ϕ-contractive multivalued mappings in complex valued metric … · φ-contractive multivalued...

Joshi et al., Cogent Mathematics (2016), 3: 1162484http://dx.doi.org/10.1080/23311835.2016.1162484

PURE MATHEMATICS | RESEARCH ARTICLE

φ-contractive multivalued mappings in complex valued metric spaces and remarks on some recent papersVishal Joshi1, Naval Singh2 and Deepak Singh3*

Abstract: The purpose of this paper is twofold. Firstly, certain common fixed point theorems are established via �-contractive multivalued mappings involving point-dependent control functions as coefficients in the framework of complex valued metric spaces. Our results improve and extend several results in the existing literature. Moreover, this section is equipped by some illustrative examples in support of our results. Secondly, we point out some slip-ups in the examples of some recent papers based on multivalued contractive mappings in complex valued metric spaces. Our observations are also authenticated with the aid of some appropriate examples. Some rectifications to correct the erratic examples are also suggested.

Subjects: Advanced Mathematics; Analysis-Mathematics; Mathematics & Statistics; Pure Mathematics; Science

Keywords: common fixed point; complex valued metric spaces; complete complex valued metric spaces; Cauchy sequence; multivalued mappings

AMS Subject Classifications: 47H10; 54H25; 30L99

*Corresponding author: Deepak Singh, Department of Applied Sciences, NITTTR, Under Ministry of HRD, Government of India, Bhopal 462002, India E-mail: [email protected]

Reviewing editor:Prasanna K. Sahoo, University of Louisville, USA

Additional information is available at the end of the article

ABOUT THE AUTHORSVishal Joshi, having more than 16 years of teaching experience, is working as an assistant professor in the Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur, India. He has qualified several national level examinations and tests. His research interests are fixed point theory, topology, and functional analysis.

Deepak Singh received his MSc and PhD degrees in Mathematics from the Barkatullah Vishwavidyalaya, Bhopal, Madhya Pradesh, India in 1993 and 2004, respectively. Currently, he is an associate professor at Department of Applied Sciences, National Institute of Technical Teachers Training and Research, Bhopal, Madhya Pradesh, India. He is associated as referees and reviewer for many journals of international repute. He has also delivered contributed /invited talks in many international conferences held in European and Asian countries. His current research interests include optimization, nonlinear functional analysis, fixed point theory and its applications. He has published 40 research papers in the journals of international repute.

PUBLIC INTEREST STATEMENTIn recent times, the notion of complex valued metric spaces is one of the developing areas in mathematical analysis. The fixed point results concerning rational contractive conditions cannot be extended in cone metric spaces, whereas in complex valued metric spaces, one can find the fixed point of mappings via rational contractive conditions. The results in this space can be utilized to find the solution of Urysohn Integral Equations., Boundary value problems and system of algebraic equations. This paper is devoted to a package of multivalued mappings with control functions, which may be useful for the researcher to find the solution for future problems specially mentioned above.

Received: 25 October 2015Accepted: 26 February 2016First Published: 07 March 2016

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.

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Deepak Singh

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1. IntroductionIn 2011, Azam, Fisher, and Khan (2011) introduced the notion of complex valued metric spaces and established some fixed point results for a pair of mappings for contraction condition satisfying a rational expression. This idea is intended to define rational expressions which are not meaningful in cone metric spaces and thus many such results of analysis cannot be generalized to cone metric spaces but to complex valued metric spaces.

After the establishment of complex valued metric spaces, Rouzkard and Imdad (2012) established some common fixed point theorems satisfying certain rational expressions in this spaces to general-ize the result of Azam et al. (2011). Subsequently, Sintunavarat and Kumam (2012) obtained com-mon fixed point results by replacing the constant of contractive condition to control functions. Sitthikul and Seajung (2012) established some fixed point results by generalizing the contractive conditions in the context of complex valued metric spaces. Recently, Sintunavarat, Cho, and Kumam (2013) introduced the notion of C-Cauchy sequence and C-completeness in complex valued metric spaces and applied it to obtain the common solution of Urysohn integral equations. Very recently, Singh, Singh, Badal, and Joshi (in press) established certain fixed point theorems which generalized numerous preceding results in the setting of complex valued metric spaces.

Ahmad, Klin-Eam, and Azam (2013) established the existence of common fixed point for multival-ued mappings under generalized contractive condition in complex valued metric spaces. Afterward Azam, Ahmad, and Kumam (2013) and then Kutbi, Ahmad, Azam, and Al-Rawashdeh (2014) im-proved the contractive condition of the result of Ahmad et al. (2013) and proved some common fixed point results for multivalued mappings in complex valued metric space.

In what follows, we recall some notations and definitions that will be used in our subsequent discussion.

Let C be the set of complex numbers and z1, z2∈ C. Define a partial order ≾ on C as follows:

z1≾ z

2 if and only if Re(z

1) ≤ Re(z

2) and Im(z

1) ≤ Im(z

2).

It follows that ≾ if one of the followings conditions is satisfied.

(C1) Re(z1) = Re(z

2) and Im(z

1) = Im(z

2);

(C2) Re(z1) < Re(z

2) and Im(z

1) = Im(z

2);

(C3) Re(z1) = Re(z

2) and Im(z

1) < Im(z

2);

(C4) Re(z1) < Re(z

2) and Im(z

1) < Im(z

2).

