−r .

Proof. The Proof of Theorem 2.5 is complete by setting L = 0 in the Proof of Theorem2.4.

Remark 2.1. Our results generalize and extend analogous ones in the literature. Forinstance, see [18, 21] and some others.

Example 2.1. Suppose that E = IR2, P = (x, y) ∈ IR2 | x, y ≥ 0 ⊆ IR2, and letX = [0, 1]. We define d : X ×X → X by d(x, y) = (|x− y|, |x− y|).

Therefore, (X, d) is a complete cone metric space. Define the mapping T : X ×X → Xby T (x, y) = 3x+2y

10. Then, T satisfies the (k, µ)−Lipschitzian contractive condition (1.2) with

k = 310

and µ = 15

in the following sense:

d(T (x, y), T (u, v)) = |T (x, y)− T (u, v)|= |3x+2y

10− 3u+2v

10| = |3(x−u)

10+ 2(y−v)

10|

310|x− u|+ 1

5|y − v| = 3

10d(x, u) + 1

5d(y, v),

where k = 310

and µ = 15.

Acknowledgements. Some part of this research work was carried out while the author wasa Postdoctoral Fellow at the Centre for Advanced Studies in Mathematics, Lahore Universityof Management Sciences, Lahore, Pakistan. Author is also thankful to the referee for hisvaluable suggestions.

References[1] M. Abbas and I. Beg, Coupled random fixed points of random multivalued operators

on ordered Banach spaces, Communications on Applied Nonlinear Analysis, 13(4)(2006), 31-42.

[2] R.P. Agarwal, D. O’Regan and D.R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications - Topological Fixed Point Theory 6, SpringerScience+Bussiness Media (www.springer.com) (2009).

[3] V. Berinde, Iterative Approximation of Fixed Points, Springer-Verlag Berlin Heidel-berg, 2007.

[4] T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially orderedmetric spaces and applications, Nonlinear Analysis: Theory, Methods & Applications,65(7) (2006), 1379-1393.

[5] I. Beg, Structure of the set of fixed points of nonexpansive mappings on convex metricspaces, Ann. Univ. Marie Curie-Sklodowska Sec. A LII (1998), 7-14.

[6] I. Beg, Inequalities in metric spaces with applications, Topological Methods inNonlinear Analysis, 17 (2001), 183-190.

48

[7] I. Beg, A. Latif, R. Ali and A. Azam, Coupled fixed point of mixed monotone operatorson probabilistic Banach spaces, Archivum Math., 37(1) (2001), 1-8.

[8] S.S. Chang and Y.H. Ma, Coupled fixed point of mixed monotone condensing operatorsand existence theorem of the solution for a class of functional equations arising indynamic programming, J. Math. Anal. Appl., 160 (1991), 468-479.

[9] S.S. Chang, Y.J. Cho and N.J. Huang, Coupled fixed point theorems with applications,J. Korean Math. Soc. 33(3) (1996), 575-585.

[10] L. Ciric and V. Lakshmikantham, Coupled random fixed point theorems for nonlinearcontractions in partially ordered metric spaces, Stochastic Analysis and Applications27, (2009), 1246-1259.

[11] H. Duan and G. Li, Coupled fixed point theorems for a class of mixed monotoneoperators and their applications, Acta Anal. Funct. Appl., 8(4) (2006), 335-340.

[12] M.D. Guay, K.L. Singh and J.H.M. Whitfield, Fixed point theorems for nonexpansivemappings in convex metric spaces, Proceedings, Conference on Nonlinear Analysis(S.P. Singh and J.H. Barry, Eds.), Vol. 80, Marcel Dekker Inc., New York (1982),179-189.

[13] D. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators withapplications, Nonlinear Anal. Theory, Methods & Appl., 11 (1987), 623-632.

[14] S. Guu, On some coupled quasi-fixed point theorems, J. Math. Anal. Appl., 204(2)(1996), 444-450.

[15] L.G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractivemappings, Journal of Math. Anal. Appl., 332(2) (2007), 1468-1476.

[16] V. Lakshmikantham and L. Ciric, Coupled fixed point theorems for nonlinearcontractions in partially ordered metric spaces, Nonlinear Analysis: Theory, Methods& Applications, 70(12) (2009), 4341-4349.

[17] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 44 (1953),506-510.

[18] M.O. Olatinwo, Coupled fixed point theorems in cone metric spaces, Ann. Univ.Ferrara, 57 (2011), 173-180 (DOI 10.1007/s11565-010-0111-3).

[19] M.O. Olatinwo, Stability of coupled fixed point iteration and the continuous depen-dence of coupled fixed points, Communications on Applied Nonlinear Analysis, 19(2)(2012), 71-83.

[20] M.O. Olatinwo, Coupled common fixed points of contractive mappings in metric spaces,Journal of Advanced Research in Pure Mathematics, 4(2) (2012), 11-20.

[21] F. Sabetghadam, H.P. Masiha and A.H. Sanatpour, Some coupled fixed point theoremsin cone metric spaces, Fixed Point Theory and Applications, 2009 (2009), Article ID125426, 8 Pages.

[22] H. Schaefer, Uber die methode sukzessiver approximationen, Jahresber. Deutsch. Math.Verein, 59 (1957), 131-140.

[23] W. Takahashi, A convexity in metric spaces and nonexpansive mapping I, Kodai Math.Sem. Rep., 22 (1970), 142-149.

49

Jnanabha, Vol. 49(1) (2019), 50-66

EXISTENCE THEOREMS FOR A PBVP OF FIRST ORDER FUNCTIONALRANDOM INTEGRODIFFERENTIAL INCLUSIONS

ByBapurao C. Dhage

“Kasubai”, Gurukul Colony, Thodga RoadAhmedpur - 413515, Dist. Latur, Maharashtra, India


(Received : May 11, 2019 ; Revised: May 24, 2019)

Abstract

In this paper, some existence theorems for a periodic boundary value problemof first order ordinary functional random integrodifferential inclusions are provedfor convex and nonconvex cases of random multi-valued functions involved in theinclusion. The existence theorems for extremal random solutions are also proved undercertain monotonic conditions of the multi-valued function. The multi-valued randomfixed point theoretic approach of Dhage (2011) is used while establishing the resultsconcerning the existence of extremal random solutions.2010 Mathematics Subject Classifications: 47H07, 47H10, 60H25.Keywords and phrases: Random integro-differential inclusion; Random fixed pointtheorem; Existence theorem.

1 Statement of the ProblemLet (Ω,A, µ) be a complete σ-finite measure space. For a given Banach space X, let P(X)denote the class of all subsets of X, called the power set of X. Denote

Pp(X) = A ⊂ X | A is non-empty and has the property p.Here, p may be p =closed (in short cl) or p =convex (in short cv) or p =bounded (in short bd)or p =compact (in short cp). Thus Pcl(X), Pcv(X), Pbd(X) and Pcp(X) denote respectively,the classes of all closed, convex, bounded and compact subsets of X. Similarly, Pcl,bd(X) andPcv,cp(X) denote respectively, the classes of closed-bounded and compact-convex non-emptysubsets of the Banach space X.

Let R be the real line and let J = [0, T ] be a closed and bounded interval in R for someT > 0. Now, consider the first order random periodic boundary value problem (in shortRPBVP) of integro-differential inclusion,

x′(t, ω) + λx(t, ω) ∈ F(t, x(θ(t), ω),

∫ σ(t)

0

k(t, τ, x(η(τ), ω), ω) dτ, ω

)a.e. ω ∈ Ω,

x(0, ω) = x(T, ω)

(1.1)

for all t ∈ J , where θ, σ, η : J → J are continuous, k : J × J × R × Ω → R andF : J × R× R× Ω→ Pp(R).

By a random solution of the RPBVP (1.1) on J × Ω we mean a measurable functionx : Ω → C(J,R) satisfying for each ω ∈ Ω, x′(t, ω) + λx(t, ω) = v(t) for some functionv ∈ L1(J,R) such that

v(t) ∈ F

(t, x(θ(t), ω),

∫ σ(t)

0

k(t, τ, x(η(τ), ω), ω) dτ, ω

)a.e. ω ∈ Ω,

50

for all t ∈ J , where C(J,R) and L1(J,R) are respectively the Banach spaces of continuousand Lebesgue integrable real-valued functions defined on J .

The RPBVP (1.1) includes several known random differential inclusions already studiedin the literature as special cases. The special case in the form of differential inclusion

x′(t, ω) + λx(t, ω) ∈ F (t, x(t, ω), ω) a.e. ω ∈ Ω,

x(0, ω) = x(T, ω),

(1.2)

for all t ∈ J , where F : J × R × Ω → Pp(R) is a special case of the RPBVP (1.1) and canbe discussed for various aspects of the solutions via multi-valued fixed point techniques. SeeDeimling [1], Hu and Papapgeorgiou [13], Papapgeorgiou [18] and the reference therein. Inthis chapter we prove the existence theorems for RPBVP (1.1) under convex and nonconvexcase of the multi-valued function F involved in RPBVP (1.1).

2 Auxiliary ResultsLet M(J,R) denote the class of real-valued measurable functions on J and let L1(J,R)denote the Banach space of Lebesgue integrable functions on J with norm ‖ · ‖L1 defined by

‖x‖L1 =

∫ T

0

|x(t)| dt.

Definition 2.1. Given a measurable space (Ω,A) and given a Banach space E, let βE denotethe class of all Borel subsets of E. A single-valued mapping f : Ω→ E is called measurableif for any B ∈ βE, we have

f−1(B) =ω ∈ Ω | f(ω) ∈ B

∈ A.

It is known that scalar multiplication, addition and product of single-valued measurablefunctions are measurable in a separable Banach space.

Let F : J × R × R × Ω → Pp(R) be a multi-valued mapping. Then for any functionx : Ω →∈ C(J,R), let S1

F : Ω × C(J,R) → Pp(L1(J,R)) be a multi-valued operator definedby

S1F (ω)(x) =

v ∈ L1(J,R)

∣∣∣v(t) ∈ F

(t, x(θ(t), ω),

∫ σ(t)

0

k(t, s, x(η(t), ω), ω) ds, ω

)a.e. ω ∈ Ω

.

(2.1)

This is our set of selection functions for the multi-valued function F on J × R× R× Ω.When there is no confusion, we denote S1

F (ω)(x) = S1F (ω)(y), where y(t, ω) = x(θ(t), ω) for

some continuous function θ : J → J . The integral of the random multi-valued function F isdefined as ∫ T

0

F(s,x(θ(s), ω),

∫ σ(s)

0

k(s, τ, x(η(τ), ω), ω)dτ, ω)ds

=

∫ T

0

v(s) ds : v ∈ S1F (ω)(x)

.

Furthermore, if the integral on the left hand side of above expression exists for everymeasurable function x : Ω→ C(J,R), then we say that multi-valued mapping F is Lebesgueintegrable on J .

We need the following definitions in the sequel.

51

Definition 2.2. A multi-valued mapping β : Ω→ Pcp(R) is said to be measurable if for anyy ∈ R, the function ω 7→ d(y, β(ω)) = inf|y − x| : x ∈ β(ω) is measurable.

Definition 2.3. A multi-valued mapping β : J×R×R×Ω→ Pcp(R) is called strong randomCaratheodory if

(i) ω 7→ β(t, x, y, ω) is measurable for each t ∈ J and x, y ∈ R, and(ii) (t, x, y) 7→ β(t, x, y, ω) is jointly Hausdorff continuous for almost everywhere ω ∈ Ω,

Again, a strong random Caratheodory multi-valued function β is called strong randomL1-Caratheodory if

(iii) for each real number r > 0 there exists a measurable function hr : Ω → L1(J,R)such that for almost everywhere ω ∈ Ω,

‖β(t, x, y, ω)‖P = sup|u| : u ∈ β(t, x, y, ω) ≤ hr(t, ω),

for all t ∈ J and x, y ∈ R with |x| ≤ r and |y| ≤ r.

Then we have the following lemmas which are well-known in the literature.

Lemma 2.1 ([15, 16]). Let E be a Banach space. If dim (E) <∞ and β : J×E×Ω→ Pcp(E)is strong L1-Caratheodory, then S1

β(x) 6= ∅ for each x ∈ E.

Lemma 2.2 ([15, 16]). Let E be a Banach space. If β : J × E → Pcp(E) is strongCaratheodory, then the multi-valued mapping t 7→ β(t, x(t)) is measurable for any measurablefunction x : J → E.

Let X be a metric space and let Q : X → Pp(X) be a multi-valued mapping. Q iscalled bounded if

⋃Q(S) is bounded subset of X for all bounded subsets S of X. Q is

called compact if Q(X) =⋃x∈X Qx is a compact subset of X. Again, Q is called totally

bounded if Q(S) =⋃x∈S Qx is totally bounded subset of X for all bounded sets S in X.

It is clear that every compact mapping is totally bounded, but the converse may not betrue. However, these two notions are equivalent on bounded subsets of X. Q is called anupper semi-continuous at x ∈ X if for each open set V in X containing Q(x), there exists aneighborhood N(x) in X such that

⋃Q(N(x)) ⊂ V . Q is called upper semi-continuous on X

if it is upper semi-continuous at each point of X. Finally, Q is called completely continuouson X if it is upper semi-continuous and totally bounded on X. It is known that if Q is aclosed multi-valued mapping with compact values on X, then for any sequences xn andyn in X such that xn → x∗, yn → y∗ and yn ∈ Qxn, n ∈ N, we have that y∗ ∈ Qx∗. Theconverse of this statement holds if Q is a compact multi-valued mapping on X. The detailsof all these definitions appear in Deimling [1] and Hu and Papageorgiou [13].

A multi-valued mapping Q : Ω → Pp(X) is called measurable (respectively weaklymeasurable) if

Q−(B) = ω ∈ Ω | Q(ω) ∩B 6= ∅ ∈ A (2.2)

for all closed (respectively open) subsets B in X. A multi-valued mapping Q : Ω × X →Pp(X) is called a multi-valued random operator if Q(·, x) is measurable for each x ∈ X,and we write Q(ω, x) = Q(ω)x. A measurable function ξ : Ω → X is called a random

52

fixed point of the multi-valued random operator Q(ω) if ξ(ω) ∈ Q(ω)ξ(ω) for a.e. ω ∈ Ω.The set of all random fixed points of the multi-valued random operator Q(ω) is denoted byFQ(ω). A multi-valued random operator Q : Ω×X → Pp(X) is called bounded resp. totallybounded, compact, closed, completely continuous) if the multi-valued mapping Q(ω, ·) isbounded (resp. totally bounded, compact, closed, completely continuous) almost everywherefor ω ∈ Ω.

We employ the following well-known random fixed point theorem for compact andcontinuous multi-valued random mappings in a separable Banach spaces or Polish space.See Dhage [7, 8, 9] and references therein.

Theorem 2.1 (Dhage [8]). Let (Ω,A) be a measurable space and let Br(0) and Br(0) berespectively the open and closed balls in a separable Banach space X centered at origin ofradius r. Let Q : Ω×Br(0)→ Pcp,cv(X) be continuous and condensing multi-valued randomoperator. If there does not exist a function u : Ω→ ∂Br with ‖u(ω)‖ = r such that λ(ω)u ∈Q(ω)u for all ω ∈ Ω and for all measurable functions λ : Ω → R satisfying λ(ω) > 1 on Ω,then Q(ω) has a random fixed point in Br(0).

Remark 2.1. It is known that the compact and totally bounded multi-valued randomoperators are condensing, but the converse may not be true.

Corollary 2.1 (Dhage [8]). Let (Ω,A) be a measurable space and let Br(0) and Br(0) berespectively the open and closed balls in a separable Banach space X centered at origin ofradius r. Let Q : Ω × Br(0) → Pcp,cv(X) be continuous and compact multi-valued randomoperator. If there does not exist a function u : Ω→ ∂Br with ‖u(ω)‖ = r such that λ(ω)u ∈Q(ω)u for all ω ∈ Ω and for all measurable functions λ : Ω → R satisfying λ(ω) > 1 on Ω,then Q(ω) has a random fixed point in Br(0).

3 Existence ResultsWe seek the random solutions of RPBVP (1.1) in the Banach space C(J,R) with usualsupremum norm given by

‖x‖ = supt∈J|x(t)|. (3.1)

Clearly, C(J,R) becomes a separable Banach space with respect to the above norm withsome nice algebraic and topological properties in C(J,R).

We need the following definition a growth function in what follows.

Definition 3.1. A scalar function ψ : R+ → R+ is called a submultiplicative D-functionif

(i) ψ is continuous,(ii) ψ is nondecreasing , and

(iii) ψ is scalarly submultiplicative, that is, ψ(λr) ≤ λψ(r) for all λ ≥ 0 and r ∈ R+.

The class of all D-functions on R+ is denoted by Ψ. There do exist D-functions on R.Indeed, the function ψ : R+ → R+ defined by ψ(r) = ` r, ` > 0 satisfies the conditions(i) − (iii) mentioned above and hence a D-function on R+. Note also that if ψ ∈ Ψ, thenψ(0) = 0.

We consider the following set of hypotheses in what follows.

53

(H0) The functions θ, σ, η : J → J are continuous and the single-valued mapping q : Ω→ Ris measurable.

(H1) The single-valued mapping k : Ω→ C(J × J × R,R) is measurable and there exists ameasurable function α : Ω→ L1(J,R+) such that∣∣∣∣∣

∫ σ(t)

0

k(t, s, y, ω) ds

∣∣∣∣∣ ≤ α(t, ω)|y| a.e. ω ∈ Ω,

for all t, s ∈ J and y ∈ R, where the measurability of the function α is understood inthe sense of Definition 2.1.

(H2) For each (t, x, y) ∈ J × R × R, F (t, x, y, ω) is a compact-convex subset of R almosteverywhere for ω ∈ Ω.

(H3) F is strong random Caratheodory.(H4) There exists a measurable function γ : Ω→ L1(J,R) with γ(t, ω) > 0 a.e. ω ∈ Ω and

a function ψ ∈ Ψ such that

‖F (t, x, y, ω)‖P ≤ γ(t, ω)ψ(|x|+ |y|) a.e. ω ∈ Ω,

for all t ∈ J and x, y ∈ R.

We frequently make use of the following estimate concerning the multi-valued functionF (t, x, y, ω) in the sequel. If the hypotheses (H1) and (H4) hold, then for any measurablefunctions x, y : Ω → C(J,R) with ‖x(ω)‖ ≤ r and ‖y(ω)‖ ≤ r, we obtain a bound of themulti-valued function F independent of x and y and in terms of the growth functions ψ, γand α.

We need the following well-known result in what follows. See Dhage [7], Nieto and Lopez[17] and references therein.

Lemma 3.1. For any function σ ∈ L1(J,R), x is a solution to the differential equation

x′(t) + λx(t) = σ(t), t ∈ J,x(0) = x(T ),

(3.2)

if and only if it is a solution of the integral equation

x(t) =

∫ T

0

Gλ(t, s)σ(s) ds, t ∈ J, (3.3)

where, the Green’s function G(t, s) is given by

Gλ(t, s) =

eλs−λt+λT

eλT − 1, if 0 ≤ s ≤ t ≤ T,

eλs−λt

eλT − 1, if 0 ≤ t < s ≤ T.

(3.4)

Notice that the Green’s function Gλ is continuous and nonnegative on J×J and therefore,the number

MGλ := max |Gλ(t, s)| : t, s ∈ [0, T ]

exists for all λ ∈ R+. For the sake of convenience, we write Gλ(t, s) = G(t, s) and MGλ = MG.

54

Theorem 3.1. Assume that the hypotheses (H0)− (H4) hold. Furthermore, if there exists areal number r > 0 such that

r ≥∫ T

0

γ(t, ω)[1 + α(t, ω)]ψ(r) dt (3.5)

for all ω ∈ Ω, then the RPBVP (1.1) has a random solution in C(J,R) defined on J × Ω.

Proof. Let,∆1 =

ω ∈ Ω | The condition in hypothesis (H1) is true

,

∆2 =ω ∈ Ω | The condition in hypothesis (H2) is true

,


and


.

SetΣ = ∆1 ∩∆2 ∩∆3 ∩∆4,

so that,

Σc = ∆c1 ∪∆c

2 ∪∆c3 ∪∆c

4.

Therefore,

µ (Σc) = µ(∆c1) + µ(∆c

2) + µ(∆c3) + µ(∆c

4) = 0.

Set X = C(J,R). Define a multi-valued operator Q : Ω×X → Pp(X) by

Q(ω)x =

u ∈ X

∣∣u(t, ω) =

∫ T

0

G(t, s)v(s) ds, t ∈ J

and v ∈ S1F (ω)(x)

=(L S1

F (ω))(x)

(3.6)

where L : L1(J,R)→ C(J,R) is a continuous operator defined by

Lv(t) =

∫ T

0

G(t, s) v(s) ds. (3.7)

Clearly, the operator Q(ω) is well defined in view of hypothesis (H3). We shall show thatQ(ω) satisfies all the conditions of Theorem 2.1.

Step I : First, we show thatQ(ω) is a multi-valued random operator onX. First, we showthat the multi-valued mapping (ω, x) 7→ S1

F (ω)(x) is jointly measurable. Let f ∈ L1(J,R)be arbitrary. Then we have

d(f, S1F (ω)(x)) = inf‖f − h‖L1 : h ∈ S1

F (ω)(x)

= inf

∫ T

0

|f(t)− h(t)| dt : h ∈ SF (ω)(x)

=

∫ T

0

inf|f(t)− z| : z ∈ F

(t, x(θ(t), ω),

∫ σ(t)

0

k(t, s, x(η(s), ω), ω) ds, ω)

dt

=

∫ T

0

d(f(t), F

(t, x(θ(t), ω),

∫ σ(t)

0

k(t, s, x(η(s), ω), ω) ds, ω))

dt.

55

Let

y(t, ω) =

∫ σ(t)

0

k(t, s, x(η(s), ω), ω) ds, (t, ω) ∈ J × Ω.

Since the function x(t, ω) is measurable in ω and continuous in t, by Caratheodory theorem,the mapping (t, ω) 7→ x(η(t), ω) is jointly measurable. Again since the mapping k(t, s, x, ω)is measurable in ω and continuous in t, s and x, the mapping

(t, s, x, ω) 7→ k(t, s, x(η(s), ω), ω)

is jointly measurable from J × J ×X × Ω into R. Now the integral∫ σ(t)

0

k(t, s, x(η(s), ω), ω) ds

is the limit of the finite sum of measurable functions, so it is measurable, and consequentlythe mapping (t, ω) 7→ y(t, ω) is jointly measurable.

Next, the multi-valued mapping F (t, x, y, ω) is measurable in ω and dH-continuous in t,x and y, so the multi-valued mapping ω 7→ F (t, x(t, ω), y(t, ω), ω) is measurable in view ofCaratheodory theorem. This further implies that the mapping

ω 7→ F

(t, x(θ(t), ω),

∫ σ(t)

0

k(t, s, x(η(s), ω), ω) ds, ω

)is measurable for all t and x, y ∈ X. Now the function z 7→ d(z, F (t, x, y, ω)) is continuous,and, so, the the mapping

(t, x, ω, f) 7→ d

(f(t), F

(t, x(θ(t), ω),

∫ σ(t)

0k(t, s, x(η(s), ω), ω) ds, ω

))

is jointly measurable from J ×X × Ω× L1(J,R) into R+. Now the integral is the limit ofthe finite sum of measurable functions, and so, d(f, S1

F (ω)(x)) is measurable. As a result,the multi-valued mapping (·, ·)→ S1

F (·)(·) is jointly measurable.

Define the multi-valued map φ on J ×X × Ω by

φ(t, x, ω) =

∫ T

0

G(t, s)F(s, x(θ(s), ω),

∫ σ(s)

0

k(s, τ, x(η(τ), ω), ω) dτ, ω)ds.

We shall show that φ(t, x, ω) is continuous in t in the Hausdorff metric on R. Let tnbe a sequence in J converging to t ∈ J . Then, we have

dH(φ(tn, x, ω), φ(t, x, ω))

= dH

(∫ T

0

G(tn, s)F(s, x(θ(s), ω),

∫ σ(s)

0

k(s, τ, x(η(τ), ω), ω) dτ, ω)ds,

∫ T

0

G(t, s)F(s, x(θ(s), ω),

∫ σ(s)

0

k(s, τ, x(η(τ), ω), ω) dτ, ω)ds

)

=

∫ T

0

|G(tn, s)−G(t, s)|∥∥∥F(s, x(θ(s), ω),

∫ σ(s)

0

k(s, τ, x(η(τ), ω), ω) dτ, ω)∥∥∥Pds

=

∫ T

0

|G(tn, s)−G(t, s)| γ(s, ω)ψ(|x(η(s), ω)|

56

+∣∣∣ ∫ σ(s)

0

k(s, τ, x(η(τ), ω), ω), dτ)∣∣∣) ds

=

∫ T

0

|G(tn, s)−G(t, s)| γ(s, ω)ψ(|x(θ(s), ω)|+ α(s, ω)|x(η(s), ω)|

)ds

→ 0 as n→∞.Thus, the multi-valued map t 7→ φ(t, x, ω) is continuous and hence, by Lemma 2.2, the

map

(t, x, ω) 7→∫ t

0

G(t, s)F(s, x(η(s), ω),

∫ σ(s)

0

k(s, τ, x(η(τ), ω), ω) dτ, ω)ds

is measurable. Consequently, Q(ω) is a random multi-valued random operator on X intoitself.

