Applied Mathematics and Computation 232 (2014) 542–547

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Some new dynamic Opial type inequalities and applicationsfor second order integro-differential dynamic equations on timescales q

http://dx.doi.org/10.1016/j.amc.2014.01.1360096-3003/� 2014 Elsevier Inc. All rights reserved.

q This research was supported by the NNSF of China (10971139), the NSF of Shandong Province (ZR2012AL03) and the Shandong EducationCollege Scientific Research (J11LA51).⇑ Corresponding author at: School of Science, Jiangnan University, Wuxi 214122, Jiangsu, PR China.

E-mail addresses: llz3497@163.com (L. Li), mahan@shnu.edu.cn (M. Han).

Lianzhong Li a,b,⇑, Maoan Han c

a School of Science, Jiangnan University, Wuxi 214122, Jiangsu, PR Chinab School of Mathematics and Statistics, Taishan University, Tai’an, Shandong 271021, PR Chinac Department of Mathematics, Shanghai Normal University, Shanghai 200234, PR China

a r t i c l e i n f o a b s t r a c t

Keywords:Opial type inequalityIntegro-differential dynamic equationTime scaleDisfocalDisconjugate

In this paper, we give some new Opial type inequalities on time scales, at the same time westudy the similar problems as disfocal and disconjugacy properties of solutions to secondorder integro-differential dynamic equations on time scales.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Since its discovery more than four decades ago, Opial’s inequality has found many interesting applications. In fact, Opial’sinequality and its several generalizations, extensions and discretizations, play a fundamental role in the theories of differ-ential and difference equations. The readers may find very interesting results about continuous and discrete version of Opi-al’s inequality in book [1].

In the recent years, theory of time scales has received a lot of attention which was introduced by Stefan Hilger in his Ph.D.thesis in 1988 in order to unify continuous and discrete analysis. A book on the subject of time scales, by Bohner and Pet-erson [2] summarizes and organizes much of time scales calculus, for the details, can also see the book by Bohner and Pet-erson [3] for advances in dynamic equations on time scales.

Bohner and Kaymakçalan [4] initiated the time scales unification of a continuous and a discrete analog of a version ofOpial’s inequality and illustrated some applications of it to dynamic equations on time scales. It may be asserted that thissubject will continue to play a very important part in the future of applied mathematics.

2. Other mathematician’s work on dynamic Opial type inequalities

In 2001, Agarwal and Bohner [5] establish the Opial type inequality on time scale.

Theorem 2.A (Dynamic Opial Inequality). Let T be a time scale. For delta differentiable x : ½0;h�T

T! R with xð0Þ ¼ 0, we have

Fund for

L. Li, M. Han / Applied Mathematics and Computation 232 (2014) 542–547 543

Z h

0jðxþ xrÞxDjðtÞDt 6 h

Z h

0jxDj2ðtÞDt: ð2:1Þ

In 2010, Karpuz et al. [6] give the following Opial type inequality.

Theorem 2.B. Let a; b 2 T and x; f 2 C1rdð½a; b�T;RÞ with f ðaÞ ¼ 0, then we have

Z b

axðgÞj½f ðgÞ þ f rðgÞ�f DðgÞjDg 6 Kxðb; aÞ

Z b

ajf DðgÞj2Dg; ð2:2Þ

where

Kxðt; sÞ :¼ 2Z t

s½xðgÞ�2½rðgÞ � s�Dg

� �1=2

for s; t 2 ½a; b�T:

They also give other forms of the Opial inequalities (see paper [6] and references therein), and study the disfocal and discon-jugacy properties of solutions to second order dynamic equations on time scales, we do not list all of them (one can also see[7,8]).

In the following, we need the Hölder’s inequality on time scale.

Theorem 2.C (Dynamic Hölder’s inequality [4]). Let T a time scale, a; b 2 T. For f ; g 2 Crdð½a; b�T;RÞ, we have

Z b

ajf ðxÞgðxÞjDx 6

Z b

ajf ðxÞjpDx

!1p Z b

ajgðxÞjqDx

!1q

; ð2:3Þ

where p > 1 and 1=pþ 1=q ¼ 1.In this paper, we point out some mistakes of Opial type inequalities on time scales in Wong’s paper [9], we revise and give

some new generalizations of them. We also study the similar problems as disfocal and disconjugacy properties (see [10]) ofsolutions to second order integro-differential dynamic equations on time scales.

3. Opial type inequalities on time scales

Unlike the above inequalities, in a paper [9] published 2008, Wong et al. established the following result.