In particular, we will write z1⋨ z

2 if z

1≠ z

2 and one of (C2), (C3), and (C4) is satisfied and z

1≺ z

2

if only (C4) is satisfied.

Definition 1.1 (Azam et al., 2011) Let X be a non empty set. A mapping d : X × X → C is called a complex valued metric on X if the following conditions are satisfied:

(CM1) 0 ≾ d(x, y) for all x, y ∈ X and d(x, y) = 0⇔ x = y;

(CM2) d(x, y) = d(y, x) for all x, y ∈ X;

(CM3) d(x, y) ≾ d(x, z) + d(z, y) for all x, y, z ∈ X.

In this case, we say that (X, d) is a complex valued metric space.

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Example 1.2 Let X = C be a set of complex number. Define d : C × C → C by

where z1= x

1+ iy

1 and z

2= x

2+ iy

2. Then (C, d) is a complex valued metric space.

Example 1.3 Let X = C. Define a mapping d : X × X → C by d(z1, z2) = eik|z

1− z

2|, where k ∈ [0,

�

2].

Then (X, d) is a complex valued metric space.

Definition 1.4 (Azam et al., 2011) Suppose that (X, d) is a complex valued metric space.

(1) We say that a sequence {xn} is a Cauchy sequence if for every 0 ≺ c ∈ C there exists an integer N such that d(xn, xm) ≺ c for all n,m ≥ N.

(2) We say that {xn} converges to an element x ∈ X if for every 0 ≺ c ∈ C there exists an integer N such that d(xn, x) ≺ c for all n ≥ N . In this case, we write

d

xn → x.

(3) We say that (X, d) is complete if every Cauchy sequence in X converges to a point in X.

Lemma 1.5 (Azam et al., 2011) Let (X, d) be a complex valued metric space and let {xn} be a sequence in X. Then {xn} converges to x if and only if |d(xn, x)| → 0 as n→ ∞.

Lemma 1.6 (Azam et al., 2011) Let (X, d) be a complex valued metric space and let {xn} be a sequence in X. Then {xn} is a Cauchy sequence if and only if |d(xn, xn+m)| → 0 as n→ ∞.

Ahmad et al. (2013) introduced the notion of multivalued mappings as follows.

Let (X, d) be a complex valued metric space. Denote the family of nonempty, closed, and bounded subsets of a complex valued metric space by CB(X). From now on, denote s(z

1) = {z

2∈ C : z

1≾ z

2}

for z1∈ C, and

For A,B ∈ CB(X), denote

Definition 1.7 (Ahmad et al. 2013) Let (X, d) be a complex valued metric space. Let T : X → CB(X) be a multivalued map. For x ∈ X and A ∈ CB(X), define

Thus, for x, y ∈ X,

Definition 1.8 (Ahmad et al. 2013) Let (X, d) be a complex valued metric space. A subset A of X is called bounded from below if there exists some z ∈ X such that z ≾ a for all a ∈ A.

Definition 1.9 (Ahmad et al. 2013) Let (X, d) be a complex valued metric space. A multivalued map-ping F : X → 2

C is called bounded from below if for each x ∈ X there exists zx ∈ C such that zx ≾ u for all u ∈ Fx.

d(z1, z2) = |x

1− x

2| + i|y

1− y

2|,

s(a,B) =⋃

b∈B

s(d(a, b)) =⋃

b∈B

{z ∈ C : d(a, b) ≾ z} for a ∈ X and B ∈ CB(X).

s(A,B) =

(⋂

a∈A

s(a,B)

)⋂

(⋂

b∈B

s(b,A)

).

Wx(A) = {d(x, a) : a ∈ A}.

Wx(Ty) = {d(x,u) : u ∈ Ty}.

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Definition 1.10 (Ahmad et al. 2013) Let (X, d) be a complex valued metric space. The multivalued mapping T : X → CB(X) is said to have the lower bound property (l.b property) on (X, d), if the for any x ∈ X, the multivalued mapping F

x: X → 2

C defined by

is bounded from below. That is, for x, y ∈ X, there exists an element lx(Ty) ∈ C such that

for all u ∈Wx(Ty), where lx(Ty) is called a lower bound of T associated with (x, y).

Definition 1.11 (Ahmad et al. 2013) Let (X, d) be a complex valued metric space. The multivalued mapping T : X → CB(X) is said to have the greatest lower bound property (g.l.b property) on (X, d) if a greatest lower bound of Wx(Ty) exists in C for all x, y ∈ X. Denote d(x, Ty) by the g.l.b of Wx(Ty). That is,

Definition 1.12 Let Ψ be a family of non-decreasing functions, � : C → C such that �(0) = 0 and 𝜙(t) ≺ t, when 0 ≺ t.

2. Main resultWe start this section with the following observation.

Proposition 2.1 Let (X, d) be a complex valued metric space and S, T : X → CB(X). Let x0∈ X and

defined the sequence {xn} by

Assume that there exists a mapping � : X → [0, 1) such that �(u) ≤ �(x) and �(v) ≤ �(x), ∀x ∈ X,u ∈ Sx, v ∈ Tx. Then

Proof Let x ∈ X and n = 0, 1, 2,…. Then we have

i.e. �(x2n) ≤ �(x

0).

Similarly, we have

The subsequent example illustrates the preceding proposition. □

Example 2.2 Let X = {1,1

2,1

3,1

4,1

5,…}. Define d : X × X → C as d(x, y) = i|x − y| then clearly (X, d) is a

complex valued metric space. Also define multivalued mappings S and T by

Fx(Ty) =Wx(Ty),

lx(Ty) ≾ u,

d(x, Ty) = inf {d(x,u) : u ∈ Ty}.