Step II : Next, we show that Q(ω) is compact and continuous for each fixed ω ∈ Σ.First, we show that Q(ω) is a continuous multi-valued random operator on X. Let xnbe a given sequence of points in X converging to a point x. Then by dH-continuity of themulti-valued mapping F (t, x, y, ω) in x and y and by the dominated convergence theorem,we obtain

limn→∞

Q(ω)xn(t)

= limn→∞

∫ T

0

G(t, s)F(s, xn(θ(s), ω),

∫ σ(s)

0

k(s, τ, xn(η(τ), ω), ω) dτ, ω)ds

=

∫ T

0

G(t, s) limn→∞

F(s, xn(θ(s), ω),

∫ σ(s)

0

k(s, τ, xn(η(τ), ω), ω) dτ, ω)ds

= Q(ω)x(t)

for all t ∈ J and ω ∈ Σ. This shows that Q(ω) is a Hausdorff continuous multi-valuedrandom operator on X.

Next we how that Q(ω) is compact operator on X for each fixed ω ∈ Σ. If S be abounded set in X, then there is a constant r > 0 such that ‖x‖ ≤ r for all x ∈ S. Letyn(ω) be a sequence sequence in

⋃Q(ω)(S) for above fixed ω ∈ Σ. We will show that

yn(ω) has a cluster point. This is achieved by showing that yn(ω) is uniformly boundedand equi-continuous sequence in X.

Case I : First, we show that yn(ω) is uniformly bounded sequence. By the definitionof yn(ω), we have a vn ∈ S1

F (ω)(xn) for some xn ∈ S such that

yn(t, ω) =∫ T

0G(t, s)vn(s) ds, t ∈ J.

Therefore,

|yn(t, ω)| ≤∫ T

0

G(t, s)|vn(s)| ds

≤∫ T

0

G(t, s)∥∥∥F(s, x(θ(s), ω),

∫ σ(s)

0


≤∫ T

0

G(t, s)γ(s, ω)ψ(|x(θ(s), ω)|+ α(s, ω)|x(η(s), ω)|) ds

≤MG ‖δ(ω)‖L1ψ(r)

57

for all t ∈ J , where δ(t, ω) = γ(t, ω)(1 +α(t, ω)) for all (t, ω) ∈ J ×Σ. Taking the supremumover t in the above inequality yields,

‖yn(ω)‖ ≤MG ‖δ(ω)‖L1ψ(r)

which shows that yn(ω) is a uniformly bounded sequence in Q(ω)(X).

Case II : Next we show that yn(ω) is an equi-continuous sequence in Q(ω)(S). Lett, τ ∈ J . Then, for each fixed ω ∈ Σ, we have

|yn(t, ω)− yn(τ, ω)|

=∣∣∣∫ T

0

G(t, s)vn(s) ds−∫ T

0

G(τ, s)vn(s) ds∣∣∣

≤∣∣∣∫ T

0

∣∣G(t, s)−G(τ, s)∣∣ |vn(s)| ds

∣∣∣≤∣∣∣∫ T

0

∣∣G(t, s)−G(τ, s)∣∣ ∥∥∥F(s, x(θ(s), ω),

∫ σ(s)

0

k(s, τ, x(η(τ), ω), ω) dτ, ω)∥∥∥Pds∣∣∣

≤∣∣∣∫ T

0

∣∣G(t, s)−G(τ, s)∣∣γ(s, ω)ψ(‖x(ω)‖+ α(s, ω)‖x(ω)‖) ds

∣∣∣≤∣∣∣∫ T

0

∣∣G(t, s)−G(τ, s)∣∣ γ(s, ω)[1 + α(s, ω)]ψ(r) ds

∣∣∣,for all n ∈ N. From the above inequality, it follows that

|yn(t, ω)− yn(τ, ω)| → 0 as t→ τ.

uniformly for all n ∈ N. This shows that yn(ω) is an equi-continuous sequence in Q(ω)(S).Now yn(ω) is a uniformly bounded and equi-continuous sequence for each ω ∈ Σ, so it hasa cluster point in view of Arzela-Ascoli theorem. Thus Q is a continuous and compact andhence completely continuous multi-valued random operator on Ω×X into X.

Step III : Next, we show that Q(ω) has convex values on X for each ω ∈ Σ. Again, letu1, u2 ∈ Q(ω)x. Then there are v1, v2 ∈ S1

G(ω)(x) such that

u1(t) =

∫ T

0

G(t, s)v1(s) ds, t ∈ J,

and

u2(t) =

∫ T

0

G(t, s)v2(s) ds, t ∈ J.

Now for any λ ∈ [0, 1], one has

λu1(t) + (1− λ)u2(t)

= λ

(∫ T

0

G(t, s)v1(s)) ds

)+ (1− λ)

(∫ T

0

G(t, s)v2(s) ds

)=

∫ T

0

G(t, s)[λv1(s) + (1− λ)v2(s)

]ds.

Since S1F (ω) has convex values on X (because F has convex values), we have that v(t) =

λv1(t) + (1 − λ)v2(t) ∈ S1F (ω)(x)(t) for all t ∈ J . Hence, λu1 + (1 − λ)u2 ∈ Q(ω)x and

consequently Q(ω)x is a convex subset of X for each x ∈ X. As a result, Q(ω) defines amulti-valued random operator Q : Ω×X → Pcp,cv(X).

58

Step IV : Finally, let r be a fixed positive real number and consider the closed ballB[0, r] in the separable Banach space C(J,R). Let there exists an u : Ω→ ∂Br be such that‖u‖ = r and λu(t, ω) ∈ Q(ω)u(t, ω) on J × Ω for all λ > 1. Then there is a v ∈ S1

F (ω)(u)such that

λu(t, ω) =

∫ T

0

G(t, s)v(s) ds

for all t ∈ J and ω ∈ Ω. Therefore,

λ |u(t, ω)| ≤∫ T

0

|G(t, s)| |v(s)| ds

≤∫ T

0

G(t, s)∥∥∥F(s, u(θ(s), ω),

∫ σ(s)

0

k(s, τ, u(η(τ), ω), ω) dτ, ω)∥∥∥Pds

≤∫ T

0

γ(s, ω)ψ(|u(θ(t), ω)|+ α(s, ω)|u(η(s), ω)|) ds

≤∫ T

0

γ(s, ω)ψ(‖u(ω)‖+ α(s, ω)‖u(ω)‖) ds

≤∫ T

0

γ(s, ω)[1 + α(s, ω)

]ψ(‖u(ω)‖) ds

≤∫ T

0

γ(s, ω)[1 + α(s, ω)

]ψ(r) ds (3.8)

for all t ∈ J and ω ∈ Ω. Taking the supremum over t in the above inequality (3.8), we obtain

λ‖u(ω)‖ ≤∫ T

0

γ(s, ω)[1 + α(s, ω)

]ψ(r) ds

or

λr ≤∫ T

0

γ(s, ω)[1 + α(s, ω)

]ψ(r) ds.

This is a contraction since λ > 1 and the inequality∫ T

0

γ(s, ω)[1 + α(s, ω)

]ψ(r) ds ≤ r

for all s ∈ J and ω ∈∑

. Now a direct application of Corollary 2.1 yields the desired result.

4 Non-Convex CaseIn this section, we obtain the existence of the random solutions for nonconvex case of RPBVP(1.1) defined on J ×Ω. This is achieved under certain monotonicity conditions of the multi-valued function F with respect to certain order relation in C(J,R). Define an order relation in C(J,R) by

x y ⇐⇒ y − x ∈ K (4.1)

where, the order cone K in C(J,R) defined by

K = x ∈ C(J,R) | x(t) ≥ 0 for all t ∈ J. (4.2)

Clearly, C(J,R) becomes an ordered Banach space with the order cone K which is normalin it. Let a, b ∈ X = C(J,R) be such that a b. Then by an order interval [a, b] we mean aset in X defined by

[a, b] = x ∈ X | a x b. (4.3)

59

Let a, b : Ω→ X be two measurable functions. By a b on Ω, we mean a(ω) b(ω) forall ω ∈ Ω. Then the sector [a, b] defined by

[a, b] = x ∈ X | a(ω) x b(ω) for all ω ∈ Ω=

⋂ω∈Ω

[a(ω), b(ω)]

is called the random order interval in X.

Now we define the different notions of order relations in Pp(X) as follows. These ordertheoretic notions are useful to define different monotonic concepts of the multi-valued randommappings on the ordered Banach space X.

Let A,B ∈ Pp(X). Then by Ai

B we mean “for every element a ∈ A there exists an

element b ∈ B such that a b.” Again Ad

B means “for each b ∈ B there exists a ∈ Asuch that a b”. Further, we have A

id

B ⇐⇒ Ai

B and Ad

B. Finally, A B impliesthat a b for all a ∈ A and b ∈ B. Note that if A A, then it follows that A is a singletonset.

Definition 4.1. A multi-valued random operator Q : Ω × X → Pcl(X) is called right

monotone increasing if x, y ∈ X, x y, then SQ(ω)(x)i

SQ(ω)(y) a.e. ω ∈ Ω.

We employ the following random fixed point theorem of Dhage [2, 3, 4, 5, 6] for rightmonotone increasing multi-valued random operators in ordered Banach spaces in whatfollows.

Theorem 4.1. Let (Ω,A) be a measurable space and let [a, b] be a random interval in aseparable ordered Banach space X. If Q : Ω × [a, b] → Pcp([a, b]) is a compact, upper semi-continuous, right monotone increasing multi-valued random operator and the cone K in Xis normal, then Q(ω) has a random fixed point in [a, b].

Definition 4.2. A multi-valued F : J ×R×R×Ω→ Pcp(R) is called random Caratheodoryif

(i) ω 7→ F (t, x, y, ω) is measurable for each t ∈ J and x, y ∈ R, and(ii) (t, x, y) 7→ F (t, x, y, ω) is an upper semi-continuous almost everywhere for ω ∈ Ω.

Again, a random Caratheodory multi-valued function F is called random L1-Caratheodory if

(iii) for each real number r > 0 there exists a measurable function hr : Ω → L1(J,R) suchthat for a.e. ω ∈ Ω,

‖F (t, x, y, ω)‖P = sup|u| : u ∈ F (t, x, y, ω) ≤ hr(t, ω)

for all t ∈ J and x, y ∈ R with |x| ≤ r and |y| ≤ r.

Then we have the following lemmas which are well-known in the literature.

Lemma 4.1. Let E be a Banach space. If dim (E) < ∞ and F : J × E × Ω → Pcp(E) israndom L1-Caratheodory, then S1

F (x) 6= ∅ for each x ∈ E.

60

Lemma 4.2. Let E be a Banach space, F a Caratheodory classical multi-valued operatorwith S1

F 6= ∅ and let L : L1(J,E) → C(J,E) be a continuous linear mapping. Then thecomposite operator

L S1F : C(J,E)→ Pbd,cl(C(J,E))

is a closed graph operator on C(J,E)× C(J,E).

We need the following definition in what follows.

Definition 4.3. A measurable function a : Ω → C(J,R) is a strict lower random solutionfor the RPBVP (1.1) if for all v ∈ S1

F (ω)(a), we have

a′(t, ω) + λa(t, ω) ≤ v(t), a(0, ω) ≤ a(T, ω) a.e. ω ∈ Ω,

for all t ∈ J . Similarly, a strict upper random solution for the RPBVP (1.1) on J × Ω isdefined.

We frequently use the following fundamental results while establishing the existence andapproximation theorems for random solution of the RPBVP (1.1) in what follows.

Lemma 4.3 (Dhage [10, 11]). If there exists a differentiable function u ∈ C(J,R) such that

u′(t) + λu(t) ≤ σ(t), t ∈ J,u(0) ≤ u(T ),

(4.4)

for all t ∈ J , where λ ∈ R, λ > 0 and σ ∈ L1(J,R), then

u(t) ≤∫ T

0

G(t, s)σ(s) ds, (4.5)

for all t ∈ J , where G(t, s) is a Green’s function given by the expression (3.4) on J × J .

Lemma 4.4 (Dhage [10, 11]). If there exists a differentiable function v ∈ C(J,R) such that

v′(t) + λv(t) ≥ σ(t), t ∈ J,v(0) ≥ v(T ),

(4.6)

for all t ∈ J , where λ ∈ R, λ > 0 and σ ∈ L1(J,R), then

v(t) ≥∫ T

0

G(t, s)σ(s) ds, (4.7)

for all t ∈ J , where G(t, s) is a Green’s function given by expression (3.4) on J × J .

We consider the following set of hypotheses in the sequel.

(H5) F is random L1-Caratheodory.(H6) The multi-valued mapping x 7→ S1

F (ω)(x) is right monotone increasing in C(J,R).(H7) RPBVP (1.1) has a strict lower random solution a and a strict upper random solution

b with a b defined on J × Ω.

Hypotheses (H5) is common in the literature. Some nice sufficient conditions forguarantying S1

F (ω) 6= ∅ appear in Deimling [32,33] and references therein. A mild hypothesisof (H5) has been used in Halidias and Papageorgiou [60]. Hypotheses (H6) and (H7) arerelatively new to the literature, but the special forms have already been appeared in theworks of several authors.

61

Theorem 4.2. Assume that the assumptions (H0)-(H1) and (H5)-(H7) hold. Then theRPBVP (1.1) has a random solution in [a, b] defined on J × Ω.

Proof. Let,Γ1 =

ω ∈ Ω | The condition in hypothesis (H1) is true

,

Γ5 =ω ∈ Ω | The condition in hypothesis (H5) is true

,


and


.

SetΓ = Γ1 ∩ Γ5 ∩ Γ6 ∩ Γ7,

so that,

Γc == Γc1 ∪ Γc5 ∪ Γc6 ∪ Γc7.

Therefore,

µ (Γc) = µ(Γc1) + µ(Γc5) + µ(Γc6) + µ(Γc7) = 0.

Let X = C(J,R). Define a random order interval [a, b] in X which is well defined in viewof hypothesis (H7). Now the RPBVP (1.1) is equivalent to the random integral inclusion

x(t, ω) ∈∫ T

0

G(t, s)F(s, x(η(s), ω),

∫ σ(s)

0

k(s, τ, x(η(τ), ω), ω) dτ, ω)ds (4.8)

for all t ∈ J .Define a multi-valued operator Q : Ω × [a, b] → Pp(X) by (3.6). Clearly, the operator

Q(ω) is well defined in view of hypothesis (H5). We shall show that Q(ω) satisfies all theconditions of Theorem 4.1.

Step I : First, we show that Q is closed valued multi-valued random operator onΩ× [a, b]. Let ω ∈ Γ be fixed. Observe that the operator Q(ω) = F 1(ω). To show Q(ω) hasclosed values, it then suffices to prove that the composition operator L S1

F (ω) has closedvalues on [a, b]. Let x ∈ [a, b] be arbitrary and let vn be a sequence in S1

F (ω)(x) convergingto v in measure. Then, by the definition of S1

F (ω),

vn(t) ∈ F(t, x(θ(t), ω),

∫ σ(t)

0

k(t, τ, x(η(τ), ω), ω) dτ, ω

)a.e. ω ∈ Ω.

for all t ∈ J . Since F(t, x(θ(t), ω),

∫ σ(t)

0

k(t, τ, x(η(τ), ω), ω) dτ, ω)

is closed, we have that

v(t) ∈ F(t, x(θ(t), ω),

∫ σ(t)

0

k(t, τ, x(η(τ), ω), ω) dτ, ω

)a.e. ω ∈ Ω

for all t ∈ J . Hence, v ∈ S1F (ω)(x). As a result, S1

F (ω)(x) is a closed set in L1(J,R) foreach ω ∈ Σ. From the continuity of L, it follows that (L S1

F (ω)(x) is a closed set in X.Therefore, Q(ω) is a closed-valued multi-valued operator on [a, b] for each ω ∈ Γ.

Next, proceeding with the arguments as in Theorem 4.2, it can be shown that Q(ω) is amulti-valued random operator on [a, b] into X.

62

Step II: Secondly, we show that Q(ω) is right monotone increasing and multi-valuedrandom operator on [a, b] into itself. Let Let ω ∈ Γ be fixed and let x, y ∈ [a, b] be such that

x y. Since (H6) holds, we have that S1F (ω)(x)

i

S1F (ω)(y). Hence Q(ω)(x)

i

Q(ω)(y) forall ω ∈ Σ. From Lemmas 4.3 and 4.4 it follows that a Q(ω)a and Q(ω)b b for all ω ∈ Γ.Therefore, Q(ω) is a right monotone increasing, so we have for almost everywhere ω ∈ Ω,

a Q(ω)ai

Q(ω)xi

Q(ω)b b

for all x ∈ [a, b]. Hence Q defines a right monotone increasing multi-valued random operatorQ : Ω× [a, b]→ Pcp([a, b]).

Step III : Next, we show that Q(ω) is completely continuous random multi-valued mapon Ω × [a, b]. First, we show that Q(ω)([a, b]) is compact for each ω ∈ Γ. Let yn(ω) bea sequence in Q(ω)([a, b]) for above fixed ω ∈ Γ. We will show that yn(ω) has a clusterpoint. This is achieved by showing that yn(ω) is uniformly bounded and equi-continuoussequence in X.

Case I : First, we show that yn(ω) is uniformly bounded sequence. Since the coneK in X is normal, the random order interval [a, b] is norm-bounded. Hence there is a realnumber r > 0 such that ‖yn(ω)‖ ≤ r for all n ∈ N. By the definition of yn(ω), we have avn(ω) ∈ S1

F (ω)(x) for some x ∈ [a, b] such that

yn(t, ω) =

∫ T

0

G(t, s)vn(s) ds, t ∈ J.

Therefore,

|yn(t, ω)| ≤∫ T

0

G(t, s)|vn(s)| ds

≤∫ T

0

G(t, s)∥∥∥F(s, x(θ(s), ω),

∫ σ(s)

0


≤MG

∫ T

0

hr(s, ω) ds

≤MG‖hr(ω)‖L1

for all t ∈ J , where r = ‖a(ω)‖+‖b(ω)‖. Taking the supremum over t in the above inequalityyields,

‖yn(ω)‖ ≤MG‖hr(ω)‖L1

which shows that yn(ω) is a uniformly bounded sequence in Q(ω)([a, b]).

Next we show that yn(ω) is an equi-continuous sequence in Q(ω)([a, b]). Let t, τ ∈ J .Then we have

|yn(t, ω)− yn(τ, ω)| ≤∣∣∣∫ T

0

G(t, s)vn(s) ds−∫ T

0

G(τ, s)vn(s) ds∣∣∣

≤∣∣∣∫ T

0

|G(t, s)−G(τ, s)| |vn(s)| ds∣∣∣

≤∣∣∣∫ T

0

|G(t, s)−G(τ, s)|hr(s, ω) ds∣∣∣

63

→ 0 as t→ τ

for each n ∈ N and ω ∈ Ω.This shows that yn(ω) is an equi-continuous sequence in Q(ω)([a, b]). So yn(ω) has a

cluster point in view of Arzela-Ascoli theorem. As a result, Q(ω) is a compact multi-valuedrandom operator on Ω× [a, b] into Pcp([a, b]).

Case II : Next, let xn(ω) be a sequence in [a, b] such that xn(ω)→ x∗(ω). Let yn(ω)be a sequence such that yn(ω) ∈ Q(ω)xn and yn(ω)→ y∗(ω). We show that y∗(ω) ∈ Q(ω)x∗.Since yn(ω) ∈ Q(ω)xn, there exists a vn ∈ S1

F (ω)(xn) such that

yn(t, ω) =

∫ T

0

G(t, s)vn(s) ds, t ∈ J.

From lemma 4.2, it follows that L S1F (ω) is a closed graph operator. Also from the

definition of L, we haveyn(t, ω) ∈

(L S1

F (ω))(xn).

Since yn(ω)→ y∗(ω), there is a point v∗ ∈ S1F (x∗) such that

y∗(t, ω) =

∫ T

0

G(t, s)v∗(s) ds, t ∈ J.

This shows that Q(ω) is a upper semi-continuous multi-valued random operator on[a, b]. Therefore, Q(ω) defines an upper semi-continuous and compact and hence completelycontinuous multi-valued random operator on [a, b]. Now an application of Theorem 4.1yields that Q(ω) has a random fixed point which further implies that the RPBVP (1.1) hasa random solution on J × Ω. This completes the proof.

Next, we prove the existence of the extremal random solutions for the RPBVP (1.1) onJ × Ω.

Definition 4.4. A multi-valued random operator Q : Ω × X → Pp(X) is called strictmonotone increasing if x, y ∈ X x y, x 6= y, then Q(ω)x Q(ω)y a.e. ω ∈ Ω. Similarly,the multi-valued random operator Q(ω) is called strict monotone decreasing if x, y ∈ Xx y, x 6= y, then Q(ω)x Q(ω)y a.e. ω ∈ Ω. Finally, Q(ω) is called strict monotone ifit is either a strict monotone increasing or strict monotone decreasing multi-valued randomoperator on X.

Remark 4.1. We remark that every strict monotone increasing multi-valued randomoperator in an ordered Banach space with compact values is right monotone increasing, butthe converse may not be true.

Below we prove a random fixed point theorem for strict monotone increasing multi-valuedrandom operators on a separable ordered Banach space into itself. We use the the followingmulti-valued random fixed point theorem of Dhage [6] in what follows.

Theorem 4.3 (Dhage [6]). Let (Ω,A) be a measurable space and let [a, b] be a randominterval in a separable ordered Banach space X. If Q : Ω× [a, b] → Pcp([a, b]) is completelycontinuous and strict monotone increasing multi-valued random operator and the cone K in

64

X is normal, then Q(ω) has a least random fixed point x∗(ω) and a greatest random fixedpoint y∗(ω) in [a, b] and the sequences xn(ω) and yn(ω) defined by

x0(ω) = a(ω), xn+1(ω) ∈ Q(ω)xn, n = 0, 1, 2, ...,

andy0(ω) = b(ω), yn+1(ω) ∈ Q(ω)yn, n = 0, 1, 2, ...,

converge to x∗(ω) and y∗(ω) respectively.

We consider the following hypothesis in the sequel.

(H8) For each t, s ∈ J , the single-valued mapping k(t, s, x, ω) is nondecreasing in x almosteverywhere for ω ∈ Ω.

(H9) For each t ∈ J , the multi-valued map (F (t, x, y, ω) is strict monotone increasing in xand y almost everywhere for ω ∈ Ω.

Theorem 4.4. Assume that (H5), (H7) and (H8)-(H9) hold. Then the RPBVP (1.1) has aminimal random solution and a maximal random solution in [a, b] defined on J × Ω.

Proof. The proof is quite similar to that of Theorem 4.2. We briefly sketch the outline ofthe proof. Let,

Υ5 =ω ∈ Ω | The condition in hypothesis (H5) is true

,


,


and


.

SetΥ = Υ5 ∩Υ7 ∩Υ8 ∩Υ9,

so that,

Υc = Υc5 ∪Υc

7 ∪Υc8 ∪Υc

9.

Therefore,

µ (Υc) = µ(Υc5) + µ(Υc

7) + µ(Υc8) + µ(Υc

9) = 0.

Set X = C(J,R) and consider the random order interval [a, b] in X which does existin view of hypothesis (H7). Here S1

F (ω)(x) 6= ∅ for each x ∈ [a, b] and ω ∈ Υ in view ofhypothesis (H5). Also by hypotheses (H8) and (H9), the multi-valued mapping x 7→ S1

F (ω)(x)is strict monotone increasing on [a, b]. Consequently, the multi-valued random operator Q(ω)defined by (3.6) is strict monotone increasing on [a, b]. The rest of the proof is similar toTheorem 4.2 and now the desired result follows by an application of Theorem 4.3.

Acknowledgement. Author is thankful to the referee for his valuable suggestions toimprove the paper in its present form.

65

References[1] K. Deimling, Multi-valued Differential Equations, De Gruyter, Berlin 1998.[2] B.C. Dhage, Some algebraic fixed point theorems for multi-valued operators with

applications, Discuss. Math. Differ. Incl. Control & Optim., 26 (2006), 5-55.[3] B.C. Dhage, Fixed point theorem for discontinuous multi-valued operators on ordered

spaces with applications, Comput. & Math. Appl., 51 (3-4) (2006), 589-604.[4] B.C. Dhage, Monotone iterative technique for Caratheodory theory of nonlinear

functional random integral equations, Tamkang J. Math., 33 (4)(2002), 341-351.[5] B.C. Dhage, Multi-valued operators and fixed point theorems in Banach algebras I,

Taiwanese J. Math., 10 (4) (2006), 1025-1045.[6] B.C. Dhage, Monotone increasing multi-valued random operators and differential

inclusions, Nonlinear Funct. Anal. & Appl., 12 (2007), 399-419.[7] B.C. Dhage, Periodic boundary value problems of first order Caratheodory and

discontinuous differential equations, Nonlinear Funct. Anal. & Appl., 13(2) (2008),323-352.

[8] B.C. Dhage, Multi-valued condensing random operators and functional random integralinclusions, Opuscula Math., 31 (1) (2011), 27-48.

[9] B.C. Dhage, Existence theory for first order functional random integrodifferentialinclusions, Nonlinear Studies, 24(2) (2017), 309-328.

[10] B.C. Dhage, Dhage iteration method for PBVPs of nonlinear first order hybrid integro-differential equations, Int. J. Nonlinear Anal. Appl., 8 (2017), 95-112.

[11] B.C. Dhage, Dhage iteration method in the theory of ordinary nonlinear PBVPs offirst order functional differential equations, Commun. Optim. Theory, 2017 (2017),Article ID 32, pp. 22.