Theorem 3.D [9]. Let f : ½a; b� ! R with f ðaÞ ¼ 0 be delta differentiable, h 2 Crdð½a; b�; ½1;1ÞÞ :¼ fhjh : ½a; b� ! ½1;1Þis a rd-continuous functiong; p P 0 and q P 1. Then,

Z b

ahðxÞjf ðxÞjpjf DðxÞjqDx 6

qpþ q

ðb� aÞpZ b

ahðxÞjf DðxÞjðpþqÞDx: ð3:4Þ

In fact, we find the above Theorem 3.D is not correct, so the main results (Theorem 2.2, 2.3, 2.4 [9]) are not true since theyare based on Theorem 3.D (Theorem 2.1 in [9]), we illustrate this by a simple example.

Example 3.1. Let T ¼ R; a ¼ 0; b > 0; p ¼ q ¼ 1, for f ðxÞ ¼ x; hðxÞ ¼ 1þ x be real functions, obviously all the conditions inTheorem 3.D satisfied, but

Z b

ahðxÞjf ðxÞjpjf DðxÞjqDx ¼

Z b

0ð1þ xÞxdx ¼ b2 1

2þ 1

3b

� �;

qpþ q

ðb� aÞpZ b

ahðxÞjf DðxÞjðpþqÞDx ¼ 1

2bZ b

0ð1þ xÞdx ¼ b2 1

2þ 1

4b

� �:

Since b > 0, then

Z b

ahðxÞjf ðxÞjpjf DðxÞjqDx >

qpþ q

ðb� aÞpZ b

ahðxÞjf DðxÞjðpþqÞDx:

So Theorem 3.D is not correct. In fact, from the proof in paper [9], we can see that the results are true whenhðtÞ � 1; t 2 ½a; b�T, it is to say, we have the following results.

544 L. Li, M. Han / Applied Mathematics and Computation 232 (2014) 542–547

3.1. The case with hðtÞ � 1; t 2 ½a; b�T

Theorem 3.1. Let T be a time scale, a; b 2 T. For x : ½a; b�T ! R with xðaÞ ¼ 0 be delta differentiable, p P 0; q P 1be realconstants. Then,

Z b

ajxðtÞjpjxDðtÞjqDt 6

qpþ q

ðb� aÞpZ b

ajxDðtÞjðpþqÞDt: ð3:5Þ

Theorem 3.2. Suppose T; a; b; p; q be as in Theorem 3.1. For x : ½a; b�T ! R be n-times delta differentiable withxðaÞ ¼ xDðaÞ ¼ � � � ¼ xDn�1 ðaÞ ¼ 0. Then,

Z b

ajxðtÞjpjxDn ðtÞjqDt 6

qpþ q

ðb� aÞnpZ b

ajxDnðtÞjðpþqÞDt: ð3:6Þ

Theorem 3.3. Suppose T; a; b; p; q be as in Theorem 3.1. For x; y : ½a; b�T ! R be n�times delta differentiable withxðaÞ ¼ xDðaÞ ¼ � � � ¼ xDn�1 ðaÞ ¼ 0 and yðaÞ ¼ yDðaÞ ¼ � � � ¼ yDn�1 ðaÞ ¼ 0. Then,

Z b

ajxðtÞjpjyDn ðtÞjq þ jyðtÞjpjxDn ðtÞjqh i

Dt 62q

pþ qðb� aÞnp

Z b

ajxDnðtÞjðpþqÞ þ jyDn ðtÞjðpþqÞh i

Dt: ð3:7Þ

Theorem 3.4. Suppose T; a; b; p; q be as in Theorem 3.1. For x; y : ½a; b�T ! R be n�times delta differentiable withxðaÞ ¼ xDðaÞ ¼ � � � ¼ xDn�1 ðaÞ ¼ 0; xðbÞ ¼ xDðbÞ ¼ � � � ¼ xDn�1 ðbÞ ¼ 0; yðaÞ ¼ yDðaÞ ¼ � � � ¼ yDn�1 ðaÞ ¼ 0 andyðbÞ ¼ yDðbÞ ¼ � � � ¼ yDn�1 ðbÞ ¼ 0. If ðaþbÞ

2 2 ½a; b�T, then,

Z b

ajxðtÞjpjyDn ðtÞjq þ jyðtÞjpjxDn ðtÞjqh i

Dt 62q

pþ qb� a

2

� �np Z b

ajxDnðtÞjðpþqÞ þ jyDn ðtÞjðpþqÞh i

Dt: ð3:8Þ

3.2. The case with arbitrary hðtÞ; t 2 ½a; b�T

Now we give some Opial type inequalities on time scales including arbitrary function hðtÞ.