(2.1)x2n+1 ∈ Sx2n,

x2n+2 ∈ Tx2n+1, ∀ n = 0, 1, 2,…

�(x2n) ≤ �(x

0)and �(x

2n+1) ≤ �(x1).

�(x2n) = �(u

1) ≤ �(x

2n−2), for u

1∈ Sx

2n−1

= �(u2) ≤ �(x

2n−2), for u

2∈ Sx

2n−2

⋮

≤ �(x0)

�(x2n+1) ≤ �(x

1).

S(

1

n + 1

)=

[0,

1

n + 2

]= T

(1

n + 1

), n = 0, 1, 2, 3,…

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Choosing sequence {xn} as xn =1

n+1,n = 0, 1, 2, 3,… Then x

0= 1 ∈ X.

Clearly, x2n+1 ∈ Sx2n and x

2n+2 ∈ Tx2n+1.

Consider a mapping � : X → [0, 1) by �(x) = x

6, for all x ∈ X.

Undoubtedly

for all x, y ∈ X

Consider

that is �(x2n) ≤ �(x

0),n = 0, 1, 2…. Also consider

that is �(x2n+1) ≤ �(x

1),n = 0, 1, 2…, ∀x ∈ X.

Thus Proposition 2.1 is verified.

Our main theorem runs as follows.

Theorem 2.3 Let (X, d) be a complete complex valued metric space and S, T : X → CB(X) be multival-ued mapping with g.l.b. property. Then there exist mappings �, �, �, �, �,�, � : X → [0, 1) such that,

(i) �(u) ≤ �(x), �(u) ≤ �(x), �(u) ≤ �(x), �(u) ≤ �(x), �(u) ≤ �(x),�(u) ≤ �(x) and �(u) ≤ �(x), for all u ∈ Sx and ∀x ∈ X;

(ii) �(v) ≤ �(x), �(v) ≤ �(x), �(v) ≤ �(x), �(v) ≤ �(x), �(v) ≤ �(x),�(v) ≤ �(x) and �(v) ≤ �(x), for all v ∈ Tx and ∀x ∈ X;

(iii) 𝜆(x) + 𝜂(x) + 𝛿(x) + 2𝜉(x) + 𝜈(x) + 𝜇(x) + 𝛾(x) < 1,∀x ∈ X;

(iv)

for all x, y ∈ X and � ∈ Ψ.

Then S and T have a common fixed point.

Proof Let x0 be an arbitrary point in X and x

1∈ Sx

0. From (2.2) with x = x

0 and y = x

1, we get

This yields that

�(u) ≤ �(x)and �(v) ≤ �(x), ∀x ∈ X,u ∈ Sx, v ∈ Tx

�(x2n) =

1

6(2n + 1)≤1

6= �(x

0),

�(x2n+1) =

1

6(2n + 2)≤

1

2

6= �(x

1),

(2.2)

�

(�(x)d(x, y) + �(x)d(x, Sx) + �(x)d(y, Ty) + �(x)

d(x, Sx)d(x, Ty)

1 + d(x, y)

+�(x)d(y, Sx)d(y, Ty)

1 + d(x, y)+ �(x)

d(x, Sx)d(y, Ty)

1 + d(x, y)+ �(x)

d(y, Sx)d(x, Ty)

1 + d(x, y)

)∈ s(Sx, Ty),

�

(�(x

0)d(x

0, x

1) + �(x

0)d(x

0, Sx

0) + �(x

0)d(x

1, Tx

1) + �(x

0)d(x

0, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

0, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

)

∈ s(Sx0, Tx

1).

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This implies that

for all x ∈ Sx0. Now since x

1∈ Sx

0, one can have

So there exists some x2∈ Tx

1, such that

Therefore

Utilizing the greatest lower bound property (g.l.b. property) of S and T, we obtain

�

(�(x

0)d(x

0, x

1) + �(x

0)d(x

0, Sx

0) + �(x

0)d(x

1, Tx

1) + �(x

0)d(x

0, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

0, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

)

∈⋂

x∈Sx0

s(Sx0, Tx

1).

�

(�(x

0)d(x

0, x

1) + �(x

0)d(x

0, Sx

0) + �(x

0)d(x

1, Tx

1) + �(x

0)d(x

0, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

0, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

)

∈ s(x, Tx1),

(2.3)

�

(�(x

0)d(x

0, x

1) + �(x

0)d(x

0, Sx

0) + �(x

0)d(x

1, Tx

1) + �(x

0)d(x

0, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

0, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

)

∈ s(x1, Tx

1),

⇒ �

(�(x

0)d(x

0, x

1) + �(x

0)d(x

0, Sx

0) + �(x

0)d(x

1, Tx

1) + �(x

0)d(x

0, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

0, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

)

∈⋃

x∈Tx1

s(d(x1, x)).