[12] S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for DiscontinuousNonlinear Differential Equations, Pure and Applied Mathematics, Marcel Dekker, NewYork, 1994.

[13] S. Hu and N.S. Papageorgiou, Hand book of Multivalued Analysis, Vol. I: Theory,Kluwer Academic Publishers Dordrechet / Boston / London 1997.

[14] S. Itoh, Random fixed point theorems with applications to random differentialequations in Banach spaces, J. Math. Anal. Appl., 67 (1979), 261-273.

[15] A. Lasota and Z. Opial, An application of the Kakutani–Ky Fan theorem in the theoryof ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phy.,13 (1965), 781-786.

[16] K. Kuratowskii, and C. Ryll-Nardzewskii, A general theorem on selectors, Bull. Acad.Polon. Sci. Ser. Math. Sci. Astron. Phys., 13 (1965), 397–403.

[17] J.J. Nieto, R. Rodriguez-Lopez, Existence and approximation of solution for nonlineardifferential equations with periodic boundary conditions, Comput. Math. Appl., 40(2000), 435-442.

[18] N.S. Papageorgiou, Random fixed points and random differential inclusions, Intern. J.Math. & Math. Sci., 11 (3) (1988), 551-560.

66

Jnanabha, Vol. 49(1) (2019), 67-79

A COMMON FIXED POINT THEOREM FOR WEAKLY RECIPROCALLYCONTINUOUS SYSTEMS OF MAPS SATISFYING A GENERAL

CONTRACTIVE CONDITION OF INTEGRAL TYPEBy

Deepak Khantwal* and U.C. Gairola*Department of Mathematics,

Graphic Era Hill University, Dehradun-248002, Uttarakhand, IndiaDepartment of Mathematics,

H.N.B. Garhwal University, BGR Campus, Pauri Garhwal-246001, Uttarakhand, IndiaEmail:[email protected]*, [email protected]

(Received : May 15, 2019 ; Revised: May 24, 2019)

Abstract

In this paper we prove a common fixed point theorem for weakly reciprocallycontinuous systems of maps satisfying a general contractive inequality of integral typeon the finite product of metric spaces. Our result generalizes the results of Branciari[2], Rhoades [25], Matkowski [17] and Gairola [5].2010 Mathematics Subject Classifications: 47H10, 54H25.Keywords and phrases: Fixed point, coordinatewise commuting maps, weaklycommuting maps, asymptotically commuting maps, weak reciprocal continuous mapsand product space.

1 Introduction and PreliminariesBranciari [2] generalized the celebrated Banach contraction principle for a single-valued self-map using an integral type contraction on a complete metric space. This result has beenextended and generalized among others by Rhoades [25] , Vijayaraju et al. [35] , Suzuki[32], Gairola [5] , Gairola-Rawat [12], Vetro [34], Samet-Vetro [26], Stojakovic et al. [31] andothers. On the other hand Matkowski [17] -[18] generalized the Banach contraction principalfor a system of n maps on the finite product of metric spaces. Several authors have extendedand generalized the result of Matkowski [op. cit.] for systems of sigle-valued as well assystems of multi-valued maps on product space (cf. Czerwik [3]-[4], Reddy- Subrahmanyam[23]-[24], Singh-Kulshrestha [30], Singh-Gairola [28]-[29], Baillon-Singh [1], Matkowski-Singh[19], Gairola et al. [13], [14], Gairola-Jangwan [6]-[7], Gairola-Khantwal [8]-[10] and others).Recently Gairola [5] combined the idea of Matkowski [op. cit.] and Branciari [op. cit.] andproved a fixed point theorem for a system of single-valued maps on the finite product ofmetric spaces. In this paper we extend and generalize the result of Gairola [op. cit.] andprove a common fixed point theorem for two systems of maps on finite product of metricspaces using coordinatewise weak reciprocal continuity. Our result also generalizes the resultsof Branciari [2], Rhoades [25], Matkowski [17] and others in the literature.

Throughout this paper, we shall follow the following notations and definitions.Let aik be non-negative numbers for i, k = 1, . . . , n and c

(t)ik square matrix defined in

Matkowski [17, 18] (see also [3]).

c(0)ik =

aik, i 6= k

1− aik, i = ki, k = 1, . . . , n (1.1)

67

c(t+1)ik =

c

(t)11 c

(t)i+1,k+1 + c

(t)i+1,1c

(t)1,k+1, i 6= k

c(t)11 c

(t)i+1,k+1 − c

(t)i+1,1c

(t)1,k+1, i = k

(1.2)

t = 0, 1, . . . , n− 2, i, k = 1, . . . , n− t− 1, n ≥ 2.

Let (Xi, di), i = 1, . . . , n, be metric spaces,

X = X1 × · · · ×Xn,

x = (x1, . . . , xn) ∈ X,fi, Ti : X → Xi, i = 1, . . . , n and

xm = (xm1 , . . . , xmn ), m ∈ N (natural numbers) be a sequence in X.

Definition 1.1. [29] Two systems of maps f1, . . . , fn and T1, . . . , Tn are coordinatewisecommuting at a point x ∈ X if and only if fi(T1x, . . . , Tnx) = Ti(f1x, . . . , fnx), i =1, . . . , n. Two systems of maps are coordinatewise commuting on X if and only if theyare coordinatewise commuting at every point of X.

Definition 1.2. [29] Two systems of maps f1, . . . , fn and T1, . . . , Tn are coordinatewiseweakly commuting at a point x ∈ X if and only if di(fi(T1x, . . . , Tnx), Ti(f1x, . . . , fnx)) ≤di(fix, Tix), i = 1, . . . , n. Two systems of maps are coordinatewise weakly commuting on Xif and only if they are coordinatewise weakly commuting at every point of X.

Remark 1.1. Evidently two coordinatewise commuting systems of maps are coordinatewiseweakly commuting. However, the weakly commuting systems of maps need not to becommuting (see [28],[29]).

Definition 1.3. [14] Two systems of maps f1, . . . , fn and T1, . . . , Tn are coordinatewiseasymptotically commuting or, following the terminology of Jungck [15], coordinatewisecompatible, if and only if

limm→∞

di(fi(T1xm, . . . , Tnx

m), Ti(f1xm, . . . , fnx

m)) = 0,

whenever limm→∞

fixm = lim

m→∞Tix

m = ui for some ui ∈ Xi, i = 1, . . . , n.

Definition 1.4. [8] Two systems of maps f1, . . . , fn and T1, . . . , Tn are coordinatewisereciprocal continuous if and only if lim

m→∞fi(T1x

m, . . . , Tnxm) = fiz and lim

m→∞Ti(f1x

m, . . . , fnxm) =

Tiz, whenever there exist a sequence xm in X such that limm→∞

fixm = lim

m→∞Tix

m =

zi for all i = 1, . . . , n.

If two systems of maps f1, . . . , fn and T1, . . . , Tn are continuous then they arecoordinatewise reciprocal continuous but the converse need not be true (see Example 1.2[8]).

Definition 1.5. [9] Two systems of maps f1, . . . , fn and T1, . . . , Tn are said to becoordinatewise weak reciprocal continuous if and only if lim

m→∞fi(T1x

m, . . . , Tnxm) = fiz

or limm→∞

Ti(f1xm, . . . , fnx

m) = Tiz, whenever there exist a sequence xm in X such that

limm→∞

fixm = lim

m→∞Tix

m = zi for all i = 1, . . . , n.

68

Remark 1.2. Notice that the above definitions with n = 1 are standard ones for commuting,weakly commuting (see [16] and [27]), asymptotically commuting (see, [33]) (also calledcompatible [15]), reciprocal continuous maps ([20] and see also [22]) and weak reciprocalcontinuous maps (see [21]).

Remark 1.3. Asymptotically commuting (or compatible) class of maps includes commutingand weakly commuting maps. Commuting maps are necessarily weakly and asymptoticallycommuting both (see, for instance, [15], [27], [29], [33]).

Remark 1.4. The commutativity, weak commutativity and asymptotic commutativity (orcompatibility) are equivalent at the point of coincidence of two (or two systems of) maps(see, [1], [13], [15]).

The following Lemma is due to Matkowski [op. cit.] (see also [3], [30]).

Lemma 1.1. Let c(0)ik ≥ 0, i, k = 1, . . . , n, n ≥ 2, then the system of inequalities

n∑k=1

aikrk < ri, i = 1, . . . , n (1.3)

has a positive solution r1, . . . , rn if and only if the following inequalities hold:

c(t)ii > 0, i = 1, . . . , n− t; t = 0, . . . , n− 1, n ≥ 2. (1.4)

2 Main ResultNow we state our main result.

Theorem 2.1. Let (Xi, di), i = 1, . . . , n, be complete metric spaces and Ti, fi : X → Xi, besuch that

Ti(X) ⊂ fi(X), i = 1, . . . , n. (2.1)

The systems of maps (T1, . . . , Tn) and (f1, . . . , fn) are coordinatewise weakly reciprocallycontinuous and coordinatewise asymptotically commuting on X.

(2.2)If there exist non-negative numbers b < 1 and aik, i, k = 1, . . . , n, such that (1.1),(1.2),(1.3)and the following hold:∫ di(Tix,Tiy)

0

φ(ξ)dξ ≤ maxi

n∑k=1

aik

∫ dk(fkx,fky)

0

φ(ξ)dξ, b

∫ Ni(x,y)

0

φ(ξ)dξ

(2.3)

where Ni(x, y) = maxdi(fix, Tix), di(fiy, Tiy), di(fix,Tiy)+di(fiy,Tix)

2

for all x, y ∈ X and

ξ : R+ → R+ is a Lebesgue-integrable mappings which is summable, non-negative and suchthat ∫ ε

0

φ(ξ)dξ > 0 for each ε > 0, (2.4)

then there exists unique point z ∈ X such that

fiz = zi = Tiz, i = 1, . . . , n. (2.5)

Proof. First we note that the system (1.3) and

69

n∑k=1

aikrk < ri, i = 1, . . . , n,

are equivalent for some positive numbers r1, . . . , rn. Further if we put

h = max

r−1i

n∑k=1

aikrk

then h ∈ (0, 1) and we may choose positive numbers r1, . . . , rn such that

n∑k=1

aikrk ≤ hri, i = 1, . . . , n.

Pick x0i in Xi, i = 1, . . . , n. Since (2.1) holds, we choose a sequence xm in X such that

fixm+1 = Tix

m, i = 1, . . . , n, m = 0, 1, . . . .

If at any stage fixm+1 = fix

m+2 then fixm+1 = Tix

m+1 that is, fi and Ti have a coincidencepoint at xm+1. Without loss of generality, we may assume that∫ di(fix

2,fix1)

0

φ(ξ)dξ ≤ ri, i = 1, . . . , n.

Then by (2.3), we have∫ di(fix3,fix

2)

0

φ(ξ)dξ =

∫ di(Tix2,Tix

1)

0

φ(ξ)dξ

≤ max

n∑k=1

aik

∫ dk(fkx2,fkx

1)

0

φ(ξ)dξ, b

∫ Ni(x2,x1)

0

φ(ξ)dξ

(2.6)

where

Ni(x2, x1) = max

di(fix

2, Tix2), di(fix

1, Tix1),

di(fix2, Tix

1) + di(fix1, Tix

2)

2

= max

di(fix

2, fix3), di(fix

1, fix2),

di(fix1, fix

3)

2

= max

di(fix

2, fix3), di(fix

1, fix2).

Substituting the value of Ni(x2, x1) in (2.6), we get∫ di(fix

3,fix2)

0

φ(ξ)dξ ≤ max

n∑k=1

aik∫ dk(fkx

2,fkx1)

0φ(ξ)dξ,

b∫ di(fix2,fix3)

0φ(ξ)dξ, b

∫ di(fix1,fix2)

0φ(ξ)dξ

≤ max

n∑k=1

aik∫ dk(fkx

2,fkx1)

0φ(ξ)dξ, b

∫ di(fix1,fix2)

0φ(ξ)dξ

≤ max

n∑k=1

aikrk, bri

≤ max hri, bri = cri, where c = maxh, b .

Similarly ∫ di(fix4,fix

3)

0

φ(ξ)dξ ≤ c2ri.

70

Inductively ∫ di(fixm+1,fix

m)

0

φ(ξ)dξ ≤ cm−1ri,

which implies thatdi(fix

m+1, fixm)→ 0 as m→∞. (2.7)

Now we prove that fixm is a Cauchy sequence in Xi, i = 1, . . . , n. Assume that it is nottrue then for each i = 1, . . . , n and positive integer s, there exist a εi > 0 and positiveintegers pi(s), qi(s) with s < pi(s) < qi(s) such that

di(fixpi(s), fix

qi(s)) ≥ εi, i = 1, . . . , n. (2.8)

Let qi(s) be the least integer exceeds pi(s) and satisfies (2.8), for each positive integer s.Then it is clear that

di(fixpi(s), fix

qi(s)) ≥ εi and di(fixpi(s), fix

qi(s)−1) < εi,∀s ∈ N, i = 1, . . . , n. (2.9)

Now from (2.7) and (2.9), we have

εi ≤ di(fpi(s)i x, f

qi(s)i )x)

≤ di(fpi(s)i x, f

qi(s)−1i x) + di(f

qi(s)−1i x, f

qi(s)i x).

Making s→∞, it follows that

lims→∞

di(fpi(s)i x, f

qi(s)i )x = εi.

Using triangular inequality,

di(fpi(s)i x, f

qi(s)i x) ≤ di(f

pi(s)i x, f

pi(s)−1i x) + di(f

pi(s)−1i x, f


qi(s)−1i x, f

qi(s)i x)

≤ 2di(fpi(s)i x, f

pi(s)−1i x) + di(f

pi(s)i x, f


qi(s)−1i x, f

qi(s)i x).

Making s→∞ and using (2.7), (2.9), we deduce that

lims→∞

di(fixpi(s)−1, fix

qi(s)−1) = εi = lims→∞

di(fpi(s)i x, f

qi(s)i x). (2.10)

Now we may assume that ∫ εi

0

φ(ξ)dξ ≤ ri, i = 1, . . . , n.

Then from (2.3),∫ di(fixpi(s),fix

qi(s))

0

φ(ξ)dξ =

∫ di(Tixpi(s)−1,Tix

qi(s)−1)

0

φ(ξ)dξ

≤ max

n∑k=1

aik∫ dk(fkx

pi(s)−1,fkxqi(s)−1)

0φ(ξ)dξ,

b∫ Ni(xpi(s)−1,xqi(s)−1)

0φ(ξ)dξ

(2.11)

where

Ni(xpi(s)−1, xqi(s)−1) = max

di(fix

pi(s)−1, Tixpi(s)−1), di(fix

qi(s)−1, Tixqi(s)−1),

di(fixpi(s)−1,Tix

qi(s)−1)+di(fixqi(s)−1,Tix

pi(s)−1)2

= max

di(fix

pi(s)−1, fixpi(s)), di(fix

qi(s)−1, fixqi(s)),

di(fixpi(s)−1,fix

qi(s))+di(fixqi(s)−1,fix

pi(s))2

71

≤ max

di(fix

pi(s)−1, fixpi(s)), di(fix

qi(s)−1, fixqi(s)),

di(fixpi(s)−1,fix

qi(s)−1)+di(fixqi(s)−1,fix

qi(s))2

+di(fix

qi(s)−1,fixpi(s)−1)+di(fix

pi(s)−1,fixpi(s))

2

.

Making s→∞ and using (2.7), (2.10), we deduce that

lims→∞

Ni(xpi(s)−1, xqi(s)−1) ≤ εi. (2.12)

Taking limit s→∞ both sides in (2.11) and using (2.10), (2.12), we get∫ εi

0

φ(ξ)dξ ≤ max

n∑k=1

aik

∫ εk

0

φ(ξ)dξ, b

∫ εi

0

φ(ξ)dξ

≤ max

n∑k=1

aikrk, bri

≤ max hri, bri

= cri, where c = maxh, b < 1.

Inductively ∫ εi

0

φ(ξ)dξ ≤ cmri.

Making m→∞, we get ∫ εi

0

φ(ξ)dξ ≤ 0,

which contradict the condition (2.4). Hence fixm and Tixm are Cauchy sequences inXi, i = 1, . . . , n, and since Xi is a complete metric space therefore there exist a point ti (say)in Xi such that the both sequences fixm and Tixm converges to ti .

Since systems of maps (T1, . . . , Tn) and (f1, . . . , fn) are coordinatewise weakly reciprocallycontinuous, so for each i = 1, . . . , n,

either limm→∞

fi(T1xm, . . . , Tnx

m) = fit or limm→∞


m) = Tit.

Case (I): Let us suppose that limm→∞


m) = fit, i = 1, . . . , n then coordinate-

wise asymptotic commutativity of systems of maps (T1, . . . , Tn) and (f1, . . . , fn) gives

di(Ti(f1xm, . . . , fnx

m), fi(T1xm, . . . , Tnx

m))→ 0, as m→∞,that is

limm→∞


m) = limm→∞


m) = fit. (2.13)

Since

fixm+1 = Tix

m

therefore

fi(f1xm+1, . . . , fnx

m+1) = fi(T1xm, . . . , Tnx

m)

andlimm→∞

fi(f1xm+1, . . . , fnx

m+1) = fit. (2.14)

From (2.3), with fxm+1 := (f1xm+1, ..., fnx

m+1),∫ di(Tifxm+1,Tit)

0

φ(ξ)dξ ≤ max

n∑k=1

aik∫ dk(fkfx

m+1,fkt)

0φ(ξ)dξ,

b∫ Ni(fxm+1,t)

0φ(ξ)dξ

(2.15)

72

where

Ni(fxm+1, t) = max

di(fifx

m+1, Tifxm+1), di(fit, Tit),

di(fifxm+1,Tit)+di(fit,Tifx

m+1)2

.

Making m→∞ and using (2.13) and (2.14), we get

limm→∞

Ni(fxm+1, t) = max

di(fit, fit), di(fit, Tit),

di(fit, Tit) + di(fit, fit)

2

= di(fit, Tit). (2.16)

Taking limit m→∞ both sides in (2.15) and using (2.16), we have∫ di(fit,Tit)

0

φ(ξ)dξ ≤ max

n∑k=1

aik∫ dk(fkt,fkt)

0φ(ξ)dξ, b

∫ di(fit,Tit)0

φ(ξ)dξ

= b

∫ di(fit,Tit)

0

φ(ξ)dξ,

implies thatdi(fit, Tit) = 0.

This gives fit = Tit, i = 1, . . . , n. Thus the systems of maps (f1, . . . , fn) and (T1, . . . , Tn) hasa coincidence point t = (t1, . . . , tn) in X. Since coordinatewise asymptotic commutativity ofsystems (f1, . . . , fn) and (T1, . . . , Tn) is equivalent to thier coordinatewise commutativity ata coincidence point t in X. Therefore

fi(T1t, . . . , Tnt) = Ti(f1t, . . . , fnt) = fi(f1t, . . . , fnt) = Ti(T1t, . . . , Tnt).

Now we may assume that∫ di(Tit,Ti(T1t,...,Tnt))

0

φ(ξ)dξ ≤ ri, i = 1, . . . , n.

From (2.3), with Tt := (T1t, . . . , Tnt), we obtain∫ di(Tit,TiTt)

0

φ(ξ)dξ ≤ max

n∑k=1

aik

∫ dk(fkt,fkTt)

0

φ(ξ)dξ, b

∫ Ni(t,T t)

0

φ(ξ)dξ

, (2.17)

where

Ni(t, T t) = max

di(fit, Tit), di(fiTt, TiTt),

di(fit, TiTt) + di(fiTt, Tit)

2

= max

di(Tit, Tit), di(TiTt, TiTt),

di(Tit, TiTt) + di(TiTt, Tit)

2

= di(Tit, TiTt).

Substituting the value of Ni(t, T t) in (2.17), we get∫ di(Tit,TiTt)

0

φ(ξ)dξ ≤ max

n∑k=1

aik

∫ dk(Tkt,TkTt)

0

φ(ξ)dξ, b

∫ di(Tit,TiTt)

0

φ(ξ)dξ

≤ max

n∑k=1

aikrk, bri

≤ maxhri, bri


73

Inductively ∫ di(Tit,TiTt)

0

φ(ξ)dξ ≤ cmri.

Making m→∞, we getdi(Tit, TiTt) = 0,

that is, Tit = Ti(T1t, . . . , Tnt). We also have Tit = Ti(T1t, . . . , Tnt) = fi(T1t, . . . , Tnt). HenceTit is a solution of systems of equation (2.5).

Case (II): Let us assume that limm→∞


m) = Tit, i = 1, . . . , n, then

coordinatewise asymptotic commutativity of systems of maps (T1, . . . , Tn) and (f1, . . . , fn)yields

limm→∞


m) = limm→∞


m) = Tit. (2.18)

Since

fixm+1 = Tix

m

therefore

fi(f1xm+1, . . . , fnx

m+1) = fi(T1xm, . . . , Tnx

m)

andlimm→∞

fi(f1xm+1, . . . , fnx

m+1) = Tit. (2.19)

In view of (2.1), there exist a point v = (v1, . . . , vn) ∈ X such that

Tit = fiv, i = 1, . . . , n. (2.20)

By (2.3), with fxm+1 := (f1xm+1, . . . , fnx

m+1),∫ di(Tifxm+1,Tiv)

0

φ(ξ)dξ ≤ max

n∑k=1

aik∫ dk(fkfx

m+1,fkv)

0φ(ξ)dξ,

b∫ Ni(Txm+1,v)

0φ(ξ)dξ

, (2.21)

where

Ni(fxm+1, v) = max

di(fifx

m+1, Tifxm+1), di(fiv, Tiv),

di(fifxm+1,Tiv)+di(fiv,Tifx

m+1)2

.

Making m→∞ and using (2.18), (2.19) and (2.20), we have

limm→∞

Ni(fxm+1, v) = max

di(Tit, Tit), di(fiv, Tiv), di(Tit,Tiv)+di(fiv,Tit)

2

= di(Tit, Tiv). (2.22)

Now taking limit m→∞ both side in (2.21) and using (2.22), we get∫ di(Tit,Tiv)

0

φ(ξ)dξ ≤ max

n∑k=1

aik∫ dk(Tkt,fkv)

0φ(ξ)dξ, b

∫ di(Tit,Tiv)

0φ(ξ)dξ

= b

∫ di(Tit,Tiv)

0

φ(ξ)dξ.

This givesdi(Tit, Tiv) = 0, (2.23)

which implies

74

Tit = Tiv.

From (2.20) and (2.23)Tiv = fiv, i = 1, . . . , n.

This proves that the systems of maps (f1, . . . , fn) and (T1, . . . , Tn) have a coincidence pointv in X. Since coordinatewise asymptotic commutativity of systems of maps (f1, . . . , fn)and (T1, . . . , Tn) implies their coordinatewise commutativity at a coincidence point v in X.Therefore

fi(T1v, . . . , Tnv) = Ti(f1v, . . . , fnv) = fi(f1v, . . . , fnv) = Ti(T1v, . . . , Tnv).

Again we may assume that∫ di(Tiv,Ti(T1v,...,Tnv))

0

φ(ξ)dξ ≤ ri, i = 1, . . . , n.

From (2.3), with Tv := (T1v, . . . , Tnv), we obtain∫ di(Tiv,TiTv)

0

φ(ξ)dξ ≤ max

n∑k=1

aik

∫ dk(fkv,fkTv)

0

φ(ξ)dξ, b

∫ Ni(v,Tv)

0

φ(ξ)dξ

, (2.24)

where

Ni(v, Tv) = max

di(fiv, Tiv), di(fiTv, TiTv),

di(fiv, TiTv) + di(fiTv, Tiv)

2

= max

di(Tiv, Tiv), di(TiTv, TiTv),

di(Tiv, TiTv) + di(TiTv, Tiv)

2

= di(Tiv, TiTv).

Substituting the value of Ni(v, Tv) in (2.24), we get∫ di(Tiv,TiTv)

0

φ(ξ)dξ ≤ max

n∑k=1

aik

∫ dk(Tkv,TkTv)

0

φ(ξ)dξ, b

∫ di(Tiv,TiTv)

0

φ(ξ)dξ

≤ max

n∑k=1

aikrk, bri

≤ maxhri, bri


Inductively ∫ di(Tiv,TiTv)

0

φ(ξ)dξ ≤ cmri.

Making m→∞,di(Tiv, TiTv) = 0,

implies , Tiv = Ti(T1v, . . . , Tnv). We also have Tiv = Ti(T1v, . . . , Tnv) = fi(T1v, . . . , Tnv).Hence Tiv is a solution of systems of equation (2.5).

Now suppose that the system (2.5) have two distinct solutions z and z in X such that

fiz = zi = Tiz and fiz = zi = Tiz, i = 1, . . . , n.

If zi 6= zi, i = 1, . . . , n, then we may assume that

75

∫ di(zi,zi)0

φ(ξ)dξ ≤ ri, i = 1, . . . , n.

From (2.3),∫ di(zi,zi)

0

φ(ξ)dξ = =

∫ di(Tiz,Tiz)

0

φ(ξ)dξ

≤ max

n∑k=1

aik∫ dk(fkz,fk z)

0φ(ξ)dξ, b

∫ Ni(z,z)0

φ(ξ)dξ

(2.25)

where

Ni(z, z) = max

di(fiz, Tiz), di(fiz, Tiz),

di(fiz, Tiz) + di(fiz, Tiz)

2

= di(fiz, fiz) = di(zi, zi).