Theorem 3.5. Suppose T; a; b; p; q; x be as in Theorem 3.1, moreover, suppose q P p, h 2 Crdð½a; b�T;RÞ :¼fhjh : ½a; b�T ! R is a rd-continuous functiong. Then,

Z b

ahðtÞjxðtÞjpjxDðtÞjqDt 6

pq

Kp�q

q

Z b

ajhðtÞj2ðt � aÞ

2pðq�1Þþqq Dt

!12 Z b

ajxDðtÞj2qDt

þ q� pq

Kpq

Z b

ajhðtÞj2ðt � aÞ

2pðq�1Þq Dt

!12 Z b

ajxDðtÞj2qDt

!12

ð3:9Þ

for any constant K > 0.Before giving the proof, we discuss the special cases of the theorem. case I: p ¼ 0; q ¼ 1. Using (2.3) with indices 2 and 2,

the conclusion will be

Lemma 3.1. Suppose T; a; b; x;h be as in Theorem 3.5, then

Z b

ahðtÞjxDðtÞjDt 6

Z b

ajhðtÞjjxDðtÞjDt 6

Z b

ajhðtÞj2Dt

!12 Z b

ajxDðtÞj2Dt

!12

: ð3:10Þ

case II: p ¼ 1; q ¼ 1.

Lemma 3.2. Suppose T; a; b; x;h be as in Theorem 3.5, then

Z b

ahðtÞjxðtÞjjxDðtÞjDt 6

Z b

ajhðtÞj2ðt � aÞDt

!12 Z b

ajxDðtÞj2Dt: ð3:11Þ

Proof. Consider yðtÞ ¼R t

a jxDðsÞjDs, then yDðtÞ ¼ jxDðtÞj and jxðtÞj 6 yðtÞ, so

Z b

ahðtÞjxðtÞjjxDðtÞjDt 6

Z b

ajhðtÞjyðtÞyDðtÞDt:

L. Li, M. Han / Applied Mathematics and Computation 232 (2014) 542–547 545

Since yðtÞ ¼R t

a yDðsÞDs 6 ðt � aÞ12R t

ajyDðsÞj2Ds� �1

2, we have

Z b

ahðtÞyðtÞyDðtÞjDt 6

Z b

ajhðtÞjðt � aÞ

12

Z t

ajyDðsÞj2Ds

� �12

yDðtÞDt 6Z b

ajyDðtÞj2Dt

!12 Z b

ajhðtÞðt � aÞ

12jyDðtÞDt

6

Z b

ajxDðtÞj2Dt

!12 Z b

ajhðtÞðt � aÞ

12j2Dt

!12 Z b

ajxDðtÞj2Dt

!12

¼Z b

ajhðtÞj2ðt � aÞDt

!12 Z b

ajxDðtÞj2Dt;

where we have used (2.3) and (3.10), then the proof is completed. h

Now we prove Theorem 3.5.

Proof. Using (2.3) with indices q and qq�1, we have

jxðtÞj 6Z t

ajxDðsÞjDs 6 ðt � aÞ

q�1q

Z t

ajxDðsÞjqDs

� �1q

:

Let yðtÞ ¼R t

a jxDðsÞjqDs, then,

yDðtÞ ¼ jxDðtÞjq; jxðtÞjp 6 ðt � aÞpðq�1Þ

q ypqðtÞ;

and

Z b

ahðtÞjxðtÞjpjxDðtÞjqDt 6

Z b

ajhðtÞjðt � aÞ

pðq�1Þq y

pqðtÞyDðtÞDt 6

Z b

ajhðtÞjðt � aÞ

pðq�1Þq

pq

Kp�q

q yðtÞ þ q� pq

Kpq

� �yDðtÞDt

6pq

Kp�q

q

Z b

ajhðtÞj2ðt � aÞ

2pðq�1Þþqq Dt

!12 Z b

ajxDðtÞj2qDt

þ q� pq

Kpq

Z b

ajhðtÞj2ðt � aÞ

2pðq�1Þq Dt

!12 Z b

ajxDðtÞj2qDt

!12

for any constant K > 0, where we have used (3.10), (3.11) and the inequality

apq 6

pq

Kp�q

q aþ q� pq

Kpq for any constant K > 0;

where a P 0; q P p > 0. The proof is completed. h

Similar discussions can give the following result, we omit the proof.