�

(�(x

0)d(x

0, x

1) + �(x

0)d(x

0, Sx

0) + �(x

0)d(x

1, Tx

1) + �(x

0)d(x

0, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

0, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ �(x0)d(x

1, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

)

∈ s(d(x1, x

2)).

d(x1, x

2) ≾ 𝜙

(𝜆(x

0)d(x

0, x

1) + 𝜂(x

0)d(x

0, Sx

0) + 𝛿(x

0)d(x

1, Tx

1)

+ 𝜉(x0)d(x

0, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

+ 𝜈(x0)d(x

1, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ 𝜇(x0)d(x

0, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ 𝛾(x0)d(x

1, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

)

≺ 𝜆(x0)d(x

0, x

1) + 𝜂(x

0)d(x

0, Sx

0) + 𝛿(x

0)d(x

1, Tx

1)

+ 𝜉(x0)d(x

0, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

+ 𝜈(x0)d(x

1, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ 𝜇(x0)d(x

0, Sx

0)d(x

1, Tx

1)

1 + d(x0, x

1)

+ 𝛾(x0)d(x

1, Sx

0)d(x

0, Tx

1)

1 + d(x0, x

1)

.

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So that

Inductively, using Proposition 2.1, we can construct a sequence xn in X such that for n = 0, 1, 2,…

for x2n+1 ∈ Sx2n and x

2n+2 ∈ Tx2n+1, with 𝛼 =𝜆(x

0)+𝜂(x

0)+𝜉(x

0)

1−𝛿(x0)−𝜉(x

0)−𝜇(x

0)< 1.

Next for m > n, we get

Thus we have

Which on making m,n→ ∞, yields

This reflects that {xn} is a Cauchy sequence in X. Since X is complete then there exists p ∈ X such that xn → p as n→ ∞.

Now, we show that p ∈ Tp and p ∈ Sp. From (2.2), with x = x2k and y = p, we have

(2.4)

d(x1, x

2) ≾ 𝜆(x

0)d(x

0, x

1) + 𝜂(x

0)d(x

0, x

1) + 𝛿(x

0)d(x

1, x

2) + 𝜉(x

0)d(x

0, x

1)d(x

0, x

2)

1 + d(x0, x

1)

+ 𝜈(x0)d(x

1, x

1)d(x

1, x

2)

1 + d(x0, x

1)

+ 𝜇(x0)d(x

0, x

1)d(x

1, x

2)

1 + d(x0, x

1)

+ 𝛾(x0)d(x

1, x

1)d(x

0, x

2)

1 + d(x0, x

1)

= 𝜆(x0)d(x

0, x

1) + 𝜂(x

0)d(x

0, x

1) + 𝛿(x

0)d(x

1, x

2) + 𝜉(x

0)d(x

0, x

1)d(x

0, x

2)

1 + d(x0, x

1)

+ 𝜇(x0)d(x

0, x

1)d(x

1, x

2)

1 + d(x0, x

1)

.

|d(x1, x

2)| ≤ 𝜆(x

0)|d(x

0, x

1)| + 𝜂(x

0)|d(x

0, x

1)| + 𝛿(x

0)|d(x

1, x

2)|

+ 𝜉(x0)|d(x

0, x

2)||

d(x0, x

1)

1 + d(x0, x

1)| + 𝜇(x

0)|d(x

1, x

2)||||

d(x0, x

1)

1 + d(x0, x

1)

|||

≤ 𝜆(x0)|d(x

0, x

1)| + 𝜂(x

0)|d(x

0, x

1)| + 𝛿(x

0)|d(x

1, x

2)|

+ 𝜉(x0)|d(x

0, x

2)| + 𝜇(x

0)|d(x

1, x

2)|

≤ 𝜆(x0)|d(x

0, x

1)| + 𝜂(x

0)|d(x

0, x

1)| + 𝛿(x

0)|d(x

1, x

2)|

+ 𝜉(x0)|d(x

0, x

1) + d(x

1, x

2)| + 𝜇(x

0)|d(x

1, x

2)|

⇒ |d(x1, x

2)| ≤

𝜆(x0) + 𝜂(x

0) + 𝜉(x

0)

1 − 𝛿(x0) − 𝜉(x

0) − 𝜇(x

0)|d(x

0, x

1)|

= 𝛼|d(x0, x

1)|, where 𝛼 =

𝜆(x0) + 𝜂(x

0) + 𝜉(x

0)

1 − 𝛿(x0) − 𝜉(x

0) − 𝜇(x

0)< 1.

|d(xn, xn+1)| ≤ �n|d(x

0, x

1)|,

|d(xn, xm)| ≤ |d(xn, xn+1)| + |d(xn+1, xn+2)| + ... + |d(xm−1, xm)|

≤ (�n+ �

n+1+ ... + �

m−1)|d(x

0, x

1)|

≤

[�n

1 − �

]|d(x

0, x

1)|.

|d(xn, xm)| ≤[

�n

1 − �

]|d(x

0, x

1)|.

|d(xn, xm)| → 0.

�

(�(x

2k)d(x2k, p) + �(x2k)d(x2k, Sx2k) + �(x

2k)d(p, Tp) + �(x2k)d(x

2k, Sx2k)d(x2k, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(x

2k, Sx2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(x2k, Tp)

1 + d(x2k, p)

)

∈ s(Sx2k, Tp).

of 14


This implies that

for all x ∈ Sx2k. Since x

2k+1 ∈ Sx2k, we have

There exists some pk ∈ Tp, such that

Thus, we have

Utilizing Proposition 2.1 and also using greatest lower bound property of S and T, we have

�

(�(x

2k)d(x2k, p) + �(x2k)d(x2k, Sx2k) + �(x

2k)d(p, Tp) + �(x2k)d(x

2k, Sx2k)d(x2k, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(x

2k, Sx2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(x2k, Tp)

1 + d(x2k, p)