Substituting this value in (2.25), we get∫ di(z,zi)

0

φ(ξ)dξ ≤ max

n∑k=1

aik∫ dk(zk,zk)

0φ(ξ)dξ, b

∫ di(zi,zi)0

φ(ξ)dξ

≤ max

n∑k=1

aikrk, bri

= maxhri, bri

= cri, where c = maxh, b.Inductively ∫ di(zi,zi)

0φ(ξ)dξ ≤ cmri.

Making m→∞, we get

di(zi, zi) = 0,

which gives zi = zi, i = 1, . . . , n. This completes the proof.

If we take φ(ξ) = 1 in the Theorem 2.1, we get following result as a special case ofTheorem 2.1.

Corollary 2.1. Let (Xi, di), i = 1, . . . , n, be complete metric spaces and Ti, fi : X → Xi, besuch that

Ti(X) ⊂ fi(X), i = 1, . . . , n.

The systems of maps (T1, . . . , Tn) and (f1, . . . , fn) are coordinatewise weakly reciprocallycontinuous and coordinatewise asymptotically commuting on X.

If there exist non-negative numbers b < 1 and aik, i, k = 1, . . . , n, such that (1.1), (1.2),(1.4) and the following hold:

di(Tix, Tiy) ≤ maxi

n∑k=1

aikdk(fkx, fky), bNi(x, y)

where Ni(x, y) = max

di(fix, Tix), di(fiy, Tiy), di(fix,Tiy)+di(fiy,Tix)

2

for all x, y ∈ X. Then

there exists unique point z ∈ X such that fiz = zi = Tiz, i = 1, . . . , n.

Remark 2.1. If we assume fix = xi, fiy = yi, i = 1, . . . , n and b = 0 in Corollary 2.1, weobtain result of Matkowski [17].

76

Corollary 2.2. Let T and f : Y → Y are weakly reciprocally continuous and asymptoticallycommuting (or compatible) self-maps in a complete metric space (Y, d) such that T (Y ) ⊂f(Y ) and satisfying ∫ d(Tx,Ty)

0

φ(ξ)dξ ≤ k

∫ N(x,y)

0

φ(ξ)dξ

where N(x, y) = maxd(fx, fy), d(fx, Tx), d(fy, Ty), d(fx,Ty)+d(fy,Tx)

2

for all x, y ∈ Y and

ξ : R+ → R+ is a Lebesgue-integrable mappings which is summable, non-negative and suchthat ∫ ε

0

φ(ξ)dξ > 0 for each ε > 0.

Then there exists unique point z ∈ Y such that fz = z = Tz.

Proof. Proof may be completed by putting (Y, d) = (Xi, di), T = Ti, f = fi, i = 1, . . . , n andn = 1, k = maxa11, b in the proof of Theorem 2.1.

Remark 2.2. The result of Rhoades [25] is obtained from Corollary 2.2 by taking f as anidentity map. Similarly we can obtain the result of Branciari [2] by assuming N(x, y) =d(fx, fy) and f as an identity map in Corollary 2.2.

Conflicts of Interest: Both the Authors declare no conflict of interest.Funding: This research received no external funding.Ethical approval: This article does not contain any studies with human participants oranimals performed by any of the authors.

Acknowledgement. Authors are thankful to the referee for his valuable suggestions.

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79

Jnanabha, Vol. 49(1) (2019), 80-88

GENERALIZED HERMITE POLYNOMIAL FAMILIESBy

Paolo Emilio RicciSection of Mathematics, International Telematic University UniNettuno

Corso Vittorio Emanuele II, 39, 00186 - Roma, ItaliaEmail:[email protected]

(Received : May 29, 2019)

Abstract

In recent papers, new sets of Sheffer and Brenke polynomials based on higherorder Bell numbers, and several integer sequences related to them have been studied.In this article we find families of Sheffer polynomials, including as a particularcase generalized Hermite polynomial families, deriving their recurrence relations anddifferential equations, which are sometimes of fractional type.2010 Mathematics Subject Classifications: 33C45, 11B83, 12E10, 34A08Keywords and phrases: Appell polynomials, Sheffer polynomials, Hermite-Kampede Feriet polynomials, Gould Hopper polynomials, Monomiality principle.

1 IntroductionIn recent articles [8, 18], new sets of Sheffer [21] and Brenke [7] polynomials based on higherorder Bell numbers [15, 16, 17, 18] have been studied. Furthermore, several integer sequencesassociated with the considered polynomials sets both of exponential and logarithmic typehave been introduced [8].In this article we find families of Sheffer polynomials, including as a particular casegeneralized Hermite polynomial families.The classical Hermite polynomials have been previously considered by Laplace and sub-sequently studied by Chebyshev and Hermite. V.A. Steklov proved their density in theweighted space L2

w(x)(−∞,+∞), with w(x) = e−x2.

It is worth to recall that these polynomials have been introduced by the beginning in themulti-dimensional case, and were deeply analyzed by Appell and Kampe de Feriet in classicalbook [1] and widely studied and applied by G. Dattoli and his collaborators (see e.g. [9, 11]).Many extensions of the Hermite polynomials have been proposed in literature by severalauthors, so that it seems quite impossible to list all of them. We just remember the mostimportant contributions, limiting ourselves to the one-dimensional case [12, 14, 6, 20, 23].

2 Sheffer and Appell polynomial familiesWe start recalling the particular meaning of the term set in the framework of polynomialtheory.

Definition 2.1. A polynomial family Pn(x)n≥0 is called a polynomial set

iff ∀n, degPn = n.

In what follows, we are dealing with polynomial families that, in several cases, do notsatisfy the above condition.

80

The Appell polynomials An(x) are introduced [21] by means of the exponentialgenerating function [24] of the type:

A(t) exp(xt) =∞∑n=0

An(x)tn

n!, (2.1)

where

A(t) =∞∑n=0

antn

n!, (a0 6= 0) . (2.2)

The Sheffer polynomials [21] sn(x) generalize the Appell family, by considering thegenerating function [24]:

A(t) exp(xH(t)) =∞∑n=0

sn(x)tn

n!, (2.3)

where

A(t) =∞∑n=0

antn

n!, (a0 6= 0) , H(t) =

∞∑n=0

hntn

n!, (h0 = 0) . (2.4)

Obviously, when H(t) ≡ t, the Sheffer polynomials give back the Appell sets.According to a different characterization (see [19, p. 18]), the same Sheffer sequence

sn(x) can be defined by means of the pair (g(t), f(t)), where g(t) is an invertible series andf(t) is a delta series:

g(t) =∞∑n=0

gntn

n!, (g0 6= 0) , f(t) =

∞∑n=0

fntn

n!, (f0 = 0, f1 6= 0) . (2.5)

Denoting by f−1(t) the compositional inverse of f(t) (i.e. such that f (f−1(t)) =f−1 (f(t)) = t), the exponential generating function of the sequence sn(x) is given by

1

g[f−1(t)]exp

(xf−1(t)

)=∞∑n=0

sn(x)tn

n!, (2.6)

so that

A(t) =1

g[f−1(t)], H(t) = f−1(t) . (2.7)

When g(t) ≡ 1, the Sheffer sequence corresponding to the pair (1, f(t)) is called theassociated Sheffer sequence σn(x) for f(t), and its exponential generating function is givenby

exp(xf−1(t)

)=∞∑n=0

σn(x)tn

n!. (2.8)

A list of known Appel and Sheffer polynomial sequences and their associated ones can befound in [5].

Remark 2.1. It is well known [7, 13] that there is a natural link between the function H(t)and the degree of polynomials sn(x) in expansion (2.3). Namely,

deg sn = n iff, in equation (2.4), h1 6= 0.Actually, in what follows, if H(t) is a polynomial of degree m, we have found that deg sn ≤

[ nm

], where [·] denotes the integral part.In general, we are dealing with a Sheffer polynomial set iff the condition h1 6= 0 is

satisfied.

81

3 An extensions of the Hermite polynomialsWe put, in this Section:

A(t) = exp[α (t+ γ)m] , H(t) = β tn , (3.1)

where α, β are real numbers and m,n positive integer numbers.We consider the generalized Hermite polynomial families Hk(α, β, γ,m, n;x), defined by

the generating function

G(t, x) = exp[α (t+ γ)m + x β tn] =∞∑k=0

Hk(α, β, γ,m, n;x)tk

k!. (3.2)

Remark 3.1. Note that the fourth parameter β is unessential, since it only produces achange of scale on the x-axis. Actually it has been introduced for recovering exactly theHermite polynomial set by putting α = −1, β = 2, γ = 0, m = 2, n = 1.

The above generating function (3.2) includes, as particular cases, several other familiesof classical polynomials generalizing the Hermite set.

We find:• the Gould Hopper polynomials (G(t, x) = exp[xt + htr]), assuming α = h, β = 1, γ = 0,m = r, n = 1;• the L.R. Bragg polynomials (G(t, x) = exp[pxt − tp]), assuming α = −1, β = p, γ = 0,m = p, n = 1;• the Lahiri polynomials (G(t, x) = exp[νxt − tm]), assuming α = −1, β = ν, γ = 0,m = m, n = 1;• the two variables higher order Hermite-Kampe de Feriet polynomials, (G(t, x) = exp[xt+ytm]), assuming α = y, β = 1, γ = 0, m = m, n = 1.

4 Properties of the polynomials Hk(α, β, γ,m, n;x)4.1 A differential identityTheorem 4.1. For any k ≥ 0, the polynomials Hk(α, β, γ,m, n;x) satisfy the differentialidentity:

H′k(α, β, γ,m, n;x) = β (k)nHk−n(α, β, γ,m, n;x) , (4.1)

where we have used the falling factorial symbol (k)n := k(k − 1) · · · (k − n+ 1).

Proof. Differentiating G(t, x) with respect to x , we have:

∂G

∂x= β tnG(t, x) = β

∞∑k=0

Hk(α, β, γ,m, n;x)tk+n

k!, (4.2)

and therefore:

∂G

∂x=∞∑k=0

H′k(α, β, γ,m, n;x)tk

k!= β

∞∑k=0

[k(k − 1) · · · (k − n+ 1)]Hk−n(α, β, γ,m, n;x)tk

k!,

so that the differential identity (4.1) is proved.

82

4.2 Recurrence relationTheorem 4.2. For any k ≥ 0, the polynomials Hk(α, β, γ,m, n;x) satisfy the recurrencerelation:

Hk+1(α, β, γ,m, n;x) = αmm−1∑h=0

(m− 1

h

)γm−h−1(k)hHk−h(α, β, γ,m, n;x) +

+ βn (k)n−1 xHk−n+1(α, β, γ,m, n;x) ,

(4.3)

where (k)h := k(k − 1) · · · (k − h+ 1) and (k)n−1 := k(k − 1) · · · (k − n+ 2).

Proof. Differentiating G(t, x) with respect to t , and putting, for shortness: Hk(x) :=Hk(α, β, γ,m, n;x), we find:

∂G

∂t=[αm (t+ γ)m−1 + βnx tn−1

]G(t, x) =

∞∑k=0

Hk+1(x)tk

k!, (4.4)

and therefore∞∑k=0

Hk+1(x)tk

k!= αm

∞∑k=0

Hk(x)tk

k!

m−1∑h=0

(m− 1

h

)γm−h−1th + βn

∞∑k=0

xHk(x)tk+n−1

k!,

i.e.∞∑k=0

Hk+1(x)tk

k!= αm

∞∑k=0

Hk(x)m−1∑h=0

(m− 1

h

)γm−h−1 t

k+h

k!+ βn

∞∑k=0

xHk(x)tk+n−1

k!,

∞∑k=0

Hk+1(x)tk

k!= αm

∞∑k=0

[m−1∑h=0

(m− 1

h

)γm−h−1(k)h

]Hk−h(x)

tk

k!+

+ βn∞∑k=0

x (k)n−1Hk−n+1(x)tk

k!,

so that the recurrence (4.3) follows.

4.3 Shift operatorsWe recall that a polynomial set pn(x) is called quasi-monomial if and only if there existtwo operators P and M such that

P (pk(x)) = kpk−1(x) , M (pk(x)) = pk+1(x) , (k = 1, 2, . . . ). (4.5)

P is called the derivative operator and M the multiplication operator, as they act in thesame way of classical operators on monomials.This definition, tracing back to a paper by J.F. Steffensen [25], has been recently improvedby G. Dattoli [9] and widely used in several applications (see e.g. [10, 11]).Y. Ben Cheikh [3] proved that every polynomial set is quasi-monomial under the action ofsuitable derivative and multiplication operators. In particular, in the same article (Corollary3.2), the following result is proved

Theorem 4.3. Let (pk(x)) denote a Boas-Buck polynomial set, i.e. a set defined by thegenerating function

A(t)ψ(xH(t)) =∞∑k=0

pk(x)tk

k!, (4.6)

83

where

A(t) =∞∑k=0

aktk , (a0 6= 0) , ψ(t) =

∞∑k=0

γktk , (γk 6= 0 ∀k) , (4.7)

with ψ(t) not a polynomial, and lastly

H(t) =∞∑k=0

hk tk+1 , (h0 6= 0) . (4.8)

Let σ ∈ Λ(−) the lowering operator defined by

σ(1) = 0 , σ(xk) =γk−1

γkxk−1 , (k = 1, 2, . . . ). (4.9)

Put

σ−1(xk) =γk+1

γkxk+1, (k = 0, 1, 2, . . . ). (4.10)

Denoting, as before, by f(t) the compositional inverse of H(t), the Boas-Buck polynomialset pk(x) is quasi-monomial under the action of the operators

P = f(σ) , M =A′[f(σ)]

A[f(σ)]+ xDxH

′[f(σ)]σ−1 , (4.11)

where prime denotes the ordinary derivatives with respect to t.

Remark 4.1. It is worth to note that the above mentioned result (Corollary 3.2 in [3]), givenfor polynomial sets, never uses in proof the condition h1 6= 0. Therefore, it can be appliedeven to polynomials defined by Sheffer generating functions (2.3), i.e. to Sheffer polynomialfamilies.

Note that in our case we are dealing with a Sheffer polynomial family, so that, since wehave ψ(t) = et, the operator σ defined by equation (4.9) simply reduces to the derivativeoperator Dx. Furthermore, we have:

A′(t) = A(t)αm (t+ γ)m−1 ,A′(t)

A(t)= αm (t+ γ)m−1 ,

H ′(t) = β n tn−1 , H−1(t) = f(t) = β−1/n t1/n ,so that we have the theorem

Theorem 4.4. The generalized Hermite polynomial family Hk(α, β, γ,m, n;x) is quasi-monomial under the action of the operators

P = β−1/nD1/nx , M = αm

(β−1/nD

1/nx + γ

)m−1

+ nxβ1/nD(n−1)/nx . (4.12)

so thatMP = αmβ−1/n

(β−1/nD1/n

x + γ)m−1

D1/nx + nxDx .

4.4 Differential equationAccording to the results of monomiality principle [9, 11], the quasi-monomial polynomialspn(x) satisfy the differential equation

MP pk(x) = k pk(x) . (4.13)In the present case, we have

Theorem 4.5. Putting, for shortness: Hk(x) := Hk(α, β, γ,m, n;x), the generalizedHermite polynomials Hk(x) satisfy the differential equation[

αmβ−1/n(β−1/nD1/n

x + γ)m−1

D1/nx + nxDx

]Hk(x) = kHk(x) , (4.14)

i.e. [αm

m−1∑h=0

(m− 1

h

)β−(h+1)/nγm−h−1D(h+1)/n

x + nxDx

]Hk(x) = kHk(x) . (4.15)

84

5 The particular case when γ = 0Since all the classical extension of the Hermite polynomials recalled in Remark 2 are obtainedby letting γ = 0, it is convenient to rewrite the preceding formulas in this particular case.

We put, in this Section:

A(t) = exp(α tm) , H(t) = βtn , (5.1)

where α, β are real numbers and m,n positive integer numbers.Then, putting Hk(α, β,m, n;x) := Hk(α, β, 0,m, n;x), we consider the generalized Hermitepolynomial families Hk(α, β,m, n;x), defined by the generating function

G(t, x) = exp[α tm + x βtn] =∞∑k=0

Hk(α, β,m, n;x)tk

k!. (5.2)

By using the same techniques of the preceding Section, we find the following results.

5.1 A differential identityTheorem 5.1. For any k ≥ 0, the polynomials Hk(α, β,m, n;x) satisfy the differentialidentity:

H′k(α, β,m, n;x) = β (k)nHk−n(α, β,m, n;x) , (5.3)

where we have used the falling factorial symbol (k)n := k(k − 1) · · · (k − n+ 1).

5.2 Recurrence relationTheorem 5.2. For any k ≥ 0, the polynomials Hk(α, β,m, n;x) satisfy the recurrencerelation:

Hk+1(α, β, γ,m, n;x) = αm (k)m−1Hk−m+1(α, β,m, n;x) + βn (k)n−1 xHk−n+1(α, β,m, n;x) ,(5.4)

where (k)m−1 := k(k − 1) · · · (k −m+ 2) and (k)n−1 := k(k − 1) · · · (k − n+ 2).

5.3 Shift operatorsTheorem 5.3. The generalized Hermite polynomial family Hk(α, β,m, n;x) is quasi-monomial under the action of the operators

P = β−1/nD1/nx , M = αmβ(1−m)/nD

(m−1)/nx + nxβ1/nD

(n−1)/nx . (5.5)

so that

MP = αmβ−m/nDm/nx + nxDx .

5.4 Differential equationTheorem 5.4. The generalized Hermite polynomials Hk(α, β,m, n;x) satisfy the differ-ential equation[

αmβ−m/nDm/nx + nxDx

]Hk(α, β,m, n;x) = kHk(α, β,m, n;x) . (5.6)

Remark 5.1. Note that the equation (5.6), when n = 1, is an ordinary differential equationof order m. The corresponding polynomials constitute a standard polynomial set (for anyindex k we find a polynomial of degree n = k).

When n = 1 and m = 2 we have a three term recurrence relation so that the polynomialsare orthogonal with respect to a suitable measure.

When n = 1 and m > 2 we find again a the three term recurrence relation but the indexesare delayed.

85

When n > 1 the differential equation is of fractional order, so that the polynomial degreen does not correspond to the index k.

For example, if n = 3 and m = 1 the polynomials are arranged in groups containing eachthree polynomials, and the polynomial degree n increases every three steps, according to therule k = 3n+ j, (j = 0, 1, 2), i.e. k − j ≡ n, (mod 3).

If (n,m) = 1 the degree of polynomials appear in different order, depending on the valuesof n and m.

This can be easily checked by using the computer algebra program Alpha c©.

5.5 First few values, in particular cases, when α = β = 1 and γ = 0As an example, assuming α = β = 1, and by using a simplified notation, we consider herethe polynomials H0(1, 1, 3, 1;x) =: H0(3, 1;x) (a Gould-Hopper case):

H0(3, 1;x) = 1H1(3, 1;x) = xH2(3, 1;x) = x2

H3(3, 1;x) = x3 + 6H4(3, 1;x) = x4 + 24xH5(3, 1;x) = x5 + 60x2

H6(3, 1;x) = x6 + 120x3 + 360H7(3, 1;x) = x7 + 210x4 + 2520xH8(3, 1;x) = x8 + 336x5 + 10080x2

and the polynomials H0(1, 1, 1, 3;x) =: H0(1, 3;x):

H0(1, 3;x) = 1H1(1, 3;x) = 1H2(1, 3;x) = 1H3(1, 3;x) = 6x+ 1H4(1, 3;x) = 24x+ 1H5(1, 3;x) = 60x+ 1H6(1, 3;x) = 360x2 + 120x+ 1H7(1, 3;x) = 2520x2 + 210x+ 1H8(1, 3;x) = 10080x2 + 336x+ 1.

Remark 5.2. It is worth to note that interchanging (m, 1) (the Gould-Hopper case)with (1,m) (the fractional derivative case), the corresponding polynomials can be easilyreconstructed, since the coefficients appear in reverse order and the degree obey the increasingorder rule observed in the preceding Remark 4. This is a general phenomenon, not onlyappearing when m = 3.

5.6 First few values in a particular case, when γ = 1

Putting, for shortness, Hk(x) := Hk(1, 1, 1, 2, 1;x), we consider here the polynomials definedby the generating function

G(t, x) = exp[(t+ 1)2 + x t] =∞∑k=0

Hk(x)tk

k!.

86

We find:

H0(x) = e

H1(x) = e (x+ 2)

H2(x) = e (x2 + 4x+ 6)

H3(x) = e (x3 + 6x2 + 18x+ 20)

H4(x) = e (x4 + 8x3 + 36x2 + 80x+ 76)

H5(x) = e (x5 + 10x4 + 60x3 + 200x2 + 380x+ 312) .

Remark 5.3. Note that the sequence 1, 2, 6, 20, 76, 312, . . . appears in the Encyclopediaof Integer Sequences [22] under # A000898, namely, the sequence defined by the recurrencerelation: a(n) = 2[a(n− 1) + (n− 1)a(n− 2)], a(0) = 1. (Formerly M1648 N0645). That isthe value of the n-th derivative of exp(t2) evaluated at t = 1. – N. Calkin, Apr. 22, 2010.

References[1] P. Appell and J. Kampe de Feriet, Fonctions Hypergeometriques et Hyperspheriques.

Polynomes d’Hermite, Gauthier-Villars, Paris, 1926.[2] E.T. Bell, Exponential polynomials, Annals of Mathematics, 35 (1934), 258-277.[3] Y. Ben Cheikh, Some results on quasi-monomiality, Appl. Math. Comput., 141 (2003),

63-76.[4] R.P. Boas and R.C. Buck, Polynomials defined by generating relations, Amer. Math.

Monthly, 63 (1958), 626-632.[5] R.P. Boas and R.C. Buck, Polynomial Expansions of Analytic Functions, Springer-

Verlag, Berlin, Gottingen, Heidelberg, New York, 1958.[6] L.R. Bragg, Products of certain generalized Hermite polynomials; associated relations,

Boll. Un. Mat. Ital. (4) 1, (1968), 347-355.[7] W.C. Brenke, On generating functions of polynomial systems, Amer. Math. Monthly,

52 (1945), 297–301.[8] G. Bretti, P. Natalini and P.E. Ricci, A new set of Sheffer-Bell polynomials and

logarithmic numbers, Georgian Math. J., 2018, (to appear).[9] G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: A by-product of the

monomiality principle, in Advanced Special Functions and Applications (Proceedingsof the Melfi School on Advanced Topics in Mathematics and Physics; Melfi, 9-12 May,1999) (D. Cocolicchio, G. Dattoli and H.M. Srivastava, Editors), Aracne Editrice,Rome, 2000, pp. 147–164.

[10] G. Dattoli, P.E. Ricci and H.M. Srivastava, Editors), Advanced Special Functions andRelated Topics in Probability and in Differential Equations, (Proceedings of the MelfiSchool on Advanced Topics in Mathematics and Physics; Melfi, June 24-29, 2001), in:Applied Mathematics and Computation, 141, (No. 1) (2003), 1-230.

[11] G. Dattoli, B. Germano, M.R. Martinelli and P.E. Ricci, Monomiality and partialdifferential equations, Math. Comput. Modelling, 50 (2009), 1332-1337.

[12] H.W. Gould and A.T. Hopper, Operational formulas connected with two generaliza-tions of Hermite Polynomials, Duke Math. J., 29 (1962), 51-62.

[13] W.N. Huff, The type of the polynomials generated by f(x t)φ(t), Duke Math. J., 14(4)(1947), 1091-1104.

87

[14] M. Lahiri, On a generalisation of Hermite polynomials, Proc. Amer. Math. Soc., 27(1971), 117-121.

[15] P. Natalini and P.E. Ricci, An extension of the Bell polynomials, Comput. Math. Appl.,47 (2004), 719-725.

[16] P. Natalini and P.E. Ricci, Higher order Bell polynomials and the relevant integersequences, Appl. Anal. Discrete Math., 11 (2017), 327-339.

[17] P. Natalini and P.E. Ricci, Remarks on Bell and higher order Bell polynomials andnumbers, Cogent Mathematics, 3 (2016), 1-15.

[18] P.E. Ricci, P. Natalini and G. Bretti, Sheffer and Brenke polynomials associated withgeneralized Bell numbers, Jnanabha, Vijnana Parishad of India, 47 (No2-2017), 337-352.

[19] S.M. Roman, The Umbral Calculus, Academic Press, New York, 1984.[20] B.B. Saha, On a generating function of generalized Hermite polynomials, C.R. Acad.

Bulgare Sci. 27, (1974), 889-892.[21] I.M. Sheffer, Some properties of polynomials sets of zero type, Duke Math. J., 5 (1939),

590-622.[22] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences. Published electroni-

cally at http://oeis.org, 2016.[23] H.M. Srivastava, A Note on a Generating Function for Generalized Hermite Polyno-

mials, Indag. Math. (Proceed.), 79, (1976), 457-461.[24] H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Halsted Press

(Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester,Brisbane and Toronto, 1984.

[25] J.F. Steffensen, The poweroid, an extension of the mathematical notion of power, ActaMath., 73 (1941), 333-366.

88

Jnanabha, Vol. 49(1) (2019), 89-96

A CLASS OF TWO VARIABLES SEQUENCE OF FUNCTIONSSATISFYING ABEL’S INTEGRAL EQUATION AND THE PHASE SHIFTS

ByHemant Kumar

Department of MathematicsD. A-V. Postgraduate College Kanpur - 208001, Uttar Pradesh, India


(Received : May 31, 2019 ; Revised: June 02, 2019)

Abstract

In this paper, we introduce a class of two variables sequence of functions as satisfyingAbel’s integral equation in which unknown function is the potential function and againconsider that the Riemann - Liouville fractional integral of this class of functions equalsto the slope of that potential function and then discuss some of its oscillatory propertiesand use them to evaluate the phase shifts in terms of arcsine of the series consisting ofthe Srivastava and Daoust’s triple hypergeometric function.2010 Mathematics Subject Classifications: 34 A08, 41A10, 34A45, 33C66, 33C90.Keywords and phrases: A class of two variables sequence of functions, generalsequences, phase shifts, Riemann - Liouville fractional integral, Kampe de Ferietfunction, Srivastava and Daoust function.

1 IntroductionIn our investigation, we introduce a class of two variables functions in the form

f(r;λ, q) =∞∑p=0

∞∑n=0

ap,n(λ, q)rp+2

p+ 2!, λ 6= 0, q > 0. (1.1)

The class (1.1) consists of a general sequences ap,n(λ, q),∀p ≥ 0, n ≥ 0, λ 6= 0, q > 0.

Set ap,n(λ, q) =∏mj=1 Γ(Aj+p+n)

∏µj=1 Γ(Cj+p)

∏νj=1 Γ(Dj+n)∏m

′j=1 Γ(Bj+p+n)

∏µ′j=1 Γ(C

′j+p)

∏ν′j=1 Γ(D

′j+n)

Γ(p+3)λp

Γ(p+1)qn

n!in Eqn. (1.1), then

f(r;λ, q) = r2Fm:µ;ν

m′ :µ′ ;ν′

[(A)1,m : (C)1,µ; (D)1,ν

(B)1,m′ : (C′)1,µ′ ; (D

′)1,ν′

∣∣∣∣λrq]

(1.2)

provided that for the convergence,(i) m+ µ < m

′+ µ

′+ 1,m+ ν < m

′+ ν

′+ 1; |λr| <∞, |q| <∞, or

(ii) m+ µ = m′+ µ

′+ 1,m+ ν = m

′+ ν

′+ 1; |λr| <∞, |q| <∞, and

(iii) |λr|

1(m−m′) + |q|

1(m−m′) < 1, ifm > m

′;

max|λr|, |q| < 1, ifm ≤ m′.

(1.3)

Here in Eqn. (1.2) - (1.3), the function Fm:µ;ν

m′ :µ′ ;ν′[λr, q] is a generalized Kampe de Feriet

function, a generalization of the Appell and Lauricella functions (see, Appell and Kampe deFeriet [2], Srivastava and Panda [22], Srivastava and Manocha [21]). Therefore, by the Eqns.(1.1), (1.2) and (1.3), several one and two variables hypergeometric functions may be foundand studied.

89

The phase shifts are useful in various physical problems, for example the photo - ionizationcross - sections and a reliable calculation of the impact ionization cross - section of an atomrequires an accurate determination of the continuum wave functions in the incident and inthe exit channels. In the theory of collisions, phase shifts determine the scattering cross -sections (Pain [12], Ikot et al. [6], Mahajan and Varma [10], Raghuwanshi and Sharma [14]).Many of workers have calculated and computed the phase shifts of scattering of electronson considering the solvable potential functions for example, the exponential functions (Teitz[24], Bhattacharjie and Sudarshan [3], Ikot et al. [6]), binomial functions (Agrawal andKumar [1], Kumar, Chandel and Agrawal [8]) and other special functions as Mittag - Lefflerfunctions (Kumar and Singh [9]), Coulomb potentials (Pain [12]).

For further extensions and developments of this theory of collision and phase shifts, weintroduce some more parameters in the potential function found by Abel’s integral equationand consider an angular momentum, where, εnL is the binding energy of the excited electronin nL orbital, L the quantum number of s− wave, then, for radius vector r and K2 = 2εnL,0 < α < 1, λ 6= 0, q > 0, 0 ≤ a ≤ r ≤ b, the asymptotic solution of the s - wave Schrodingerequation

d2

dr2UL(r) + [K2 − V α(r; 0, λ, q)− L(L+ 1)

r2]UL(r) = 0 (1.4)

has been represented as (see Pain [12])

UL(Kr) = CL

√πKr

2[cosηL(K)JL+ 1

2(Kr)− sinηL(K)JL− 1

2(Kr)]. (1.5)

In Eqns. (1.4) and (1.5), CL is a constant, V α(r; 0, λ, q) the potential function, with0 < α < 1, λ 6= 0, q > 0, 0 ≤ r <∞, and ηL(K) the phase shift for the quantum number L,Jν(z), ν > −1 the classical Bessel function (see Rainville [15]).

Again, by the Eqn. (1.5), the phase shift difference formula has been given in the form(see Tietz [24])

ηL(K)− ηL+1(K) = arcsin[π

2K

∫ ∞0

rd

drV α(r; 0, λ, q)JL+ 1

2(Kr)JL+ 3

2(Kr)dr]. (1.6)

To make application of the fractional calculus techniques ([5], [16]) in our present work,we claim following definitions:

The Riemann - Liouville fractional differential operator is defined by (see Diethelm [5, p.27])

Dαa f = DmJm−αa f,m− 1 < α ≤ m,∀α ∈ R, Dmf(x) =

dm

dxmf(x), (1.7)

where m ∈ N, f be a function such that analytic on the interval [a, b].Here in Eqn. (1.7), the Riemann - Liouville integral Jαa f is given by

Jαa f =1

Γ(α)

∫ x

a

(x− ξ)α−1f(ξ)dξ, a ≤ x ≤ b. (1.8)

We also make an application of multivariable representation of the Kampe de Ferietfunction (1.2) - (1.3) and evaluate the phase shifts in terms of arcsine of the series consistingof the Srivastava and Daoust’s triple hypergeometric function.

The multivariable representation of the Kampe de Feriet function (1.2) - (1.3) is themultivariable Srivastava and Daoust’s hypergeometric function ([11], [17] - [19], [20], [21])defined by

90

SA:B(′);...;B(k)

C:D(′);...;D(k)

([(a) : θ

′, . . . , θ(k)] : ((b

′) : ϕ

′); . . . ; ((b(k)) : ϕ(k))

[(c) : ψ′, . . . , ψ(k)] : ((d

′) : δ

′); . . . ; ((d(k)) : δ(k))

|z1, . . . , zk

)

=∞∑

m1,...,mk=0

∏Aj=1 Γ(aj +m1θ

′j + . . .+mkθ

(k)j )∏B(′)

j=1 Γ(b′j +m1ϕ

′j)∏B(k)

j=1 Γ(b(k)j +mkϕ

(k)j )∏C

j=1 Γ(cj +m1ψ′j + . . .+mkψ

(k)j )∏D(′)

j=1 Γ(d′j +m1δ

′j)∏D(k)

j=1 Γ(d(k)j +mkδ

(k)j )

× zm11

m1!. . .

zmkkmk!

provided that for the convergence conditions,

1 +C∑j=1

ψ(i)j +

D(i)∑j=1

δ(i)j −

A∑j=1

θ(i)j −

B(i)∑j=1

ϕ(i)j ≥ 0, |zi| <∞∀i = 1, . . . , k. (1.9)

In the next section 2, we consider the function (1.1) as a known function of an Abel’sintegral equation (2.1) and then, utilize its solution in obtaining of the phase shifts formula interms of arcsine of the series consisting of the Srivastava and Daoust’s triple hypergeometricfunction (1.9) (for k = 3).

2 The function satisfying Abel’s integral equation and the potential functionsTo proceed our work, we suppose that the function defined in Eqn. (1.1) satisfies the Abel’sintegral equation given by

f(r − a;λ, q) = Γ(1− α)

∫ r

a

(r − ξ)−αV α(ξ; a, λ, q)dξ,

where, 0 ≤ a ≤ r ≤ b, 0 < α < 1, λ 6= 0, q > 0. (2.1)

Then, the solution of (2.1) gives us a function

V α(r; a, λ, q) =1

Γ(α)

d

dr

∫ r

a

(r − ξ)α−1f(ξ − a;λ, q)dξ, ∀0 ≤ a ≤ r ≤ b, 0 < α < 1. (2.2)

Again then due to Eqns. (1.1) and (2.2), we find

V α(r; a, λ, q) =∞∑p=0

∞∑n=0

ap,n(λ, q)(r − a)p+α+1

Γ(p+ α + 2),∀0 < α < 1, λ 6= 0, q > 0, 0 ≤ a ≤ r ≤ b.

(2.3)

Particularly, put ap,n(λ, q) = (−1)nλp Γ(p+α+2)Γ(p+1)

(n+1)p

qn+1 ∀p ≥ 0, n ≥ 0, λ 6= 0 in (2.3), and onchanging the order of summation to find that

V α(r; a, λ, q) =∞∑n=0

∞∑p=0

(−1)nλp(r − a)α+1 (n+ 1)p

qn+1

(r − a)p

Γ(p+ 1)

= (r − a)α+1

∞∑n=1

(−1)n−1

qnenλ(r−a) = (r − a)α+1 eλ(r−a)

q + eλ(r−a). (2.4)

Again, in Eqn. (2.3), set ap,n(λ, q) = Γ(p+α+2)Γ(p+1)

λpq−n to get that

V α(r; a, λ, q) =q(r − a)α+1eλ(r−a)

q − 1,∀0 < α < 1, λ 6= 0, q > 1, 0 ≤ a ≤ r ≤ b. (2.5)

Hence, the function in Eqn. (2.2) may be a general potential function.

91

Remark 2.1. It is remarkable that the Eqns. (1.1) and (2.3) give us the relation

limα→1

V α(r; a, λ, q) = f(r − a;λ, q) (2.6)

Now from (2.2), we find

d

drV α(r; a, λ, q) =

d2

dr2Wα(r; a, λ, q),

where, Wα(r; a, λ, q)

=1

Γ(α)

∫ r

a

(r − ξ)α−1f(ξ − a;λ, q)dξ, 0 < α < 1, λ 6= 0, q > 0, 0 ≤ a ≤ r ≤ b. (2.7)

Here, the function Wα(r; a, λ, q) is defined as the Riemann - Liouville fractionalderivative, given in the Eqns. (1.7) - (1.8), of the function f(.).

Again, suppose that the slope of the function V α(r; a, λ, q) is negatively proportional tothe Wα(r; a, λ, q) that is

d

drV α(r; a, λ, q) ∝ −Wα(r; a, λ, q). (2.8)

Then, on application of the Eqns. (2.7) and (2.8), we find

d2

dr2Wα(r; a, λ, q) = −µ2Wα(r; a, λ, q), µ 6= 0, (2.9)

µ2 is the proportionality constant.Again, by the Eqn. (2.7) the function Wα(r; a, λ, q) satisfies the initial condition

Wα(a; a, λ, q) = 0. (2.10)

Now, we suppose that Wα(b; a, λ, q) = 0, b > a, then, the problem (2.9) along with thecondition (2.10) becomes oscillatory in the interval [a,b].

Theorem 2.1. If Wα(r; a, λ, q) = 1Γ(α)

∫ ra

(r − ξ)α−1f(ξ − a;λ, q)dξ, 0 < α < 1, λ 6= 0, q >

0, 0 ≤ a ≤ r ≤ b and ddrV α(r; a, λ, q) = −µ2Wα(r; a, λ, q), where, V α(r; a, λ, q) is found

due to the Abel’s integral equation (2.1), then, for the conditions Wα(a; a, λ, q) = 0 =Wα(b; a, λ, q), a and b are integral values, b > a, the eigen solutions of Eqn. (2.9)

are given by Wαn (r; a, λ, q) =

√2

(b−a)sinnπr, b > a, for the corresponding eigen values

µn = nπ∀n = 1, 2, . . . on [a, b] and the Green functionG(r, ξ) = 2

(b−a)

∑∞n=1

sinnπrsinnπξn2π2 , b > a. Also, the Eqn. (2.9) has infinite number of

solutionsWα(r; a, λ, q) = γsinnπr, γ is any constant ∀n = 1, 2, 3, . . . . (2.11)

Proof. Take Riemann - Liouville fractional derivative by Eqns. (1.7) and (1.8) of the function(1.1) and then make an appeal to the definition (2.7) to find

Wα(r; a, λ, q) =1

Γ(α)

∫ r

a

(r − ξ)α−1f(ξ − a;λ, q)dξ, 0 < α < 1, λ 6= 0, q > 0, 0 ≤ a ≤ r ≤ b.

(2.12)Now, make an appeal to the Eqn. (2.2) and then the Eqns. (2.7) - (2.10) and with the

help of (2.12) and the theory of integral equations we prove the theme of the Theorem 2.1.

92

3 Application to find out the phase shiftsIn this section, we make an appeal to the Eqn. (2.9) to get that linear in r in the form∫ r

0

d2

dr2Wα(r;0,λ,q)

Wα(r;0,λ,q)dr = −µ2r, whenever Wα(r; 0, λ, q) is never zero, now to make further devel-

opments in this theory here, to introduce µ2 = K2−L(L+ 1) and difference of the integrals

is taken as∫ r

0

d2

dr2WαL (r;0,λ,q)

WαL (r;0,λ,q)

dr −∫ r

0V α(r; 0, λ, q) + (1− r2)L(L+1)

r2dr = L(L+ 1)−K2r,

then prove following theorem:

Theorem 3.1. If WαL (r; 0, λ, q) 6= 0,∀ 0 < α < 1, λ 6= 0, q > 0, 0 ≤ r < ∞ and the integral

is given by∫ r

0

[d2

dr2WαL (r; 0, λ, q)− V α(r; 0, λ, q) + (1− r2)L(L+1)

r2Wα

L (r; 0, λ, q)

WαL (r; 0, λ, q)

]dr = L(L+1)−K2r.

(3.1)Then, there exists a real number ∀K > 0, 0 < α < 1, λ 6= 0, q > 0, 0 ≤ r < ∞, and it is

equivalent to the phase shifts by the formula

ηL(K)− ηL+1(K) = arcsin[π(K)2L+1

(2)2L+3Γ(L+ 32)Γ(L+ 5

2)

×∞∑p=0

∞∑n=0

ap,n(λ, q)

Γ(p+ α + 1)

∫ ∞0

rp+2L+α+31F2[

L+ 2;L+ 5

2, 2L+ 3;

−K2r2]dr]. (3.2)

Proof. Differentiate both sides of the Eqn. (3.1) with respect to r to find the differentialequation equivalent to the Eqn. (1.4) as

d2

dr2WαL (r; 0, λ, q) + K2 − V α(r; 0, λ, q)− L(L+ 1)

r2Wα

L (r; 0, λ, q) = 0. (3.3)

Consider the formulae (1.4) - (1.6) in the Eqn. (3.3) to get sinηL(K) − ηL+1(K) =[ π2K

∫∞0r ddrV α(r; 0, λ, q)JL+ 1

2(Kr)JL+ 3

2(Kr)dr], 0 < α < 1, and then on using the Eqn. (2.3),

to introduce hereddrV α(r; 0, λ, q) =

∑∞p=0

∑∞n=0 ap,n(λ, q) (r)p+α

Γ(p+α+1), ∀ 0 < α < 1, λ 6= 0, q > 0, 0 ≤ r <∞,

to find that

sinηL(K)− ηL+1(K) = [π

2K

∞∑p=0

∞∑n=0

ap,n(λ, q)

Γ(p+ α + 1)

∫ ∞0

rp+α+1JL+ 12(Kr)JL+ 3

2(Kr)dr].

(3.4)Now use the formula [15, p. 121]

Jn(z)Jm(z) =( z2

)n+m

Γ(n+1)Γ(m+1) 2F3[12(n+m+ 1), 1

2(n+m+ 2);

n+ 1,m+ 1, n+m+ 1;− z2] in right hand side of (3.4)

to get

sinηL(K)− ηL+1(K) = π(K)2L+1

(2)2L+3Γ(L+ 32

)Γ(L+ 52

)

×∞∑p=0

∞∑n=0

ap,n(λ, q)

Γ(p+ α + 1)

∫ ∞0

rp+2L+α+31F2[

L+ 2;L+ 5

2, 2L+ 3;

−K2r2]dr. (3.5)

The Eqn. (3.5) immediately gives the result (3.2).

93

Remark 3.1. In the result (3.2), in right side of the integrand, the generalized hypergeo-metric function 1F2[.] has in the denominator more parameters than its numerator, so thatit is an entire function (see Titchmarsh [25, p. 285, Ex. 6 ]) and thus due to Weierstrassfactorization theorem (see Titchmarsh [25, p. 247]) one of its factor may be in the formexp[−Kα+2rα+2] cosKr H(K) and then by Csordas and Varga [4], Pathan and Kumar [13],the integral has Polya class, and hence, due to this integral in the right hand side has somereal zeros for K > 0, r ∈ (0,∞) and thus the phase shift in Eqn. (3.2) has some real zeros(see also Kumar [7]). Hence, certainly all phase shifts coincide at a real point.

4 An application of Srivastava and Daoust’s triple hypergeometric function tocompute the phase shifts

In this section, we make an application of the Theorem 3.1 and obtain the phase shifts interms of the arcsine of the series containing the Srivastava and Daoust function (1.9).

Theorem 4.1. For the sequence ap,n(λ, q) = (−1)n+pλp (n+1)p

qn+1 , λ 6= 0, q > 0,∀p =0, 1, 2, . . . ;n = 1, 2, 3, . . . , there exists the phase shift difference formula

ηL(K)− ηL+1(K) = arcsin[√π

2(K)2L+1

(β)2L+α+4

∑∞n=1

(−1)n−1

qn

×S1:1;−;1−:1;−;2(

[2L+ α + 4 : 1, 1, 2] : (1 : 1); (− : −); (L+ 2 : 1)[− : −,−,−] : (1 + α : 1); (− : −); (L+ 5

2: 1), (2L+ 3 : 1)

|−nλβ, 1, −K

2

β2 )]

provided that

0 6= λ <γ1β

n,∀n = 1, 2, 3, . . . ; 0 < K <

√γ2β; β > 0, γ1 <∞, γ2 <∞, q > 0, 0 < α < 1.

(4.1)

Proof. Consider the sequence ap,n(λ, q) = (−1)n+pλp (n+1)p

qn+1 , λ 6= 0, q > 0, ∀p = 0, 1, 2, . . . ;n =

1, 2, 3, . . . in Eqn. (3.2), then, for 0 < α < 1, and on changing the order of summation to getthat in the form

sinηL(K)− ηL+1(K) =π(K)2L+1

(2)2L+3Γ(L+ 32)Γ(L+ 5

2)

×∞∑n=1

∞∑p=0

(−1)n−1(−λn)p

qnΓ(p+ α + 1)

∫ ∞0

rp+2L+α+31F2[

L+ 2;L+ 5

2, 2L+ 3;

−K2r2]dr. (4.2)

Now, from the Eqn. (4.2), for β > 0, we may write

sinηL(K)− ηL+1(K) =1

Γ(1 + α)

π(K)2L+1(2)−2L−3

Γ(L+ 32)Γ(L+ 5

2)

∞∑n=1

(−1)n−1(1

q)n

×∞∑p=0

∞∑m=0

(1)p(−λn)p

(1 + α)pp!

βm

m!

∫ ∞0

e−βrr2L+3+p+m+α1F2[

L+ 2;L+ 5

2, 2L+ 3;

−K2r2]dr. (4.3)

Then, on simplifying the Eqn. (4.3) and on using the definition (1.9) to find the result

sinηL(K)− ηL+1(K) =

√π

2

(K)2L+1

(β)2L+α+4

∞∑n=1

(−1)n−1

qn

× S1:1;−;1−:1;−;2(

[2L+ α + 4 : 1, 1, 2] : (1 : 1); (− : −); (L+ 2 : 1)[− : −,−,−] : (1 + α : 1); (− : −); (L+ 5

2: 1), (2L+ 3 : 1)

|−nλβ

, 1,−K2

β2),

(4.4)

94

provided that 0 6= λ < γ1βn, ∀n = 1, 2, 3, . . . ; 0 < K <

√γ2β; β > 0, γ1 < ∞, γ2 < ∞, q >

0, 0 < α < 1.The result (4.4) immediately gives us the formula (4.1).

Concluding RemarksThe result (4.1) on specializing of the parameters, under the given convergence conditions,

may help for computation of, a physical quantity, the phase shifts as using MATLABto analyse the photo - ionization cross - sections and a reliable calculation of the impactionization cross - section of an atom requires an accurate determination of the continuumwave functions in the incident and in the exit channels. In the integral operator (3.1), asclaiming the theory of Suffridge [23] on the univalent functions, the starlikeness, convexityand other geometric properties of holomorphic maps in one and higher dimensions may bestudied.

References[1] R.D. Agrawal and H. Kumar, The phase shift difference for binomial potential function,

Journal Maulana Azad College of Technology, 32 (1999), 67-75.[2] P. Appell and J. Kampe de Feriet, Fonctions Hypergeometriques et Hyperspheriques;

polynomes d’ Hermite, Gautier - Villars, Paris, 1926.[3] A. Bhattacharjie and E.C.G. Sudarshan, A class of solvable potentials, Nuovo Cim.,

25 (1962), 864 - 879.[4] G. Csordas and R.S. Varga, Integral transforms and the Laguerre Polya class, Complex

Variable, 12, (1989), 211 - 230.[5] Kai Diethelm, The Analysis of Fractional Differential Equations: An Application-

Oriented Exposition Using Differential Operators of Caputo Type, Springer, New York,2010.

[6] A.N. Ikot, H.P. Obong and T.M. Abbey, Approximate analytical solutions of the Klein- Gorden equation with an exponential type potential, New Physics: Sae Mulli, 65 (8)(Aug. 2015), 825 - 836. DOI:10.3938/NPSM.65.825.

[7] H. Kumar, Real zeros of generalized Mittag - Leffler functions of two variables throughintegral transforms, Jnanabha , 48 (2) (2018), 32 - 45.

[8] H. Kumar, R.C. Singh Chandel and R. D. Agrawal, Phase shift involving multiplehypergeometric function of Srivastava and Daoust, Jnanabha , 29 (1999), 117-122.

[9] H. Kumar and Vimal Pratap Singh, phase shifts of s - wave Schrodinger equation forMittag - Leffler function, Jnanabha , 41 (2011), 41-46.

[10] G.B. Mahajan and R.C. Varma, Determination of phase shifts for Rydberg potentialfunction, Indian Journal of Pure and Applied Physics, 13 (1975), 816 - 819.

[11] Branislav Martic, Some results involving a generalized Kampe de Feriet function,Publication de L’ Institut Mathematique Nouvelle serie, tome, 15 (29) (1973), 101- 104.

[12] J-C. Pain, Recursive determination of phase shifts for screened Coulomb po-tentials, IOP Publishing, J. Phys. Commun. 2 (2018) 025015, 1 - 11.https://doi.org/10.1088/2399 - 6528/aaa4e4.

[13] M.A. Pathan and H. Kumar, Generalized multivariable Cauchy residue theoremand non-zero zeros of multivariable and multi-parameters generalized Mittag-Leffler

95

functions, (Accepted in South Asian Bulletin of Mathematics (2018) (printed byYunnan Univ. [email protected]), 1 - 18.

[14] S.S. Raghuwanshi and L.K. Sharma, On phase shifts for a model potential, IndianJournal of Pure and Applied Physics, 17 (1979), 102 - 105.

[15] E.D. Rainville, Special Functions, The Macmillan Company, New York, 1960; Revisedby, Chalsea Publishing Co., New York, 1971.

[16] S.G. Samko, A.A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives,translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993.

[17] H.M. Srivastava and M.C. Daoust, On Eulerian integrals associated with Kampe deFeriet function, Publ. Inst. Math. (Beograd) (N. S.), 9 (23) (1969), 199 - 202.

[18] H.M. Srivastava and M.C. Daoust, Certain generalized Neumann expansions associatedKampe de Feriet function, Nederl. Akad. Wetensch. Proc. Ser. A, 72 = Indag. Math.31 (1969), 449 - 457.

[19] H.M. Srivastava and M.C. Daoust, A note on the convergence of Kampe de Feriet’sdouble hypergeometric series, Math. Nachr., 53 (1972), 151 - 159.

[20] H.M. Srivastava and P.W. Karlsson, Multiple Gaussian Hypergeometric Series, WileyHalsted, New York, 1985.

[21] H.M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, John Wileyand Sons, New York, 1984.

[22] H.M. Srivastava and R. Panda, An integral representation for the product of two Jacobipolynomials, J. London Math. Soc., 12 (2), 419 - 425.

[23] T.J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphicmaps in higher dimensions, Lecture Notes in Math., 599, 146-159, Springer-Verlag,NewYork, 1976.

[24] T. Tietz, A new method for finding the phase shifts for the Schrodinger equation, Acts,Phys. Hung., 16 (1963), 289-292.

[25] E.C. Titchmarsh, The Theory of Functions, Oxford University Press, Ely House,London W., New York, 1939.

96

Jnanabha, Vol. 49(1) (2019), 97-106

HYPERCYCLIC PAIR WHOSE COMPONENTS ARE DIRECT SUMS OFIDENTICAL HYPERCYCLIC OPERATORS

ByNeema Wilberth, Santosh Kumar and Marco Mpimbo

Department of MathematicsCollege of Natural and Applied Sciences, University of Dar es Salaam, Tanzania

Email:[email protected], [email protected] and

[email protected]

(Received : March 01, 2019 ; Revised: June 15, 2019)

Abstract

This paper provides an answer to the question raised by Feldman [15] which statesthat “If (T1, T2) is a hypercyclic pair, is (T1⊕T1, T2⊕T2) also a Hypercyclic Pair?” Someexamples are provided to support the results.2010 Mathematics Subject Classifications: 47A16, 47A15.Keywords and phrases: Hypercyclic pair, Hypercyclic criterion, Countably hyper-cyclic, Hypercyclic vector, Orbit, dense orbit.