)

∈⋂

x∈Sx2k

s(x, Tp)

⇒ �

(�(x

2k)d(x2k, p) + �(x2k)d(x2k, Sx2k) + �(x

2k)d(p, Tp) + �(x2k)d(x

2k, Sx2k)d(x2k, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(x

2k, Sx2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(x2k, Tp)

1 + d(x2k, p)

)

∈ s(x, Tp),

�

(�(x

2k)d(x2k, p) + �(x2k)d(x2k, Sx2k) + �(x

2k)d(p, Tp) + �(x2k)d(x

2k, Sx2k)d(x2k, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(x

2k, Sx2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(x2k, Tp)

1 + d(x2k, p)

)

∈ s(x2k+1, Tp)

⇒ �

(�(x

2k)d(x2k, p) + �(x2k)d(x2k, Sx2k) + �(x

2k)d(p, Tp) + �(x2k)d(x

2k, Sx2k)d(x2k, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(x

2k, Sx2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(x2k, Tp)

1 + d(x2k, p)

)

∈⋃

p1∈Tp

s(d(x2k+1, p1)).

�

(�(x

2k)d(x2k, p) + �(x2k)d(x2k, Sx2k) + �(x

2k)d(p, Tp) + �(x2k)d(x

2k, Sx2k)d(x2k, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(x

2k, Sx2k)d(p, Tp)

1 + d(x2k, p)

+ �(x2k)d(p, Sx

2k)d(x2k, Tp)

1 + d(x2k, p)

)

∈ s(d(x2k+1, pk)).

d(x2k+1, pk) ≾ 𝜙

(𝜆(x

2k)d(x2k, p) + 𝜂(x2k)d(x2k, Sx2k) + 𝛿(x

2k)d(p, Tp)

+ 𝜉(x2k)d(x

2k, Sx2k)d(x2k, Tp)

1 + d(x2k, p)

+ 𝜈(x2k)d(p, Sx

2k)d(p, Tp)

1 + d(x2k, p)

+ 𝜇(x2k)d(x

2k, Sx2k)d(p, Tp)

1 + d(x2k, p)

+ 𝛾(x2k)d(p, Sx

2k)d(x2k, Tp)

1 + d(x2k, p)

)

≾ 𝜆(x2k)d(x2k, p) + 𝜂(x

2k)d(x2k, Sx2k) + 𝛿(x2k)d(p, Tp)

+ 𝜉(x2k)d(x

2k, Sx2k)d(x2k, Tp)

1 + d(x2k, p)

+ 𝜈(x2k)d(p, Sx

2k)d(p, Tp)

1 + d(x2k, p)

+ 𝜇(x2k)d(x

2k, Sx2k)d(p, Tp)

1 + d(x2k, p)

+ 𝛾(x2k)d(p, Sx

2k)d(x2k, Tp)

1 + d(x2k, p)

.

of 14


We have by triangular inequality

Thus, one can obtain

So that

Which on letting k→ ∞, reduces to

By Lemma 1.5, we have

Since Tp is closed then p ∈ Tp.

Similarly, one can get p ∈ Sp.

Thus p ∈ Tp ∩ Sp. Therefore S and T have a common fixed point. □

Subsequent result is an easy consequence of Theorem 2.3.

Corollary 2.4 Let (X, d) be a complete complex valued metric space and S, T : X → CB(X) be multi-valued mapping with g.l.b. property. Then there exist mappings �

1, �1, �1, �1, �1,�

1, �1: X → [0, 1) such

that, ∀x ∈ X,

(i) �1(u) ≤ �

1(x), �

1(u) ≤ �

1(x), �

1(u) ≤ �

1(x), �

1(u) ≤ �

1(x), �

1(u) ≤ �

1(x),�

1(u) ≤ �

1(x) and �

1(u) ≤ �

1(x),

for all u ∈ Sx;

(ii) �1(v) ≤ �

1(x), �

1(v) ≤ �

1(x), �

1(v) ≤ �

1(x), �

1(v) ≤ �

(x), �

1(v) ≤ �

1(x),�

1(v) ≤ �

1(x) and �

1(v) ≤ �

1(x),

for all v ∈ Tx;

(iii) 𝜆1(x) + 𝜂

1(x) + 𝛿

1(x) + 2𝜉

1(x) + 𝜈

1(x) + 𝜇

1(x) + 𝛾

1(x) < 1,

(2.5)

d(x2k+1, pk) ≾ 𝜆(x

0)d(x

2k, p) + 𝜂(x0)d(x

2k, x2k+1) + 𝛿(x0)d(p, pk)

+ 𝜉(x0)d(x

2k, x2k+1)d(x2k, pk)

1 + d(x2k, p)

+ 𝜈(x0)d(p, x

2k+1)d(p, pk)

1 + d(x2k, p)

+ 𝜇(x0)d(x

2k, x2k+1)d(p, pk)

1 + d(x2k, p)

+ 𝛾(x0)d(p, x

2k+1)d(x2k, pk)

1 + d(x2k, p)

.

d(p, pk) ≾ d(p, x2k+1) + d(x2k+1, pk).

d(p, pk) ≾ d(p, x2k+1) + 𝜆(x0)d(x

2k, p) + 𝜂(x0)d(x

2k, x2k+1) + 𝛿(x0)d(p, pk)

+ 𝜉(x0)d(x

2k, x2k+1)d(x2k, pk)

1 + d(x2k, p)