1 Introduction and PreliminariesAn operator T acting on some separable completely metrizable space X is hypercyclic ifand only if for each pair of non-empty open sets (U, V ) in X, one can find n ∈ N such thatT n(U) ∩ V 6= ∅. In this case, there is a residual set of hypecyclic vectors.

The Hypercyclicity Criterion which is a sufficient condition for an operator to behypercyclic has a wide range of applications. Hypercyclic criterion has been used to showthat the direct sums of identical hypercyclic operators is hypercyclic pair.

In 1969, Rolewicz [20] gave the concepts of orbit and dense orbit which are as follows :

Definition 1.1. [20] Suppose that T is continuous linear operator on a topological vectorspace X over the field F. For an element x ∈ X, the Orbit of x under T is Orb(T ;x) =x, Tx, T 2x, . . . where x ∈ X is a fixed vector.

Definition 1.2. [18] A continuous linear operator T on topological vector space X is said tobe Hypercyclic if there is a vector x ∈ X whose orbit under T, Orb(T, x) = x, T 1x, T 2x, ...is dense in X. In this case x ∈ X is said to be Hypercyclic vector for T .

Definition 1.3. [16] An n-tuple of operators means a finite sequence of length n ofcommuting continuous linear operators.

For example, if (T1, T2, ..., Tn) are 2 × 2 matrices, then the sequence (T1, T2, ..., Tn) isn-tuple of 2× 2 matrices.

In 2009, Costakis et. al. introduced the concept of tuple and defined the term hypercyclicas follows:

Definition 1.4. [9] The tuple (T1, T2, . . . , Tn) is hypercylic if there exists a vector x ∈ Xsuch that the set T k11 , T k22 , . . . , T knn : k1, k2, . . . , kn ≥ 0 is dense in X.

97

Following is the definition of direct sum due to Tajmouati and El berrag [3].

Definition 1.5. [3] An operator T is said to be direct sum of subspaces U1, U2, ..., Uk andwe write T = (U1 ⊕ U2 ⊕ ... ⊕ Un) if for every u ∈ T there exist unique vectors ui ∈ Ti for1 ≤ i ≤ k such that t = u1 + u2 + ...+ uk.

The following concept of countably hypercyclic was defined by Feldman 2003.

Definition 1.6. [14] An Operator T is said to be countably hypercylic if there is a bounded

separate (xk)k≥1 such that∞⋃k=1

Orb(T, xk) is dense.

The following is the definition of adjoint operator given by Kreyzig 1978 as follows:

Definition 1.7. [4] Let A : X −→ Y be a bounded operator, then the adjoint operatorA∗ : Y

′ −→ X′

is given by (A∗g)(x) = g(Ax) for all g ∈ Y ′ where X′

and Y′

are the dualspace of X and Y , respectively.

The following propositions and corollary about Hypercyclicity criterion are due toFeldman [16] and will be useful in this paper.

Proposition 1.1. [16](Hypercyclicity criterion)Suppose that (T1, T2) is a pair of operators on separable Banach space Z. Suppose also thatthere exist two strictly increasing sequence of positive integers nj and kj, dense sets Xand Y in Z and function Sj : Y −→ Z such that

1. For each x ∈ X, T1njT2

kjx −→ 0 as j −→∞2. For each y ∈ Y , Sjy −→ 0 as j −→∞3. For each y ∈ Y , T1

njT2kJSjy −→ y as j −→∞ .

then (T1, T2) is a hypercyclic pair.

Corollary 1.1. [16] If a,b>1 are relatively prime integers, thenan

bk: n, k ∈ N

is dense in

R+.

Proposition 1.2. [16] If F is a set of operators on a separable Banach space X, then F ishypercyclic if for any open sets U, V there exists a T ∈ F such that T (U) ∩ V 6= ∅.

The following Theorem is due to Feldman [14] which shows the countably hypercyclicoperators.

Theorem 1.1. [14](The Countably Hypercyclic Criterion)Suppose that T ∈ B(X). If there exists two subspaces Y and Z in X, where Y is infinitedimensional and Z is dense in X such that:(i) T nx −→ 0 for every x ∈ Y , and(ii) there exists functions Bn : Z −→ X such that T nBn = 1

Zand Bnx −→ 0 for every

x ∈ Z, then T is countably hypercyclic.

98

Let T = (T1 ⊕ T1, T2 ⊕ T2) be commuting bounded linear operator on Banach space X.We define 2-tuple hypercyclic operators and establish a hypercyclicity pair that is similar tothe well known Hypercyclicity Criterion.

Let Γ = Tm1 ⊕ Tm1 , Tn2 ⊕ T n2 : m,n ≥ 0, for x ∈ X, the orbit of x under T is the set

Orb(T, x) = (Tm1 ⊕ Tm1 )(T n2 ⊕ T n2 )(x) : m,n ≥ 0.

The vector x is hypercyclic vector for T if the set Orb(T, x) is dense in X, that means

Orb(T, x) = (Tm1 ⊕ Tm1 )(T n2 ⊕ T n2 )(x) : m,n ≥ 0, i ∈ N = X.

Thus, in this paper, we will show that if T = (Tm1 ⊕ Tm1 , Tn2 ⊕ T n2 ) satisfies the

hypercyclicity criterion then it is a hypercyclic pair. The counter example is given inthe following way: we shall find some “universal objects ”for the Hypercyclicity Criterionconsisting of hypercyclic operators where the norm on the underlying space is in a specificoperator.

For more literature on this topic one can see the papers by Ansari [19], Ayadi [1],Bayart and Costakis [6], Bayart and Matheron [5], Bourdon and Feldman [17], Costakisand Hadjiloucas [10], Costakis and Peris [8], Feldman [15], Peris [2], Grosse-Erdmann [11],Javeheri [12] and Leon- Saavedra [7].

2 Main ResultsIn this section, we will extend the work due to Feldman [15,17] and Tajmouati and Berrag[3]. Feldman [17] proved the following proposition:

Proposition 2.1. [16] Let A and B be the hypercyclic operators and let C be an operatorwith dense range that commute with B. If we define

T1 = A⊕ C and T2 = I ⊕B

where I is identity operator, then (T1, T2) is a hypercyclic pair.

The extension of Proposition 2.1 from (T1, T2) to (T1 ⊕ T1, T2 ⊕ T2) is as follows:

Proposition 2.2. Let A and B be the hypercyclic operators and C be an operator with denserange which commute with B. Define

T1 ⊕ T1 = A⊕ C ⊕ A⊕ C and T2 = I ⊕B ⊕ I ⊕B

where I is identity operator. Then (T1 ⊕ T, T2 ⊕ T2) is a hypercyclic pair.

Proof. We consider the case when T = (T1 ⊕ T1, T2 ⊕ T2). Let x and y be the hypercyclicvectors for A and B respectively.

We claim that x⊕ y is hypercyclic vector for the pair (T1⊕ T1, T2⊕ T2). Let U × V be abasis open set. Notice that,

(T1 ⊕ T1)m(T2 ⊕ T2)n(x⊕ y) = ((T1 ⊕ T1)m(T2 ⊕ T2)nx)⊕ ((T1 ⊕ T1)m(T2 ⊕ T2)ny)

99

Also, Tm1 ⊕ Tm1 , Tn2 ⊕ T n2 = Am ⊕ CmBn.

Now since x is hypercyclic vector for A, then we can choose an m such that Amx ∈ U .Next, since C has dense range, then so does Cm, thus the inverse image (Cm)−1V is anon-empty open set, that is,

Cm−1V 6= 0.

Also since y is hypercyclic vector for B, then there is n such that

Bny ∈ Cm−1V where CmBky ∈ V.

Thus, (Tm1 ⊕ Tm1 ))(T n2 ⊕ T n2 )(x⊕ y) ∈ U × V .So,

(Tm1 ⊕ Tm1 , T

n2 ⊕ T n2

)= (T1 ⊕ T1, T2 ⊕ T2) is hypercyclic pair with hypercyclic vector

(x⊕ y) for n.m = 1.

The following examples support the above result of Proposition 2.2 to show that thedirect sum of hypercyclic operator is hypercyclic pair.

Example 2.1. If A is a hypercyclic operator and B is an operator which satisfies thehypercyclicty criterion and λn as n −→ ∞ is a bounded set of nonzero complex numbers,define the operators T1 and T2 as:

T1 = A⊕ λ1I ⊕ ...⊕ λnI ⊕ ... and T2 = I ⊕B⊕...⊕B ⊕ ...where, T1 ⊕ T1 = A⊕ λ1I ⊕ ...⊕ λnI ⊕ ...⊕ A⊕ λ1I ⊕ ...⊕ λnI ⊕ ...

and T2 ⊕ T2 = I ⊕B⊕...⊕B ⊕ ...⊕ I ⊕B⊕...⊕B ⊕ ...

such that I is identity operator.Then (T1 ⊕ T1, T2 ⊕ T2) is a hypercyclic pair.

Example 2.2.

1. If A is an operator such that both A and A∗ are hypercyclic operators and A has a realmatrix representation, and we let T1 = A⊕ I and T2 = A∗ ⊕ I, then (T1 ⊕ T1, T2 ⊕ T2)is a hypercyclic pair, but T1 ⊕ T1, T2 ⊕ T2 is not cyclic operator.

2. If T1 = A⊕ I ⊕A2 ⊕ I ⊕ ...⊕Ap ⊕ I and T1 = I ⊕A∗ ⊕ I ⊕A∗2 ⊕ ...⊕ I ⊕A∗p, then(T1 ⊕ T1, T2 ⊕ T2) is hypercyclic pair, but the set(Tm1 ⊕ Tm1 , T

n2 ⊕ T n2 ) : 0 ≤ m,n ≤ p does not contain any cyclic operators.

Next, we provide a proof of Corollary 2.1 which is an extension of the corollary due toFeldman [16] that shows that (T1 ⊕ T1, T2 ⊕ T2) is hypercyclic pair.

Corollary 2.1. If (T1, T2) satisfies hypercyclicity criterion, then

(T1 ⊕ T1, T2 ⊕ T2)

also satisfies the hypercylicity criterion, hence is a hypercyclic pair.

Proof. We know that

(Tm1 ⊕ Tm1 , Tn2 ⊕ T n2 ) = (T1 ⊕ T1, T2 ⊕ T2) for n,m = 1.

100

To see that (T1 ⊕ T1, T2 ⊕ T2) is hypercyclic pair, we use Proposition 1.1 [16] and let U,Vbe the two non-empty open subsets of (X, Y ) and choose x ∈ X ∩ U and y ∈ Y ∩ V , thenx + Sjy −→ x ∈ U as n −→ ∞. We choose a positive integer n so that x + Sjy ∈ U form ≥ n and j = 1, 2, ... Since T is a linear operator of 2-tuple, we have

(T1 ⊕ T1)mj(T2 ⊕ T2)nj(x+ Sjy) = (T1 ⊕ T1)mj(T2 ⊕ T2)nj(x) + (T1 ⊕ T1)mj(T2 ⊕ T2)nj(Sjy)

On simplification, one gets

(Tmj1 ⊕ T

mj1 )(T n+j

2 ⊕ T nj2 )(x+ Sjy) =(T

mj1 ⊕ T

mj1 )(T n+j

2 ⊕ T nj2 )x+ (Tmj1 ⊕ T

mj1 )(T n+j

2 ⊕ T nj2 )y −→ y.

Let zj = (x+ Sjy). Then, for large j we have zi ∈ U and

(Tmj1 ⊕ T

mj1 )(T n+j

2 ⊕ T nj2 )(zj) ∈ V .

It follows that (Tmj1 ⊕ T

mj1 )(T n+j

2 ⊕ T nj2 )(U) ∩ V is non-empty set, i.e

(Tmj1 ⊕ T

mj1 )(T n+j

2 ⊕ T nj2 )(U) ∩ V 6= 0 for every m ≥ n .

So (T1⊕ T1, T2⊕ T2) satisfies hypercycity criterion and hence (T1⊕ T1, T2⊕ T2) is the directsum of hypercyclic pair, hence is hypercyclic pair.

The following examples are very crucial to support Corollary 2.1 [16].

Example 2.3. Let T = (T1 ⊕ T1, T2 ⊕ T2) = (I1, eiθI1) where I1 is identity operator on C

and θ is an irrational multiple of π. Then T is hypercyclic on C but hypercyclic on C butalso T satisfies the hypercyclic criterion.

Proof. It follows from Corollary 1.1 [16] and the fact that einθ : n ∈ N is dense in C i.ez : |z| = 1 that T = (I1, e

iθI1) is hypercyclic on C but also satisfies the hypercyclicitycriterion, since

(T1 ⊕ T1, T2 ⊕ T2) = (I2, eiθI2)

is hypercyclic on C2 where I2 is identity operator on C2. Hence (T1⊕T1, T2⊕T2) is hypercyclicpair.

We saw that the above pair of T = (T1 ⊕ T1, T2 ⊕ T2) = (I2, eiθI2) is hypercyclic on C2.

It is easy to check that this pair satisfies the hypercyclic criterion with respect to any twosequences mj and nj where (mj − nj) −→∞, that is:

To show that T = (T1⊕T1, T2⊕T2) satisfies hypercyclicity criterion, let U and V be twonon-empty open sets in Z, then we choose x ∈ X ∩ U and y ∈ Y ∩ V and let zj = x + Sjy.Then, as j −→∞, zj −→ x, we get

(Tmj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )(zj) =

(Tmj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )x+ (T

mj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )Sjy −→ y.

Thus for large j and zj ∈ U , we have (Tmj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )(zj) ∈ V . Hence, by using

Propostion 1.2 [16] for (Tmj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )(U) ∩ V 6= ∅ this implies that T = (T1 ⊕

T1)(T2 ⊕ T2) is hypercyclic pair.

101

Example 2.4. Let T1 ⊕ T1 = 2I and T2 ⊕ T2 = 12B where B is the backward shift operator

on `2(N). Also, let S be the forward shift on `2(N). Then T = (T1⊕T1, T2⊕T2) satisfies thehypercyclcity criterion on `2(N), where `2(N) denoted as the set of square-integrable functionon natural number. Furthermore,

1. (Tmj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 ) = 2m−nBn

2. If X is the set of vectors with finite support, then for eachx ∈ X, (T

mj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )x −→ 0 as n −→∞

3. Sm,n = 2n−kSn is a right inverse for (Tmj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 ) on all of `2(N)

4. If x is a non-zero vector in `2(N), then Sm,nx −→ 0 if and only if

(m− n) −→∞.

Proof. Let mj and nj be any two strictly increasing sequences of positive integers suchthat (mj−nj) −→∞ and let X be the non-zero vector with finite support and let Y = `2(N).Then one can see that (T

mj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )(x) = 0 for all large j when x ∈ X and since

(m− n) −→∞, then Sjy = Smj ,njy −→ 0 as j −→∞.

Example 2.5. If A and B are hypercyclic operators and we let T1 ⊕ T1 = A⊕I and T2 ⊕ T2 =I ⊕ B, then T = (T1 ⊕ T1, T2 ⊕ T2) is hypercyclic pair but neither T1 ⊕ T1 nor T2 ⊕ T2 iscyclic.

Proof. Let x and y be the hypercyclic vectors for A and B respectively. We notice that(T

mj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )(x ⊕ y) = Amx⊕Bny. It follows that (x ⊕ y) is hypercyclic vector

for a pair T = (T1⊕T1, T2⊕T2). Since A and B are both hypercyclic then (T1⊕T1)(T2⊕T2) =(A⊕B) is cyclic.

The following theorem is due to Tajmouati and Ed berrag [3] to test the hypercyclicityof operaters.

Theorem 2.1. [3] Let T = (T1, T2, ..., Tn) be an n-tuple of commuting continuous linearoperators acting on an infinite dimensional separable Banach space X, d > 0, and let x ∈ Xsatisfying that for each y ∈ X there is the sequence mj

ij, for i = 1, ..., n, of natural numbers

with ||T1m1

jT2m2

j ...Tnmnjx− y|| < d. Then, T = (T1, T2, ..., Tn) is hypercyclic.

The extension of Theorem 2.1 to direct sum of operators is given below. The result isstated for the case of 2-tuple.

Theorem 2.2. Let T = (T1 ⊕ T1, T2 ⊕ T2) be an 2-tuple of commuting continuous linearoperators acting on an infinite dimensional separable Banach space X, d > 0, and let x ∈ Xsatisfying that for each y ∈ Y there is the sequence mj

ij, for i = 1, ..., n, of natural numberswith ||(Tmj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )(x)− y|| < d. Then, T = (T1 ⊕ T1, T2 ⊕ T2) is hypercyclic.

Proof. We take n = 2.Let U, V be non-empty open subsets of X. We need to show that

(Tmj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )U ∩ V 6= ∅,

for some mj, nj ∈ N for each j = 0, 1.... We choose u ∈ U and v ∈ V and ε > 0 such thaty ∈ X : ||y − u|| < ε ⊂ U and y ∈ X : ||y − v|| < ε ⊂ V .Let x′ = xε

4d. First we will show that the set

102

Orb(T, x′) = (T1 ⊕ T1)m

1j(T2 ⊕ T2)m

2j(x), j ∈ N

intersect each open ball with radius ε4. If y ∈ X, then there is an mj, nj ∈ N such that

||(Tmj1 ⊕ Tmj1 )(T

nj2 ⊕ T

nj2 )x− 4dy

ε|| < d. Therefore, ||(Tmj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )x

′ − y|| < ε4d.

So, ||(Tmj1 ⊕ Tmj1 )(T

nj2 ⊕ T

nj2 )x− 4dy

ε|| < ε

4.

Next, we will show that Orb(T, x′) intersects each open ball with radius ε in infinite set.

Suppose that on contrary there is y ∈ X such that the set

||(Tmj1 ⊕ Tmj1 )(T

nj2 ⊕ T

nj2 )x

′ − y|| < ε : mj, nj, j ∈ N is finite.

Since the ball B(y, ε2) cannot be covered by a finite number of balls of radii ε

4, hence there

is a point

y1 ∈ B(y, 2ε4

) =⇒ y1 ∈ B(y, ε2),

such that disty1, Orb(T, x′) ≥ ε

4. Thus Orb(T, x

′) ∩B(y1,

ε4) = ∅, contradiction. Hence

B(v, ε) ∩ (Tmj1 ⊕ Tmj1 )(T

nj2 ⊕ T

nj2 )x

′, j ∈ N.

Therefore, there exists mj, nj ∈ N satisfying mj′> mj and nj

′> nj such that

(Tmj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )x

′ ∈ B(u, ε) ⊂ U , (Tmj1 ⊕ T

mj1 )(T

nj2 ⊕ T

nj2 )x

′ ∈ B(v, ε) ⊂ V

and (T! ⊕ T1)mj′−mj(T2 ⊕ T2)nj

′−nj((T1 ⊕ T1)mj(T2 ⊕ T2)nj

)∈ V .

Hence, (T! ⊕ T1)mj′−mj(T2 ⊕ T2)nj

′−nj(U) ∩ V 6= ∅. Therefore, T = (T1 ⊕ T1, T2 ⊕ T2) ishypercyclic pair and it satisfies hypercyclicity criterion.

Notice that if an operator T = (T1 ⊕ T1, T2 ⊕ T2) has a set with dense orbit, then anynon-zero multiple vector of that set also has dense orbit. Thus T has a bounded set withdense orbit that is hypercylic, if and only if the unit ball has dense orbit and B(T, r) hasdense orbit for any r > 0.

The following proposition is due to Feldman [14] on countably hypercyclic criterion.

Proposition 2.3. [14] Suppose that T1 and T2 are bounded linear operators.(i) If T1 and T2 satisfy the Countably Hypercyclic Criterion, then T1 ⊕ T2 also satisfies theCountably Hypercyclic criterion.(ii) If T1 satisfies the Countably Hypercyclic Criterion and the spectrum of T2 is containedin (C\D), then T1 ⊕ T2 satisfies the Countably Hypercyclic criterion.

In particular, we extend the Proposition 2.3 [14] in direct sum of the same operators ifit satisfies the hypercylic criterion.

Proposition 2.4. Suppose that T1 ⊕ T1 and T2 ⊕ T2 are bounded linear operators.(i) If T1⊕T1 and T2⊕T2 satisfy the Countably Hypercyclic Criterion, then

(T1⊕T1

)⊕(T2⊕T2

)also satisfies the Countably Hypercyclic criterion.(ii) If T1 ⊕ T1 satisfies the Countably Hypercyclic Criterion and the spectrum of T2 ⊕ T2 iscontained in (C\0), then (T1 ⊕ T1, T2 ⊕ T1) satisfies the Hypercyclic criterion.

103

Proof. (i) Let Tj act on the space Xj. Since T = (T1⊕T1, T2⊕T2) satisfies the HypercyclicCriterion, there are subspaces Uj and Vj where Uj is infinite dimensional and Vj is dense inXi for every j natural number. Now, simply let U = U1 ⊕ (0), U = U2 ⊕ (0), V = V1 ⊕X2

and V = V2 ⊕X1. According to Proposition 1.2, we have

(T1 ⊕ T1)(T2 ⊕ T2)U ∩ V 6= ∅.

Then, one can easily check that the required properties hold. The proof of (ii) is similar.

In 2015, Nareen and Adem gave the theorem of direct sum of two bounded linear operatorsand subspace-hypercyclicity which is as follows:

Theorem 2.3. [13] Let M1 and M2 be closed subspaces of X and T1 ⊕ T2 is (M1 ⊕M2)-hypercyclic, then T1 and T2 are M1-hypercyclic and M2-hypercyclic, respectively.

We extend the Theorem 2.3 to direct sum of two bounded identical linear operators andshow that (T1 ⊕ T1, T2 ⊕ T2) is hypercyclic pair.

Theorem 2.4. Let N1 and N2 be closed subspaces of Banach space X and (T1⊕T1)⊕(T2⊕T2)is (N1 ⊕ N2)-hypercyclic, then T1 ⊕ T1 and T2 ⊕ T2 are hypercyclic pair and it satisfies N1-hypercyclic criterion and N2-hypercyclic criterion, respectively.

Proof. Let (x, y) ∈ HC((T1 ⊕ T1) ⊕ (T2 ⊕ T2), N1 ⊕ N2), then there exist an ε > 0 and onincreasing sequence of positive integers nk ∈ N, where we show T1 ⊕ T1 and T2 ⊕ T2 satisfiesN1-hypercyclic criterion and N2-hypercyclic criterion respectively.

There exist the two dense sets of the form C1 ⊕ C2 and C3 ⊕ C4 in N1 ⊕N2 such that

1. ((T1 ⊕ T1)⊕ (T2 ⊕ T2))nk(x′1, x′2) −→ (0, 0) ∀x′1, x′2 ∈ C1 ⊕ C2.

2. For each (y′1, y′2) ∈ C3 ⊕ C4, there exist a sequence

(xk, yk)k∈N ⊂ N1⊕N2 such that (xk, yk) −→ 0 and ((T1⊕T1)⊕(T2⊕T2))nk(x′1, x′2) −→

(y′1, y′2).

Since C1 and C3 are dense set in N1 and C2 and C4 are dense in N2, then it easy to showthat

1. (T1 ⊕ T1)nk(x′1) −→ 0, ∀x′1 ∈ C1 and (T2 ⊕ T2)nk(x′2) −→ 0, ∀x′2 ∈ C2

2. For each y′1 ∈ C3, there exist a sequence xkk∈N ⊂ N1 such that xk −→ 0 and(T1⊕ T1)nk(x′k) −→ y′1 and for each y′2 ∈ C4, there exist a sequence ykk∈N ⊂ N2 suchthat yk −→ 0 and (T2 ⊕ T2)nk(y′k) −→ y′2.

Finally, since

(T1 ⊕ T1)nkN1 ⊆ N1 and (T2 ⊕ T2)nkN2 ⊆ N2

then

((T1 ⊕ T1)⊕ (T2 ⊕ T2))nk(N1 ⊕N2) ⊆ (N1 ⊕N2).

104

It follows that ((T1 ⊕ T1) ⊕ (T2 ⊕ T2)) satisfies N1 ⊕ N2-hypercyclic criterion and thushypercyclic.

Now it easy to show that T1⊕T1 and T2⊕T2 are hypercyclic pair. But we know that N1

and N2 are closed subspace of X, So let p ∈ N1 and q ∈ N2 and we see that,

||(T1 ⊕ T1)⊕ (T2 ⊕ T2)nk(x, y)− (p, q)||N1⊕N2 ≤ ε : k ∈ N.

It follows that

||(T1 ⊕ T1)nk(x)− p||N1 ≤ ε : k ∈ N. and ||(T2 ⊕ T2)nk(y)− q||N2 ≤ ε : k ∈ N.

Therefore, there exist an increasing sequence of positive integer nkk∈N such that||(T1 ⊕ T1)nk(x) : k ≥ 1|| and ||(T2 ⊕ T2)nk(y) : k ≥ 1|| are dense in N1 and N2 respectively.