+ 𝜈(x0)d(p, x

2k+1)d(p, pk)

1 + d(x2k, p)

+ 𝜇(x0)d(x

2k, x2k+1)d(p, pk)

1 + d(x2k, p)

+ 𝛾(x0)d(p, x

2k+1)d(x2k, pk)

1 + d(x2k, p)

.

|d(p, pk)| ≤|d(p, x2k+1)| + �(x0)|d(x

2k, p)| + �(x0)|d(x

2k, x2k+1)| + �(x0)|d(p, pk)|

+ �(x0)|d(x

2k, x2k+1)||d(x2k, pk)||1 + d(x

2k, p)|+ �(x

0)|d(p, x

2k+1)||d(p, pk)||1 + d(x

2k, p)|

+ �(x0)|d(x

2k, x2k+1)||d(p, pk)||1 + d(x

2k, p)|+ �(x

0)|d(p, x

2k+1)||d(x2k, pk)||1 + d(x

2k, p)|.

(1 − �(x

0))|d(p, pk)| → 0 or |d(p, pk)| → 0as k→ ∞.

limn→∞

pk = p.

of 14


(iv)

for all x, y ∈ X.


Proof Proof is immediate on choosing �(t) = kt, where k ∈ (0, 1) in Theorem 2.3 with �1(x) = k�(x), �

1(x) = k�(x), �

1(x) = k�(x), �(x) = k�(x), �

1(x) = k�(x),�

1(x) = k�(x), �

1(x) = k�(x). □

Now, consider the following corollary.

Corollary 2.5 Let (X, d) be a complete complex valued metric space and S, T : X → CB(X) be multi-valued mapping with g.l.b. property such that

for all x, y ∈ X and �1, �1, �1, �1, �1,�

1, and �

1 are non negative real numbers with

𝜆1+ 𝜂

1+ 𝛿

1+ 2𝜉

1+ 𝜈

1+ 𝜇

1+ 𝛾

1< 1.


Proof Proof can be obtain easily by restricting the point-dependent coefficient to constants i.e. by setting �

1(x) = �

1, �1(x) = �

1, �1(x) = �

1, �1(x) = �

1, �1(x) = �

1,�

1(x) = �

1 and �

1(x) = �

1 in Corollary 2.4

with �1, �1, �1, �1, �1�1, �1≥ 0 such that 𝜆

1+ 𝜂

1+ 𝛿

1+ 2𝜉

1+ 𝜈

1+ 𝜇

1+ 𝛾

1< 1. □

Remark 2.6

(1) If we set �1= �

1= �

1= �

1= 0 in Corollary 2.5, we will get the Theorem 9 of Ahmad et al. (2013).

(2) If we choose �1= �

1= �

1= �

1= 0 in Corollary 2.5, then Theorem 15 of Ahmad et al. (2013) is

obtained.

(3) Setting �1= �

1= 0 in Corollary 2.5, one can obtain the Theorem 9 of Kutbi et al. (2014).

Consequently all the corollaries corresponding to these results are immediate from our results.

Remark 2.7 Let (X, d) be a complex valued metric space. If C = R, then (X, d) is a metric space. Furthermore, for S, T ∈ CB(X), H(S, T) = inf s(S, T) is the Hausdorff metric induced by d.

Utilizing aforesaid Remark 2.7, we have the following corollary from Theorem 2.3.

Corollary 2.8 Let (X, d) be a complete metric space and let S, T : X → CB(X) be multivalued map-ping with g.l.b. property. Then there exist mappings �, �, �, �, �,�, � : X → [0, 1) such that,

(i) �(u) ≤ �(x), �(u) ≤ �(x), �(u) ≤ �(x), �(u) ≤ �(x), �(u) ≤ �(x),�(u) ≤ �(x) and �(u) ≤ �(x), for all u ∈ Sx and ∀x ∈ X;

(ii) �(v) ≤ �(x), �(v) ≤ �(x), �(v) ≤ �(x), �(v) ≤ �(x), �(v) ≤ �(x),�(v) ≤ �(x) and �(v) ≤ �(x), for all v ∈ Tx and ∀x ∈ X;

(iii) 𝜆(x) + 𝜂(x) + 𝛿(x) + 2𝜉(x) + 𝜈(x) + 𝜇(x) + 𝛾(x) < 1, ∀x ∈ X;

�1(x)d(x, y) + �

1(x)d(x, Sx) + �

1(x)d(y, Ty) + �

1(x)d(x, Sx)d(x, Ty)

1 + d(x, y)

+ �1(x)d(y, Sx)d(y, Ty)

1 + d(x, y)+ �

1(x)d(x, Sx)d(y, Ty)

1 + d(x, y)+ �

1(x)d(y, Sx)d(x, Ty)

1 + d(x, y)∈ s(Sx, Ty),

�1d(x, y) + �

1d(x, Sx) + �

1d(y, Ty) + �

1

d(x, Sx)d(x, Ty)

1 + d(x, y)

+ �1

d(y, Sx)d(y, Ty)

1 + d(x, y)+ �

1

d(x, Sx)d(y, Ty)

1 + d(x, y)+ �

1

d(y, Sx)d(x, Ty)

1 + d(x, y)∈ s(Sx, Ty),

of 14


(iv)

for all x, y ∈ X and � ∈ Ψ.


Remark 2.9 By choosing point-dependent control functions �, �, �, �, �,�, �, function � and mappings S and T suitably in Theorem 2.3, Corollaries 2.4, 2.5 and 2.8, one can deduce a multitude of results from the existing literature which includes the celebrated Banach fixed point theorem for multival-ued mappings in complex valued complete metric spaces as well as in complete metric spaces.

Following example substantiates the validity of hypothesis of our main Theorem 2.3.

Example 2.10 Let X = [0, 1]. Define d : X × X → C as follows:

Then (X, d) is a complete complex valued metric space. Consider the mappings S, T : X → CB(X) such that

Sx = [0,x

4] and Ty = [0,

y

4] for all x, y ∈ X.

Next, we define the functions �, �, �, �, �,�, � : X → [0, 1) by �(x) = x+1

3, �(x) = x

20, �(x) = x

10, �(x) = x

50, �(x) = x

25, �(x) = x

30 and �(x) = x

25.

Undoubtedly

(i) �(u) ≤ �(x), �(u) ≤ �(x), �(u) ≤ �(x), �(u) ≤ �(x), �(u) ≤ �(x),�(u) ≤ �(x) and �(u) ≤ �(x), for all u ∈ Sx = [0,

x

4] and ∀x ∈ [0, 1);

(ii) �(v) ≤ �(x), �(v) ≤ �(x), �(v) ≤ �(x), �(v) ≤ �(x), �(v) ≤ �(x),�(v) ≤ �(x) and �(v) ≤ �(x), for all v ∈ Tx = [0,

x

4] and ∀x ∈ [0, 1);

(iii) 𝜆(x) + 𝜂(x) + 𝛿(x) + 2𝜉(x) + 𝜈(x) + 𝜇(x) + 𝛾(x) < 1, ∀x ∈ [0, 1).

Also define the function � ∈ Ψ by �(t) = 3t

4.

Calculating the various functions involving in our contractive condition.

Now consider,

(2.6)

H(Sx, Ty) ≤�(�(x)d(x, y) + �(x)d(x, Sx) + �(x)d(y, Ty) + �(x)

d(x, Sx)d(x, Ty)

1 + d(x, y)

+ �(x)d(y, Sx)d(y, Ty)

1 + d(x, y)+ �(x)

d(x, Sx)d(y, Ty)

1 + d(x, y)+ �(x)

d(y, Sx)d(x, Ty)

1 + d(x, y)

),

d(x, y) = |x − y|ei�

6 .

d(x, y) = |x − y|ei�

6 ,

d(x, Sx) = |x − x

4|ei

�

6 ,

d(y, Ty) = |y −y

4|ei

�

6 ,

d(y, Sx) = |y − x

4|ei

�

6 ,

d(x, Ty) = |x −y

4|ei

�

6 ,

s(Sx, Ty) = s(| x4−y

4|ei

�

6

).

of 14


Clearly , for �(x) = x+1

3, �(x) = x

20, �(x) = x

10, �(x) = x

50, �(x) = x

25, �(x) = x

30 and �(x) = x

25 and for all

x, y ∈ [0, 1], we have

Since, one can easily calculate that

And all the remaining terms on the right hand side of Inequality (2.7) are non-negative for all x ∈ X. Consequently, one can obtain

Hence, all the conditions of Theorem 2.3 are satisfied and x = 0 remains fixed under mappings S and T.

3. Slip-ups in some recent papers and their remediesThe motivation of this section is to point out some slip-ups in the examples of some recent papers Ahmad et al. (2013) and Kutbi et al. (2014) in complex valued metric spaces.

In above-mentioned papers, the authors claimed that the function d : X × X → C defined by

where � = tan−1 | yx| and x, y ∈ X = [0, 1], is a complex valued metric which is not a reality.

Unfortunately the function d(x, y) described by the Equation 3.1 is not a complex valued metric in its present form. It is neither symmetric nor enjoys the triangular inequality which amounts to say that the other calculations in the examples are incorrect so that these examples do not illustrate the concerned theorems as claimed by the authors.

Now following examples are furnished which substantiate our viewpoints.

To substantiate the claim, consider x, y ∈ X = [0, 1] and define a function d : X × X → C as d(x, y) = |x − y|ei� , where � = tan−1 | y

x|. Then

and

so that

3

4

(�(x)|d(x, y)| + �(x)|d(x, Sx)| + �(x)|d(y, Ty)| + �(x)

|d(x, Sx)||d(x, Ty)||1 + d(x, y)|

+ �(x)|d(y, Sx)||d(y, Ty)|

|1 + d(x, y)|+ �(x)

|d(x, Sx)||d(y, Ty)||1 + d(x, y)|

+ �(x)|d(y, Sx)||d(x, Ty)|

|1 + d(x, y)|

),

(2.7)

| x4−y

4| ≤ 3

4

[(x + 1

3

)|x − y| +

(x

20

)|x − x

4| +

(x

10

)|y −

y

4| +

(x

50

) |x − x

4||x − y

4|

|1 + d(x, y)|

+

(x

25

) |y − x

4||y − y

4|

|1 + d(x, y)|+

(x

30

) |x − x

4||y − y

4|

|1 + d(x, y)|+

(x

25

) |y − x

4||x − y

4|

|1 + d(x, y)|

],

|x4−y

4| ≤ 3

4

(x + 1

3

)|x − y| for all x ∈ X.