Also the Orb(T1 ⊕ T1, x) = (T1 ⊕ T1)nx : n ∈ N and Orb(T2 ⊕ T2, y) = (T2 ⊕ T2)ny :n ∈ N are dense in N1 and N2 respectively. Hence (T1 ⊕ T1, T2 ⊕ T2)are hypercyclic pair.

Corollary 2.2. If T1 ⊕ T1 and T2 ⊕ T2 satisfies subspace hypercyclic criterion, then (T1 ⊕T1, T2 ⊕ T2) is subspace- hypercyclic pair.

The following are examples to support the above results.

Example 2.6. If B is unilateral backward shift operators on `2(N), then let T = (T1 ⊕T1, T2 ⊕ T2) = (4I, 1

4B) is hypercyclic on `p(N). Since 4B is hypercyclic on `p(N) .

Example 2.7. Suppose that B is the backward unilateral shift on `2(N) and T1 = 2B. If T2

is any bounded linear operator on a separable Banach space such that σ(T2) ⊂ C\0, thenT1 ⊕ T1 and T2 ⊕ T2 is hypercylic.

Acknowledgement. The authors are very much thankful to the referee for his valuablesuggestions to bring the paper in its present form.

References[1] A. Ayadi, Hypercyclic abelian semigroup of matrices on Cn and Rn and k-transitivity,

J. Appl. General Topology, 12 (1) (2011), 35-39.[2] A. Peris, Multi-hypercyclic operators are hypercyclic, Mathematische Zeitschrift, 236

(4) (2001), 779-786.[3] A. Tajmouati and M. El Berrag, Some results on hypercyclicity of tuple of operators,

Italian Journal of Pure and Applied Mathematics, 35(1) (2015), 487-492.[4] E. Kreyszig, Introductory functional analysis with applications, New York: Wiley, (1)

(1978).[5] F. Bayart and E. Matheron, Dynamics of linear operators, Cambriage university press,

Vol. 179 (2009).[6] F. Bayart and G. Costakis Hypercyclic operators and rotated orbits with polynomial

phases, J. London. Math. Soc., 89 (3) (2014), 663-679.[7] F. Leon-Saavedra and V. Muller Rotations of hypercyclic and supercyclic operators,

Integral Equations and Operator Theory, 50(3) (2004), 385-391.[8] G. Costakis and A. Peris, Hypercyclic semigroups and somewhere dense orbits, Comptes

Rendus Mathematique, 335 (11) (2002), 895-898.

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[9] G. Costakis, D. Hadjiloucas and A. Manoussos, Dynamics of tuples of matrices, Proc.Amer. Math. Soc., 137 (3) (2009), 1025-1034.

[10] G. Costakis and D. Hadjiloucas, Somewhere dense Cesaro orbits and rotations of Cesarohypercyclic operators, Studia Mathematica, Vol. 175 (2006), 249-269.

[11] K. G. Grosse-Erdmann and A. P. Manguillot, Linear chaos, Springer Science &Business Media, (2011).

[12] M. Javaheri Semigroups of matrices with dense orbits, Dynamical Systems, 26(3)(2011), 235-243.

[13] N. Bamerni and A. Kilcman, On the direct sum of two bounded linear operators andsubspace-hypercyclicity. arXiv preprint arXiv:1501.02862(2015)

[14] N. S. Feldman, Countably hypercyclic operators, Journal of Operator Theory, 50(2003),107-117.

[15] N. S. Feldman, Hypercyclic pairs of coanalytic Toeplitz operators, Integral Equationsand Operator Theory, 58(2) (2007), 153-173.

[16] N. S. Feldman, Hypercyclic tuples of operators and somewhere dense orbits, J. Math.Anal. Appl., 346 (1) (2008), 82-98.

[17] P. S. Bourdon and N. S. Feldman, Somewhere dense orbits are everywhere dense, IndianaUniversity Math. J., 52 (3) (2003), 811-819.

[18] S. I. Ansari, Existence of hypercyclic operator on topological vector spaces, J. Funct.Anal., 148 (2) (1997), 384-390.

[19] S. I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal., 128 (2) (1995), 374-383.[20] S. Rolewicz, On orbits of elements, Studia Mathematica, 32 (1) (1969), 17-22.

106

Jnanabha, Vol. 49(1) (2019), 107-127

A BRIEF STUDY OF THERMODYNAMIC QUANTITIES OF BLACKHOLES IN RAINBOW GRAVITY

BySandip Dutta∗, Ritwick Banerjee∗∗ and Ritabrata Biswas∗

∗ Department of MathematicsThe University of Burdwan, Golapbag Academic Complex, City : Burdwan-713104,

Dist.:Purba Burdwan, State : West Bengal, India∗∗ Department of Mathematics

Indian Institute of Engineering Science and Technology, Shibpur, IndiaEmail:[email protected], [email protected] and

[email protected]

(Received : June 19, 2019 ; Revised: June 22, 2019)

Abstract

In this paper, we investigate the thermodynamic properties of black holes underthe influence of rainbow gravity. In the metric of Schwarzschild, Reissner-Nordstromand Reissner-Nordstrom-de Sitter black hole surrounded by quintessence, we consider arainbow function and derive the existence of remnant and critical masses of a black hole.Using the Hawking temperature relation we derive the heat capacity and the entropyof the rainbow gravity inspired black holes and closely study the relation betweenentropy and area of the horizon for different values of n of the rainbow function. Theentropy relation is expressed in terms of the area of the event horizon of the blackhole and from the area-entropy graphs we make several meaningful interpretationsand closely analyse the effects of the different values of the rainbow parameters onthermodynamic properties. It is interesting to observe that the introduction of rainbowgravity keeps the overall physical interpretations similar to the Einstein gravity but itgives a quantum correction to the latter at the Planck scale and gives us new andinteresting insights into the thermodynamic journey of a BH.2010 Mathematics Subject Classifications: 83C75, 83D05, 83E99.Keywords and phrases: Black hole physics, Rainbow gravity, Thermodynamics

1 IntroductionThe discussion about the Lorentz symmetry at Planck scale leads us to many possibleanswers. Keeping the central physical message of theory of relativity unchanged, namely theequivalence of all inertial observers, we can propose double/deformed special relativity(DSR).Two postulates of relativity in that case can be formulated as : the equivalence of all inertialobservers are taken and secondly, assumption of two observer independent scales : one speedof light c and the other is the dimension of mass k (or length λ = k−1), identified with thePlanck mass. In the limit k →∞, DSR becomes special relativity [4, 5, 33, 10]. Now in sucha quantum phenomenological area near the Planck scale, the standard energy momentumdispersion relations are modified. Magueijo and Smolin [39, 40] have extended the DSR togeneral relativity. Their proposition was the energy of a test particle with the backgroundgeometry and consequently the modified dispersion relation is :

E2f

(E

Ep

)2

− p2g

(E

Ep

)2

= m2, (1.1)

107

here p, m and Ep are the momentum, the mass of the test particle and the Planck energy.So different background geometry will be observed by differently energised quantas. This iswhy we call it Rainbow gravity. Literature is enriched by works related to gravity at thePlanck scale [35, 18, 19, 29, 36, 37, 28, 38, 20, 21, 22, 23, 24, 41, 6, 8, 9, 43, 11, 7]. Thenature of rainbow functions have been discussed in many existing literatures. [1, 2, 25, 30,42, 31, 26, 17, 14, 13, 34, 15]. Popular forms are :Case 1.

f

(E

Ep

)= 1 , g

(E

Ep

)=

√1− η

(E

Ep

)n, (1.2)

here n is a positive integer and η is a constant of order unity. Both the functions becomeunity while E → 0.Case 2.

f

(E

Ep

)=

1

1− γ EEp

, g

(E

Ep

)= 1, (1.3)

where γ is the rainbow parameter [15].The Rainbow gravity inspired BHs give us a deeper insight of the fate of BH evaporation.

When the heat capacity vanishes we can say that the BH evaporation stops and it gives usthe remnant mass of the BH. Also it can be observed that the thermodynamic outcomesfor the Einstein and Rainbow gravity BHs are more or less similar. From this we can saythat the laws of physics are equivalent for both the cases [34]. Recently, Gim & Kim [15]have shown that Schwarzschild BH in Rainbow gravity in an isothermal cavity additionalHawking page phase transition near the event horizon apart from the standard one givingrise to the idea of existence of local BH.

It can be observed that the modification of metric by certain popular forms of rainbowfunctions, changes the thermodynamical behaviour of the different black holes. Themodification changes the temperature and the entropy of the systems and it brings forwardthe ideas of critical mass and the remnant mass of the black holes from the thermodynamicpoint of view. Thus the rainbow function in a way prevents the complete evaporation ofa black hole leaving behind a remnant mass which is exactly the same way as done by thegeneralized uncertainty principle (GUP). For the chosen rainbow function the entropy of thesystem have a lot of similarity to those derived using the GUP.

The main motivations for studying black holes under rainbow gravity are as follows. Dueto the high energy levels of black holes, it is important to study the properties of blackholes after considering the quantum corrections on the classical perspectives. The idea ofenergy dependent spacetime is one of those quantum corrections. Considering this energydependent spacetime, we can venture on the effects it brings about on the thermodynamicproperties of the black hole. One of the noticeable effect is the existence of remnant massof a black hole which can be proposed to be a major candidate for solving the informationparadox [27] and also being UV completion of Einstein gravity [40]. Also, we come acrossa critical mass which shows a second order phase transition making the stable black holeunstable. To have a better picture about the thermodynamic properties of the black holesit is essential to consider the high energy regimes. Here we consider the rainbow gravity tostudy its effects on the thermodynamic properties of the black holes.

108

In this paper, we have organised our studies as follows. In section 2 we first talk aboutthe basics of rainbow gravity and the way it modifies the metric of a Schwarzschild blackhole and the other thermodynamic constraints. Then we go on to find the critical mass,the remnant mass and finally the entropy of the system under the influence of rainbowgravity. Meanwhile in the process of deriving these meaningful constraints we try to gaindeeper insights into behaviour of the different thermodynamic constraints by plotting thetemperature vs. mass, heat capacity vs. mass and entropy vs. area of horizon of rainbowgravity inspired Schwarzschild black hole. We plot these graphs for different values of η andn which gives us a beter scope of comparison. Then we move on to section 3 and 4 where wediscuss the same results as before for Reissner Nordstrom and Reissner Nordstrom de Sitterblack hole surrounded by quintessence. Finaly, we conclude in section 5.

2 Thermodynamics of Rainbow Gravity Inspired Schwarzschild Black HoleIn this section we will discuss different thermodynamic properties of Schwarzschild BH takinginto account the effect of rainbow gravity functions. The Schwarzschild black hole metricinspired by rainbow gravity is given by (1.2)

ds2 = − 1

f 2( EEp

)

(1− 2MG

r

)dt2 +

1

g2( EEp

)

(1− 2MG

r

)−1

dr2 +r2

g2( EEp

)dΩ2. (2.1)

We relate surface gravity κ to the Hawking temperature by the relation T = κ2π

and

the surface gravity is defined by κ = limr→Rs

√−1

4grrgtt(gtt,r)2 where Rs = 2GM is the

Schwarzschild radius.From (2.1), we get gtt = −f 2( E

Ep)(1− 2MG

r

)−1, grr = g2( E

Ep)(1− 2MG

r

), (gtt,r)

2 =1

f4( EEp

)

(2MGr2

)2. Hence, the surface gravity of the Schwarzschild black hole under the effect of

rainbow gravity is given by

κ =g( E

Ep)

f( EEp

)

1

4MG. (2.2)

Case 1.

Therefore, the Hawking temperature is given by

T =1

8πG

√1

M2− η

(2GEp)n1

Mn+2. (2.3)

In the above expression we have set E = 12GM

[17]. Equation (2.3) gives us a relationbetween the Temperature and Mass of rainbow gravity inspired Schwarzschild BH. Figure 1arepresents the curve of Hawking temperature of Schwarzschild black hole in Einstein gravityvs the mass of the same. It shows that for low mass the temperature is high and as weincrease the mass, temperature graph reduces and becomes asymptotic to the M axis. Nowintroduction to rainbow gravity keeps the general trend almost same. It has been representedby figure 1b. But one thing to be noted is that if we count n to be one then for low mass,temperature is comparatively lower than the case of n = 2, 3. Now dM

T= dS. So if for the

same mass we have lower T the entropy is greater, that indicates n = 1 represents higherentropy system. So n = 1 carries higher disturbances than n = 2, 3 case.

109

Fig.1a Fig.1b

Fig.1a and 1b represent the Temperature vs Mass of the black hole curves for Schwarzschildblack holes in Einstein and Rainbow gravity respectively.

For the temperature to be a real quantity we must have1

M2− η

(2GEp)n1

Mn+2≥ 0. (2.4)

The above condition gives rise to a critical mass (Mcr). Below the critical mass, thetemperature is not a real quantity. This critical mass is given by :

Mcr =η

1n

2GEp= η

1nMp. (2.5)

where we have taken Ep = 12GMp

. From (2.3) we get

dTdM

=

((n+2)η

(2GEp)n1

Mn+3−2M3

)16πG

√1M2−

η(2GEp)n

1Mn+2

.

The heat capacity of the rainbow gravity inspired Schwarzschild BH is give by

C =dM

dT=

16πG√

1M2 − η

(2GEp)n1

Mn+2((n+2)η

(2GEp)n1

Mn+3 − 2M3

) . (2.6)

It is to be followed that where we do expect to have the critical mass, C vanishes there.The denominator vanishes at a point :

Mcr2 =(n+2

2

) 1n η

1n

2GEp

=(1 + n

2

) 1n η

1nMp > η

1nMp = Mcr.

At Mcr2 we have a second order phase transition which makes the stable black holeunstable. Figure 2 depicts this incident. If we increase n, Mcr2 decreases. i.e., the phasetransitions occur faster. If we increase or decrease η the curve completely shifts on the rightor left hand side respectively. But for high n such shifting is less.

If we set C = 0, we obtain the remnant mass Mrem(where the black hole stopsevaporating). This gives

Mrem =η

1n

2GEp= η

1nMp. (2.7)

110

Fig.2

Fig.2 represents C vs M curves for n = 1, 2, 3, 4. For η < 1 the graphs shift to the left sideand for η > 1 the graphs shift to the right side keeping their basic tendency same.

Thus we primarily can think that the remnant mass of the black hole is equal to itscritical mass. But actually Mpη

1n is such a point where the black hole starts its journey.

Before that no physical black hole is present. This is why at Mpη1n we get C = 0. Now we

can observe that if we take the rainbow gravity parameter η = 1 we get, Mrem = Mp, whereMp is the Planck mass. From the mass-temperature graph we saw that the temperature wasincreasing as the mass was decreasing, but when the mass of a BH reaches the remnant massi.e., the Planck mass, the heat capacity vanishes and the temperature suddenly becomeszero. So, we can say that at Planck scale the BH evaporation stops and prevents the BHfrom total evaporation. Hence, the rainbow gravity may solve the information loss and nakedsingularity problems of black holes. Now increase of η causes increase of Mcr as well as Mcr2 .But if n is high η

1n tends to one showing no big effect of η′s increment.

The entropy can be calculated by using the heat capacity of this black hole as follows :

S =

∫CdT

T=

∫dM

dT. (2.8)

Substituting equation (2.6) in equation (2.8) and carrying out a binomial expansionkeeping terms upto O(η4) and assuming that n ≥ 3 leads to : S = 8πG

∫dM√

1M2−

η(2GEp)n

1Mn+2

= 8πG∫ [

M + η2(2GEp)n

1Mn−1 + 3η2

8(2GEp)2n1

M2n−1 + 5η3

16(2GEp)3n1

M3n−1 + 35η4

128(2GEp)4n1

M4n−1

]dM

= 8πG[M2

2+ ηM2−n

2(2GEp)n(2−n)+ 3η2M2−2n

8(2GEp)2n(2−2n)+ 5η3M2−3n

16(2GEp)3n(2−3n)+ 35η4M2−4n

128(2GEp)4n(2−4n)

]= SBH(GM2

p )+S1−n2BH π

n2 η(GM2−n

p )

(2−n)(GEp)n+

3S1−nBH πnη2(GM2−2n

p )

8(1−n)(GEp)2n+

5S1− 3n

2BH π

3n2 η3(GM2−3n

p )

8(2−3n)(GEp)3n+

35S1−2nBH π2nη4(GM2−4n

p )

128(1−2n)(GEp)4n

= SBH(GM2p ) +

S1−n2BH π

n2 η(2nGM2

p )

(2−n)+

3S1−nBH πnη2(22nGM2

p )

8(1−n)+

5S1− 3n

2BH π

3n2 η3(23nGM2

p )

8(2−3n)+

35S1−2nBH π2nη4(24nGM2

p )

128(1−2n),

(2.9)where SBH = 4πM2

M2p

is the semi-classical Bekenstein-Hawking entropy for the Schwarzschild

black hole. The reason for assuming n ≥ 3 is that the result of the integration is not valid

111

for n = 1, 2 which can be easily seen from the integrand.In terms of the area of the horizon A = 4πR2

s = 16πG2M2 = 4l2pSBH , the above expressionfor the entropy can be put in the form :

S =(A4

)(GM2

p ) +(A4 )

1−n2 πn2 η(2nGM2

p )

(2−n)+

3(A4 )1−n

πnη2(22nGM2p )

8(1−n)+

5(A4 )1− 3n

2 π3n2 η3(23nGM2

p )

8(2−3n)

+35(A4

)1−2nπ2nη4(24nGM2

p )

128(1− 2n), (2.10)

where we have set lp = 1.We plot S for η = 1 and different values of n ≥ 3 vs A in Fig.3a and Fig.3b. We may

consider S ≥ 0 part only to be the physical black hole solutions. It shows as we increase n,black hole can even exist for lower area of event horizon. For lower n-s S increases directlywith A. Higher n breaks the curve into two parts - lower A steeply increasing S and aftercertain point increasing but with a low slope. This shows when black hole is small a changein A causes rapid change in S. But later this rapidness decreases.

Fig.3a Fig.3b

Fig.3a and 3b represents entropy(S) vs area of the horizon of the black hole curves forη = 1 and η = 0.5 respectively

For n = 1 the entropy is

S =

∫dM√

1M2 − η

2GEpM3

=1

32E2pG

2

[2EpGM

2

√−2η + 4EpGM

EpGM3(3ηEpGM)

+ 3η2 log

−η + 2EpGM

(2 +M

√−2η + 4EpGM

EpGM3

)]. (2.11)

For n = 2,

S =

∫dM√

1M2 − η

(2GEp)2M4

112

=8E3

pG3M3 − 2EpηGM + η

√−η + 4E2

pG2M2 log

2EpG

(2epGM +

√−η + 4E2

pG2M2

)8E3

pG3M2

√4M2 − η

E2pG

2M2

,

(2.12)

dSdA∼ dS

dM= 1

T. From this we can say that if η increases then dS

dAincreases and hence

temperature of the system decreases. So here η is somehow representative of the lesstemperature.Case 2.

From equation (2.2) we get the Hawking temparature,

T =1

8πMG

(1− γ

2EpMG

). (2.13)

We have set E = 12GM

in the above equation. The critical mass in this case is given byMcr = γ

2EpG. Differentiating the Hawking temparature we get,

dT

dM=

1

8πGM2

(γ

GEpM− 1

). (2.14)

Then the heat capacity is,

C =8πG2M3Epγ −GEpM

. (2.15)

The entropy of this black hole is

S =

∫dM

T= 16G2π

[γM

4E2pG

2+

M2

4EpG+γ2 log (−γ + 2EpGM)

8E3pG

3

]. (2.16)

3 Thermodynamics of Rainbow Gravity Inspired ReissnerNordstrom BlackHole

The Reissner-Nordstrom black hole metric under the effect of rainbow gravity is given by

ds2 =1

f 2(EEp

) [1− 2M

r+Q2

r2

]dt2 +

1

g2(EEp

) [1− 2M

r+Q2

r2

]−1

dr2 +r2

g2(EEp

)dΩ2. (3.1)

The surface gravity follows from (2.2) as

κ =g(EEp

)f(EEp

) ( M

R2N

− Q2

R3N

), (3.2)

where RN = M +√M2 −Q2 is the radius of the event horizon of the RN BH under rainbow

gravity. (We must keep in mind that there is another horizon for RN metric given byRN,inner = M −

√M2 −Q2.

But we take the outer most one.)Case 1.

The Hawking temperature is given by

T =1

2π

√1− η

(Ep)n1

RnN

(M

R2N

− Q2

R3N

). (3.3)

For getting the above relation we have put E = 1RN

.

113

Fig.4a Fig.4b Fig.4c

Fig 4a. represents T vs M and Q in Einstein gravity.

Fig 4b. represents T vs M and Q in Rainbow gravity with η = 0.8.Fig 4c. represents T vs M and Q in Rainbow gravity with η = 1.

Every thermodynamic quantity of RN BH primarily consists of two variables M and Q.For Einstein gravity if our BH comprises of low charge then temperature decreases withincreasing mass. But as we increase |Q| the temperature curve is broken into two phases,first increasing to a particular local maxima and then decreasing. We can speculate thateven if in Einstein gravity a phase transition may occur for highly charged RN BH. But ifwe look for Rainbow gravity effect on temperature (Fig.4c) we can see that, whatever be thevalue of |Q|, we will always get two distinct phases : firstly increasing and then decreasingafter attaining a local maxima. When charge is high the local maxima is also higher.

Since the temperature must be real quantity, so we must have

1− η

(Ep)n1

RnN

≥ 0. (3.4)

Hence the critical mass Mcr of the RN BH under the effect of rainbow gravity is given by

Mcr =1

2

Q2(ηEnp

) 1n

+

(η

Enp

) 1n

. (3.5)

From (3.3) we get

dTdM

=(EpRN )−n[Q2(6(EpRN )n−η(n+6))+R2

N (η(n+2)−2(EpRN )n)]4πR2

N(R2N−Q2)

√1−η(EpRN )−n

.

The heat capacity C of this BH is given by

C =dM

dT=

4πR2N(Q2 −R2

N)(EpRN)n√

1− η(EpRN)−n

2 (R2N − 3Q2) (EpRN)n + η [(n+ 6)Q2 − (n+ 2)R2

N ]. (3.6)

As before by putting C = 0 we get the remnant mass Mrem given by

Mrem =1

2

Q2(ηEnp

) 1n

+

(η

Enp

) 1n

. (3.7)

This is found to be the same as the critical mass. But as before this is actually thestarting mass. The first law of the charged BH is dE = TdS + ΦdQ, so we can calculate theentropy from the relation dS = dE−ΦdQ

T, where we consider the electric potential ΦdQ very

small. The entropy of this BH is now computed keeping terms upto O(η4) and assuming

114

Fig.5

Fig. 5 represents C vs M and Q for n = 1. This shows that there must be a phasetransition turning the BH from stable to unstable phase. Larger the |Q|, larger the M

where the transition to be occurred.

that n ≥ 3 leads to:

S = 2π∫

dM√1− η

(Ep)n1RnN

(M

R2N

− Q2

R3N

) = 2π∫ [

RN + ηEnp

1Rn−1N

+ 3η2

8E2np

1R2n−1N

+ 5η3

16E3np

1R3n−1N

+ 35η4

128E4np

1R4n−1N

]dRN

= 2π[R2N

2+ η

(2−n)EnpRn−2N

+ 3η2

8(2−2n)E2np R2n−2

N

+ 5η3

16(2−3n)E3np R3n−2

N

+ 35η4

128(2−4n)E4np R4n−2

N

]= SBH +

2πn2 η

(2− n)EnpS

n2−1

BH

+3πnη2

8(1− n)E2np S

n−1BH

+5π

3n2 η3

8(2− 3n)E3np S

3n2−1

BH

+35π2nη4

128(1− 2n)E4np S

2n−1BH

,

(3.8)where SBH = πR2

N is the semi-classical Bekenstein-Hawking entropy of the RN BH under

the rainbow gravity. => S =(A4

)+ 2π

n2 η

(2−n)Enp (A4 )n2−1 + 3πnη2

8(1−n)E2np (A4 )

n−1 + 5π3n2 η3

8(2−3n)E3np (A4 )

3n2 −1

+35π2nη4

128(1− 2n)E4np

(A4

)2n−1 . (3.9)

For n = 1 we have :

S = 2π

∫ [RN +

η

Ep+

3η2

8E2p

1

RN

+5η3

16E3p

1

R2N

+35η4

128E4p

1

R3N

]dRN

= SBH +2η√π√SBH

Ep+

3η2π ln(SBHπ

)8E2

p

− 5η3π32

8E3p

√SBH

− 35η4π2

128E4pSBH

=> S =

(A

4

)+

2η√π√(

A4

)Ep

+

3η2π ln

((A4 )π

)8E2

p

− 5η3π32

8E3p

√(A4

) − 35η4π2

128E4p

(A4

) . (3.10)

115

Fig.6

Fig.6 depicts the S vs A curve. S is increasing with A. If we increase n, the curve is shifteddownwards, i.e., more the ‘n’ less the entropy.

For n = 2 we have :

S = 2π

∫ [RN +

η

E2p

1

RN

+3η2

8E4p

1

R3N

+5η3

16E6p

1

R5N

+35η4

128E8p

1

R7N

]dRN .

= SBH +ηπ ln

(SBHπ

)E2p

− 3η2π2

8E4pSBH

− 5η3π3

32E6pS

2BH

− 35η4π4

384E8pS

3BH

=> S =

(A

4

)+

ηπ ln

((A4 )π

)E2p

− 3η2π2

8E4p

(A4

) − 5η3π3

32E6p

(A4

)2 −35η4π4

384E8p

(A4

)3 . (3.11)

Fig.7a Fig.7b

Figures 7a and 7b both show that S increases with A.