�

(�(x)d(x, y) + �(x)d(x, Sx) + �(x)d(y, Ty) + �(x)

d(x, Sx)d(x, Ty)

1 + d(x, y)

+ �(x)d(y, Sx)d(y, Ty)

1 + d(x, y)+ �(x)

d(x, Sx)d(y, Ty)

1 + d(x, y)+ �(x)

d(y, Sx)d(x, Ty)

1 + d(x, y)

)∈ s(Sx, Ty).

(3.1)d(x, y) = |x − y|ei� ,

(3.2)d(x, y) = |x − y|ei tan−1 | y

x|

(3.3)d(y, x) = |y − x|ei tan−1 | x

y|,

of 14


Triangular inequality for complex valued metric spaces runs as follows:

If we invoke the defined function d in above inequality, then the partial ordering due to Azam et al. (2011) does not hold good always.

For example, if we choose x = 0, y = 1, z =1

2, then

and

Thus

and similarly for some other values of x, y and z, we can show that triangular inequality is not satis-fied by d. Thus (X, d) is not a complex valued metric space.

Rectification:

In order to overcome the aforementioned drawbacks, we propose suitable but different rectifica-tions for both the papers separately. Firstly, in Example 14 of Ahmad et al. (2013) if we define d : X × X → C by

Then (X, d) is a complex valued metric space. If we consider the same mappings as in Example 14 of Ahmad et al. (2013) with the substitution � =

�

4, then this example demonstrates the validity of

the hypothesis of Theorem 9 of Ahmad et al. (2013).

Finally, Example 24 of Kutbi et al. (2014) can be repaired as follows:

Example 3.1 Let X = [0, 1]. Define dc: X × X → C by

Then dc(�, �) is a complex valued metric space. Carrying out routine calculation on the lines of Example 24 of Kutbi et al. (2014) under the restriction � =

�

5, one can demonstrate Theorem 9 of

Kutbi et al. (2014). Notice that � = 0 is a common fixed point of S and F.

d(x, y) ≠ d(y, x) as ei tan−1 | y

x|≠ e

i tan−1 | xy|, ∀x, y ∈ X and for x ≠ y.

d(x, y) ≾ d(x, z) + d(z, y), for all x, y, z ∈ X.

d(x, y) = |0 − 1|ei tan−1 | 1

0|

= ei tan−1

∞

= ei�

2 = i,

d(z, y) =|||1

2− 1

|||ei tan−1 |2|

= 0.2236 + 0.4472i

d(x, z) =|||1

2

|||ei tan−1 |∞|

=i

2.

d(x, y)≾d(x, z) + d(z, y), for all x, y, z ∈ X.

d(x, y) = |x − y|ei.�

4 , ∀x, y ∈ X = [0, 1].

dc(�, �) = |� − �|ei.�

5 , ∀�, � ∈ X.

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Remark 3.2 From the preceding discussions, we infer that in order to hold the validity of aforesaid results mentioned in Ahmad et al. (2013) and Kutbi et al. (2014), we can take many �s ranging in [0,

�

4].

AcknowledgementsThe authors are grateful to the learned referees for their accurate reading and their helpful suggestions.

FundingThe authors received no direct funding for this research.

Author detailsVishal Joshi1

E-mail: [email protected] Singh2

E-mail: [email protected] Singh3

E-mail: [email protected] Department of Mathematics, Jabalpur Engineering College,

Jabalpur, Madhya Pradesh, India.2 Government Science and Commerce College, Benazeer,

Bhopal, Madhya Pradesh, India.3 Department of Applied Sciences, NITTTR, Under Ministry of

HRD, Government of India, Bhopal 462002, India.

Citation informationCite this article as: φ-contractive multivalued mappings in complex valued metric spaces and remarks on some recent papers, Vishal Joshi, Naval Singh & Deepak Singh, Cogent Mathematics (2016), 3: 1162484.

ReferencesAhmad, J., Klin-Eam, C., & Azam, A. (2013). Common fixed

points for multivalued mappings in complex valued

metric spaces with applications. Abstract and Applied Analysis, 2013, 12, Article ID 854965.

Azam, A., Fisher, B., & Khan, M. (2011). Common fixed point theorems in complex valued metric spaces. Numerical Functional Analysis and Optimization, 32, 243–253.

Azam, A., Ahmad, J., & Kumam, P. (2013). Common fixed point theorems for multi-valued mappings in complex valued metric spaces. Journal of Inequality and Applications, 2013, Article 578, 12 p.

Kutbi, M. A., Ahmad, J., Azam, A., & Al-Rawashdeh, A. S. (2014). Generalized common fixed point results via greatest lower bound property. Journal of Applied Mathematics, 2014, Article ID 265865.

Rouzkard, F., & Imdad, M. (2012). Some common fixed point theorems on complex valued metric spaces. Computers & Mathematics with Applications, 64, 1866–1874.

Singh, N., Singh, D., Badal, A., & Joshi, V. (in press). Fixed point theorems in complex valued metric space. Journal of the Egyptian Mathematical Society.

Sintunavarat, W., & Kumam, P. (2012). Generalized common fixed point theorems in complex valued metric spaces and applications. Journal of Inequalities and Applications, 2012, Article ID 84.

Sintunavarat, W., Cho, Y. J., & Kumam, P. (2013). Urysohn integral equations approach by common fixed points in complex valued metric spaces. Advances in Difference Equations, 2013, Article 49, 14 p.

Sitthikul, K., & Saejung, S. (2012). Some fixed point theorems in complex valued metric space. Fixed Point Theory and Applications, 2012, Article ID 189.




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