116

Case 2. In this case the Hawking temparature becomes,

T =1

2π

(1− γ

RNEp

)(M

R2N

− Q2

R3N

). (3.12)

Hence the critical mass Mcr of RN BH in this case is given by

Mcr =Q2

RN

=Q2Epγ

. (3.13)

The entropy of the BH corresponding to the Hawking temparature is

S = π[2γRN + 2M2 + 2M√M2 −Q2 + γ2 log

(−γ2 + 2γM −Q2

)+ γ2 log

γ(γ2 −Q2

) (γ2 − 2γM +Q2

)]

+ π[γ2 log(2RN)− γ2 log

16γQ2 + 8Q2(RN − 2M)

− γ2RN

]. (3.14)

4 Thermodynamics of Rainbow Gravity Inspired Reissner Nordstrom de SitterBlack Hole Surrounded by Quintessence

In this section we want to study the thermodynamic properties of Reissner Nordstrom deSitter black hole. The metric of RN-de Sitter inspired by rainbow gravity is given by :

ds2 =1

f 2(EEp

) [1− rgr−∑n

(rnr

)3wq+1]dt2− 1

g2(EEp

) dr2[1− rg

r−∑

n

(rnr

)3wq+1]− r2

g2(EEp

)dΩ2

(4.1)where rg = 2M , M is the mass of the black hole, rn − s are the dimensional normalisationconstants and wq are the quintessential state parameters.

The work of Kiselev has provided a particular solution for the Reissner-Nordstrom-deSitter black hole surrounded by quintessence as :

gQdStt = gtt =1

f 2(EEp

) [1− rgr

+Q2

r2− r2

a2−(rqr

)3wq+1],

where a is de-sitter curvature [32]. This solution in its more particular case turns tomeaningful limits with no charge (Q = 0). The metric have three horizons : the BH Cauchyhorizon at r = r−, the BH event horizon at r =r + and the cosmological horizon at r = rc.The mathematical expressions for the horizons are given by [44],

r− = −√Z1 +

√Z2 +

√Z3,

r+ =√Z1 −

√Z2 +

√Z3,

rc =√Z1 +

√Z2 −

√Z3,

where √Z1 =

1√6

[1 +

√1− 12Q2

a2cos

α

3

] 12

,

√Z2 =

1√6

[1−

√1− 12Q2

a2cos(α

3+π

3

)] 12

,

117

√Z3 =

1√6

[1−

√1− 12Q2

a2cos(α

3− π

3

)] 12

and

α = arccos

−1− 54M2

a2+ 36Q2

a2(1− Q2

a2

) 32

.Here a is invariant and he event horizon and the cosmological horizon are two independent

thermodynamic system. Here we consider the BH event horizon to study the thermodynamic

quantities. The surface gravity relation is defined by : k = limr→Rs

√−1

4grrgtt(gtt,r)2 where

Rs = 2GM is the Schwarzschild radius.Using the surface gravity relation we get :

k =g( E

Ep)

f( EEp

)

1

4MG

[1

G− 8M2G2

a2+ (3wq + 1)

( rq2G

)3wq+1 1

M3wq+1

]. (4.2)

Using (2) we can obtain :Case 1.

T =1

8πG

√1

M2− η

(2GEp)n1

Mn+2

[1

G− 8M2G2

a2+ (3wq + 1)

( rq2G

)3wq+1 1

M3wq+1

]. (4.3)

In the above expression we have set E = 12GM

.

Fig.8

Fig.8 represents T vs M and wq graph.

This equation gives us a relation between the temperature and the mass. Since thetemperature has to be a real quantity, we obtain the following condition :

1

M2− η

(2GEp)n1

Mn+2≥ 0. (4.4)

118

The above condition readily leads to the existence of a critical mass (Mcr) below whichthe temperature becomes a complex quantity. This critical mass is given by :

Mcr =η

1n

2GEp= η

1nMp. (4.5)

where we have taken Ep = 12GMp

. From (3.4) we get :

dT

dM=

((n+2)η

(2GEp)n1

Mn+3 − 2M3

)16πG

√1M2 − η

(2GEp)n1

Mn+2

[1

G− 8M2G2

a2+ (3wq + 1)

( rq2G

)3wq+1 1

M3wq+1

]

+1

8πG

√1

M2− η

(2GEp)n1

Mn+2

[−16MG2

a2− (3wq + 1)2

( rq2G

)3wq+1 1

M3wq+2

].

The heat capacity reads :

C =dM

dT= 16πG

√1

M2− η

(2GEp)n1

Mn+2

[((n+ 2)η

(2GEp)n1

Mn+3− 2

M3

)×

1

G− 8M2G2

a2+ (3wq + 1)

( rq2G

)3wq+1 1

M3wq+1

+2

(1

M2− η

(2GEp)n1

Mn+2

)−16MG2

a2− (3wq + 1)2

( rq2G

)3wq+1 1

M3wq+2

]−1

The remnant mass Mrem(where the black hole stops evaporating) can be obtained bysetting C = 0. This yields :

Mrem =η

1n

2GEp= η

1nMp. (4.6)

Thus we can see that the remnant mass of the black hole is equal to its critical mass.The entropy can be calculated by using the heat capacity of this black hole given by relation(2.9). Substituting equation (3.4) in equation (2.9) we get :

S =

∫8πGdM√

1M2 − η

(2GEp)n1

Mn+2

[1G− 8M2G2

a2+ (3wq + 1)

( rq2G

)3wq+1 1M3wq+1

] . (4.7)

We take values of wq = −13,−2

3and −1 and carrying out a binomial expansion keeping

terms upto O(η4) leads to three different cases respectively.Case 1.1:

wq = −1

3S = 8πa2G2

∫ [M + η

2(2GEp)n1

Mn−1 + 3η2

8(2GEp)2n1

M2n−1 + 5η3

16(2GEp)3n1

M3n−1 + 35η4

128(2GEp)4n1

M4n−1

]dM

[a2 − 8M2G3].

Different cases for values of n are given in the appendix. (In Case 1.1(a) and 1.1(b))Case 1.2:

wq = −2

3S = 8πG2a2rq

∫M +

ηMnp

2Mn−1 +3η2M2n

p

8M2n−1 +5η3M3n

p

16M3n−1 +35η4M4n

p

128M4n−1

a2rq − 8G3M2rq − 2a2G2M.

Different cases for values of n are given in the appendix. (In Case 1.2(a) and 1.2(b))

119

Fig.9a Fig.9b

Fig.9a and 9b shows that for n = 1 the BH starts its journey faster than for the n = 2 case.Initially for n = 1 S increases slowly with increase in A but then the rate of increase keepsincreasing. In both the cases, S increases as A increase.

Case 1.3:

wq = −1S = 8πG2a2r2q

∫M +

ηMnp

2Mn−1 +3η2M2n

p

8M2n−1 +5η3M3n

p

16M3n−1 +35η4M4n

p

128M4n−1

a2r2q − 8M2G3r2

q − 8M2G3a2dM.

Different cases for values of n are given in the appendix. (In Case 1.3(a) and 1.3(b))Case 2:The Hawking temparature is given by

T =1

8πMG

(1− γ

2GMEp

)[1

G− 8M2G2

a2+ (3wq + 1)

( rq2G

)3wq+1 1

M3wq+1

]. (4.8)

The entropy corresponding to the Hawking temperature is,

S =

∫dM

T=

∫8πMGdM(

1− γ2GMEp

) [1G− 8M2G2

a2+ (3wq + 1)

( rq2G

)3wq+1 1M3wq+1

] . (4.9)

This integration can not be generalised so this is written in integral form.

5 ConclusionsIn this paper we studied the thermodynamic effects on a black hole under the influence ofrainbow gravity. Firstly we considered the rainbow surface gravity and derived the rainbowHawking temperature, and subsequently we obtained the other thermodynamic quantitiessuch as heat capacity and entropy. In the process we came across the existence of remnantmass and critical mass and also derived the critical points of the black hole thermodynamicensemble and studied its stability.

We observed that the temperature of the black hole in rainbow gravity depends on theenergy E of a probe provided by the rainbow functions. We derived the remnant mass of theblack hole and showed that the Hawking temperature becomes zero at Planck scale. Thisimplies that the divergent nature of the standard Hawking temperature is regularized bythe rainbow gravity and it also prevents the complete evaporation of the black hole which

120

Fig.10a Fig.10b

Fig.10a and 10b depicts the S vs A curves for n = 1, 2 respectively. For n = 1 case the BHstarts its journey late than the n = 2 case. In both the cases the S vs A curve is almostsimilar. We get a feasible graph only for values of rq < 0.

can be treated as a probable candidate for solving the information loss paradox and nakedsingularity problems of black holes.

We derive the entropy of the black holes using the rainbow Hawking temperature andthus the entropy expressions consists of quantum corrections. Then the entropy relation isexpressed in terms of the area of the event horizon of the black hole and from the area-entropy graphs we make several meaningful interpretations and closely analyse the effects ofthe different values of the rainbow parameters on these thermodynamic properties.

It can be stated from entropy vs. area of horizon graphs that the presence of the rainbowparameter η in the rainbow function affects the temperature of the system as increasedvalues of η decreases the temperature. Hence it can be said that the rainbow parameter ηis somehow a representative of less temperature. In case of Reissner Nordstrom de Sitterblack hole surrounded by quintessence for wq = −2

3and −1 it can be observed that, for non-

negative values of rq there can be no feasible graphical representations of S vs. A graphs.

Hence for physically meaningful interpretations the quantity( rqr

)3wq+1must be a negative

quantity in solution of the RNDS BH surrounded by the quintessence. It is interesting toobserve that the introduction of rainbow gravity keeps the overall physical interpretationssimilar to the Einstein gravity but it gives a quantum correction to the latter at the Planckscale and gives us new and interesting insights into the thermodynamic journey of a BH.AppendixCase 1.1(a) : n = 1

S = −[8πa2G2 ln(a2−8G3M2)

16G3

]+

8πa2G2η tanh−1

(2√2G

32M

a

)4√

2a(2GEp)G32

+

[8πa2G2

3η2 ln(

M2

a2−8G3M2

)16a2(2GEp)2

]

121

Fig.11a Fig.11b

Fig.11a and 11b depicts the S vs A curve for n = 1 and n = 2 respectively. The graphs arevery similar to Fig.10a and 10.2. For the n = 1 case the BH starts its journey before thanthe n = 2 case. We get a feasible graph only for values of rq < 0.

−

8πa2G25η3

[a−2√

2G32Mtanh−1

(2√2G

32M

a

)]16a3(2GEp)3M

− [8πa2G235η4

[a2+8G3M2ln

(a2−8G3M2

M2

)]256a4(2GEp)4M2

]=

5πη3G2M3p

32

(8

a2−8G3M2 − 7ηMp

M2

)+ πηMpa

√2G tanh−1

(2√

2G32M

a

)+ 1

4a2ln(

M2

a2−8G3M2

) (πη2G2M2

p

) (6a2 + 35η2G3M2

p + 5ηMp

)=

5πη3G2M3p

32

(8π

πa2−2G3M2pSBH

− 28ηπMpSBH

)+ πηMpa

√2G tanh−1

(√2SBHG

32Mp

a√π

)+ 1

4a2ln(

M2pSBH

4πa2−8G3M2pSBH

) (πη2G2M2

p

) (6a2 + 35η2G3M2

p + 5ηMp

)(5.1)

=5πη3G2M3

p

32

(8π

πa2−2G3M2p(A4 )

− 28ηπ

Mp(A4 )

)+ πηMpa

√2G tanh−1

(√(A2 )G

32Mp

a√π

)+ 1

4a2ln

(M2p(A4 )

4πa2−8G3M2p(A4 )

)(πη2G2M2

p

) (6a2 + 35η2G3M2

p + 5ηMp

). (5.2)

Case 1.1(b) : n = 2

S = −πa2

2Gln(a2 − 8G3M2) + 2η G2M2

pπ ln(

M2

a2−8G3M2

)+

(12πη2G5M4p)

a2ln(

M2

a2−8G3M2

)−3πη2G2M4

p

2M2 − 10πη3G5M6p

a2M2 +(80πη3G8M6

p)a4

ln(

M2

a2−8G3M2

)− 5πη3G2M6

p

8M4

= −35πM8pη

4G2

96M6 − 5πM6pη

3G2(a2+7M2pηG

3)8a2M4 − πM4

pη2G2(3a4+20a2M2

pηG3+140M4

pη2G6)

2a4M2

+ln(M)(4πM2

pηG2)(a6+6a4M2

pηG3+40a2M4

pη2G6+280M6

pη3G9)

a6

−π(a8+4a6M2pηG

3+24a4M4pη

2G6+160a2M6pη

3G9+1120M8pη

4G12)ln(a2−8G3M2)2a6G

122

= −70π4M2pη

4G2

3S3BH

− 10π3M2pη

3G2(a2+7M2pηG

3)a2S2

BH− 2π2M2

pη2G2(3a4+20a2M2

pηG3+140M4

pη2G6)

a4S2BH

+ln

(√SBHMp

2√π

)(4πM2

pηG2)(a6+6a4M2

pηG3+40a2M4

pη2G6+280M6

pη3G9)

a6

−π(a8+4a6M2

pηG3+24a4M4

pη2G6+160a2M6

pη3G9+1120M8

pη4G12)ln

(a2π−2G3M2

pSBHπ

)2a6G

= −70π4M2pη

4G2

3(A4 )3 − 10π3M2

pη3G2(a2+7M2

pηG3)

a2(A4 )2 − 2π2M2

pη2G2(3a4+20a2M2

pηG3+140M4

pη2G6)

a4(A4 )2

+ln

(√AMp4√π

)(4πM2

pηG2)(a6+6a4M2

pηG3+40a2M4

pη2G6+280M6

pη3G9)

a6

−π(a8 + 4a6M2

pηG3 + 24a4M4

pη2G6 + 160a2M6

pη3G9 + 1120M8

pη4G12

)ln

(a2π−G3M2

p(A2 )π

)2a6G

.

(5.3)

Case 1.2(a) : n = 1

S = −35πη4G2M4p

32M2 − πG2(140a2η4G2M4p rq+80a2η3M3

p r2q)

32a2r2qM+

πη2G2M2p log(M)(a2(35η2G4M2

p+20ηG2Mprq+12r2q)+70η2G3M2p r

2q)

4a2r2q

−π log(a2(2G2M−rq)+8G3M2rq)(4a4r2q+a2η2G3M2p(35η2G4M2


2q)

8a2r2qG+

π tanh−1

(√G(a2+8GMrq)a√

a2G+8r2q

)(−4a5r2q+a3ηGMp(35η3G6M3

p+20η2G4M2p rq+12ηG2Mpr2q+16r3q)+10aη3G4M3

p r2q(21ηG2Mp+8rq))

4a2√Gr2q√a2G+8r2q

= −35π2η4G2M2p

8SBH− 5π

32 η3G2M2

p [7ηG2Mp+4rq]4rq√SBH

+πη2G2M2

p log

(√SBHMp

2√π

)(a2(35η2G4M2


2q)

4a2r2q

−π log

(2a2G2

(√SBHMp

2√π

)−a2rq+8G3

(SBHMp

4π

)rq

)(4a4r2q+a2η2G3M2

p(35η2G4M2p+20ηG2Mprq+12r2q)+70η4G6M4

p r2q)

8a2r2qG

+

π tanh−1

a2√G+8G

32 rq

(√SBHMp2√π

)a√

a2G+8r2q

[−4a5r2q+a3ηGMp(35η3G6M3p+20η2G4M2

p rq+12ηG2Mpr2q+16r3q)+10aη3G4M3p r

2q(21ηG2Mp+8rq)]

4a2√Gr2q√a2G+8r2q

= −35π2η4G2M2p

8(A4 )− 5π

32 η3G2M2

p [7ηG2Mp+4rq]4rq

√(A4 )

+

πη2G2M2p log

√

(A4 )Mp2√π

(a2(35η2G4M2p+20ηG2Mprq+12r2q)+70η2G3M2

p r2q)

4a2r2q

−π log

2a2G2

√

(A4 )Mp2√π

−a2rq+8G3

((A4 )Mp

4π

)rq

(4a4r2q+a2η2G3M2p(35η2G4M2


2q)

8a2r2qG

+π tanh−1

a2√G+8G

32 rq

√

(A4 )Mp2√π

a√a2G+8r2q

[−4a5r2q + a3ηGMp

(35η3G6M3

p + 20η2G4M2p rq + 12ηG2Mpr

2q + 16r3

q

)

123

+10aη3G4M3p r

2q (21ηG2Mp + 8rq)

] [4a2√Gr2

q

√a2G+ 8r2

q

]−1

. (5.4)

Case 1.2(b) : n = 2Considering terms upto O(η2) we get :

S = −3πη2G2M4p

2M2 − 6πη2G4M4p

Mrq−

π tanh−1

(√G(a2+8GMrq)a√

a2G+8r2q

)(a4r2q−4a2ηG3M2

p(3ηG4M2p+r2q)−72η2G6M4

p r2q)

a√Gr2q√a2G+8r2q

+log(M)(4πηG2M2

p)(3a2ηG4M2p+r2q(a2+6ηG3M2

p))a2r2q

− π log(a2(2G2M−rq)+8G3M2rq)(a4r2q+4a2ηG3M2p(3ηG4M2

p+r2q)+24η2G6M4p r

2q)

2a2Gr2q

= −6π2η2G2M2p

SBH− 12π

32 η2G4M3

p√SBHrq

−π tanh−1

√G

(a2+4G

(Mp√SBH√π

)rq

)a√

a2G+8r2q

(a4r2q−4a2ηG3M2p(3ηG4M2

p+r2q)−72η2G6M4p r

2q)

a√Gr2q√a2G+8r2q

+log

(Mp√SBH

2√π

)(4πηG2M2

p)(3a2ηG4M2p+r2q(a2+6ηG3M2

p))a2r2q

−π log

(a2(G2Mp

√SBH√π

−rq)

+2G3M2

prqSBHπ

)(a4r2q+4a2ηG3M2

p(3ηG4M2p+r2q)+24η2G6M4

p r2q)

2a2Gr2q

= −6π2η2G2M2p

(A4 )− 12π

32 η2G4M3

p√(A4 )rq

−

π tanh−1

√G

a2+4GrqMp

√(A4 )

√π

a√

a2G+8r2q

(a4r2q−4a2ηG3M2p(3ηG4M2

p+r2q)−72η2G6M4p r

2q)

a√Gr2q√a2G+8r2q

+

log

Mp

√(A4 )

2√π

(4πηG2M2p)(3a2ηG4M2

p+r2q(a2+6ηG3M2p))

a2r2q−

π log

a2G2Mp

√(A4 )

√π

−rq

+2G3M2

prq(A4 )π

(a4r2q+4a2ηG3M2p(3ηG4M2

p+r2q)+24η2G6M4p r

2q)

2a2Gr2q.

(5.5)

Case 1.3(a) : n = 1

S = −35πη4G2M4p

32M2 − 5πη3G2M3p

2M+

√2πη√GMp tanh−1

(2√2G3/2M

√a2+r2q

arq

)(a2(5η2G3M2

p+r2q)+5η2G3M2p r

2q)

arq√a2+r2q

−π log[a2(r2q−8G3M2)−8G3M2r2q](a4(35η4G6M4p+6η2G3M2

p r2q+2r4q)+a2(70η4G6M4

p r2q+6η2G3M2

p r4q)+35η4G6M4

p r4q)

4a2Gr2q(a2+r2q)

+πG2 log(M)(a2(35η4G3M4

p+6η2M2p r

2q)+35η4G3M4

p r2q)

2a2r2q

= −35π2η4G2M2p

8SBH− 5π

32 η3G2M2

p√SBH

+


(√2G3/2√SBHMp

√a2+r2q

πarq

)(a2(5η2G3M2

p+r2q)+5η2G3M2p r

2q)

arq√a2+r2q

−π log

[a2(r2q−

2G3M2pSBHπ

)−

2G3M2pr

2qSBH

π

](a4(35η4G6M4

p+6η2G3M2p r

2q+2r4q)+a2(70η4G6M4

p r2q+6η2G3M2

p r4q)+35η4G6M4

p r4q)

4a2Gr2q(a2+r2q)

+πG2 log

(Mp√SBH

2√π

)(a2(35η4G3M4

p+6η2M2p r

2q)+35η4G3M4

p r2q)

2a2r2q

124

= −35π2η4G2M2p

8(A4 )− 5π

32 η3G2M2

p√(A4 )

+


√2G3/2√

(A4 )Mp√

a2+r2q

πarq

(a2(5η2G3M2p+r2q)+5η2G3M2

p r2q)

arq√a2+r2q

−π log

[a2

(r2q−

2G3M2p(A4 )π

)−

2G3M2pr

2q(A4 )

π

](a4(35η4G6M4

p+6η2G3M2p r

2q+2r4q)+a2(70η4G6M4

p r2q+6η2G3M2

p r4q)+35η4G6M4

p r4q)

4a2Gr2q(a2+r2q)

+

πG2 log

Mp

√(A4 )

2√π

(a2(35η4G3M4p+6η2M2

p r2q)+35η4G3M4

p r2q)

2a2r2q. (5.6)

Case 1.3(b) : n = 2Considering terms upto O(η2) we get :

S = −3πη2G2M4p

2M2 +4πηG2M2

p

a2r2qlog(M)

[a2(6ηG3M2

p + r2q

)+ 6ηG3M2

p r2q

]−π log[a2(r2q−8G3M2)−8G3M2r2q][24a4η2G6M4

p+4a2ηG3M2p r

2q(a2+12ηG3M2

p)+r4q(a4+4a2ηG3M2p+24η2G6M4

p)]2a2Gr2q(a2+r2q)

= −6π2η2G2M2p

SBH+

4πηG2M2p

a2r2qlog(Mp√SBH

2√π

) [a2(6ηG3M2

p + r2q

)+ 6ηG3M2

p r2q

]−π log

[a2(r2q−

2G3M2pSBHπ

)−(

2G3M2pr

2qSBH

π

)][24a4η2G6M4

p+4a2ηG3M2p r

2q(a2+12ηG3M2

p)+r4q(a4+4a2ηG3M2p+24η2G6M4

p)]

2a2Gr2q(a2+r2q)

= −6π2η2G2M2p

(A4 )+

4πηG2M2p

a2r2qlog

(Mp

√(A4 )

2√π

)[a2(6ηG3M2

p + r2q

)+ 6ηG3M2

p r2q

]−π log

[a2

(r2q −

2G3M2p(A4 )π

)−(

2G3M2p r

2q(A4 )

π

)] [24a4η2G6M4

p + 4a2ηG3M2p r

2q

(a2 + 12ηG3M2

p

)+r4

q

(a4 + 4a2ηG3M2

p + 24η2G6M4p

)] [2a2Gr2

q

(a2 + r2

q

)]−1(5.7)

Acknowledgements. SD Thanks Goverment of West Bengal, Department of HigherEducation, Science and Technology and Biotechnology for non-NET fellowship. R Banerjeethanks IIEST, Shibpur for institutional fellowship. R Biswas thanks the project grant ofGoverment of West Bengal, Department of Higher Education, Science and Technology andBiotechnology (File no:- ST/P/S&T/16G− 19/2017). R Biswas also thanks IUCAA, Punefor Visiting Associateship.

Authors are also thankful to the referee for his valuable suggestions to bring the paperin its present form.

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Jñānãbha –Vijñāna Parishad of India



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http://www.zbl.theta.ro/too/zs/j.html

Contents

TIME SERIES ANALYSIS OF HEAT STROKE 1-10Rashmi Bhardwaj and Varsha Duhoon

STUDY OF TRAFFIC CONGESTION FLOW USING QUEUEING MODEL

Jitendra Kumar and Vikas Shinde

NUMERICAL SOLUTION OF THE CONVECTION DIFFUSION EQUATION BY THE LEGENDRE WAVELET METHOD 26-39

Devendra Chouhan and R.S. Chandel

SOME COUPLED FIXED POINT THEOREMS IN CONVEX METRICSPACES 40-49

Memudu Olaposi Olatinwo

EXISTENCE THEOREMS FOR A PBVP OF FIRST ORDER FUNC-TIONAL RANDOM INTEGRODIFFERENTIAL INCLUSIONS 50-66

Bapurao C. Dhage

A COMMON FIXED POINT THEOREM FOR WEAKLY RECIPRO-CALLY CONTINUOUS SYSTEMS OF MAPS SATISFYING A GENERALCONTRACTIVE CONDITION OF INTEGRAL TYPE 67-79

Deepak Khantwal and U.C. Gairola

GENERALIZED HERMITE POLYNOMIAL FAMILIES 80-88Paolo Emilio Ricci

A CLASS OF TWO VARIABLES SEQUENCE OF FUNCTIONS SATISFY-ING ABEL’S INTEGRAL EQUATION AND THE PHASE SHIFTS 89-96

Hemant Kumar

HYPERCYCLIC PAIR WHOSE COMPONENTS ARE DIRECT SUMS OFIDENTICAL HYPERCYCLIC OPERATORS 97-106

Neema Wilberth, Santosh Kumar and Marco Mpimbo

A BRIEF STUDY OF THERMODYNAMIC QUANTITIES OF BLACKHOLES IN RAINBOW GRAVITY 107-127

Sandip Dutta, Ritwick Banerjee and Ritabrata Biswas

11-25

volume 49 number 1 june 2019 - …

Documents