Theorem 3.6. Suppose T; a; b; p; q; x be as in Theorem 3.5, except that xðaÞ ¼ 0 is replaced by xðbÞ ¼ 0. Then,

Z b

ahðtÞjxðtÞjpjxDðtÞjqDt 6

pq

Kp�q

q

Z b

ajhðtÞj2ðb� tÞ

2pðq�1Þþqq Dt

!12 Z b

ajxDðtÞj2qDt

þ q� pq

Kpq

Z b

ajhðtÞj2ðb� tÞ

2pðq�1Þq Dt

!12 Z b

ajxDðtÞj2qDt

!12

ð3:12Þ

for any constant K > 0.Combining Theorem 3.5 and Theorem 3.6, we get

Theorem 3.7. Suppose T; a; b; p; q; x be as in Theorem 3.5, except that xðaÞ ¼ 0 is replaced by xðaÞ ¼ xðbÞ ¼ 0. Then,

Z b

ahðtÞjxðtÞjpjxDðtÞjqDt 6

pq

Kp�q

q

Z c

ajhðtÞj2f1ðtÞDt

� �12

þZ b

cjhðtÞj2g1ðtÞDt

!12

24

35Z b

ajxDðtÞj2qDt

þ q� pq

Kpq

Z c

ajhðtÞj2f2ðtÞDt

� �12

þZ b

cjhðtÞj2g2ðtÞDt

!12

24

35 Z b

ajxDðtÞj2qDt

!12

ð3:13Þ

for any constant K > 0 and for any c 2 ða; bÞT, where f1ðtÞ ¼ ðt � aÞ2pðq�1Þþq

q ; f 2ðtÞ ¼ ðt � aÞ2pðq�1Þ

q ; g1ðtÞ ¼ ðb� tÞ2pðq�1Þþq

q ;

g2ðtÞ ¼ ðb� tÞ2pðq�1Þ

q .Especially, we have

546 L. Li, M. Han / Applied Mathematics and Computation 232 (2014) 542–547

Corollary 3.1. Suppose T; a; b; p; q; x be as in Theorem 3.7, then,

Z b

ahðtÞjxðtÞjpjxDðtÞjqDt 6 min

c2½a;b�T

pq

Kp�q

q

Z c

ajhðtÞj2f1ðtÞDt

� �12

þZ b

cjhðtÞj2g1ðtÞDt

!12

24

35

8<:

Z b

ajxDðtÞj2qDt þ q� p

qK

pq

Z c

ajhðtÞj2f2ðtÞDt

� �12

þZ b

cjhðtÞj2g2ðtÞDt

!12

24

35 Z b

ajxDðtÞj2qDt

!12

9=; ð3:14Þ

for any constant K > 0, where f1ðtÞ; f 2ðtÞ; g1ðtÞ; g2ðtÞ be as in Theorem 3.7.

Remark 3.1. Choose c ¼ a, Theorem 3.7 reduces to Theorem 3.6, while choose c ¼ b, Theorem 3.7 reduces to Theorem 3.5.

Remark 3.2. Choose p ¼ 0; q ¼ 1 or p ¼ q ¼ 1 in Theorems 3.6, 3.7, we get corollaries similar to Lemmas 3.1, 3.2, they areleft to the readers.

4. Disfocal problem and disconjugacy condition

In this section, we consider the second order integro-differential dynamic equation

yD2 þZ t

akðsÞyðsÞDs ¼ 0; t 2 ½a; b�T; ð4:15Þ

where a 6 t 6 b 2 T; k 2 Crdð½a; b�T; RÞ satisfyingR b

a jkðtÞjDt <1.By a solution of (4.15), we mean a continuous function y : ½a;2ðbÞ�T ! R, which is twice differentiable on ½a; b�T and sat-

isfying Eq. (4.15), with yD2 rd-continuous. We say y has a generalized zero at some c 2 ½a;rðbÞ�T provided that yðcÞyrðcÞ 6 0holds, and (4.15) is called disconjugate on ½a; b� if there is no nontrivial solution of (4.15) with at least two generalized zerosin ½a; b�. Finally, (4.15) is said to be disfocal on ½a;r2ðbÞ� provided there is no nontrivial solution y of (4.15) with a generalizedzero in ½a;r2ðbÞ� followed by a generalized zero of yD in ½a;rðbÞ�.

Now we study the similar problems as disfocal and disconjugacy properties of solutions to (4.15).

Theorem 4.1. Assume that y is a nontrivial solution of (4.15) which satisfies yðaÞ ¼ yDðqðbÞÞ ¼ 0, then,

1 6 ðb� aÞ3Z qðbÞ

a

Z qðbÞ

rðgÞjkðnÞjDn

����������

2

Dg: ð4:16Þ

Proof. Multiplying (4.15) by yr and integrating from a to qðbÞ we get

�Z qðbÞ

ayD2ðtÞyrðtÞDt ¼

Z qðbÞ

ayrðtÞ

Z t

akðsÞyðsÞDs

� �Dt; ð4:17Þ

considering the boundary conditions, we have

�Z qðbÞ

ayD2ðtÞyrðtÞDt ¼ �yDðqðbÞÞyðqðbÞÞ þ yDðaÞyðaÞ þ

Z qðbÞ

aðyDðtÞÞ2Dt ¼

Z qðbÞ

aðyDðtÞÞ2Dt: ð4:18Þ�

Let hðtÞ ¼Z qðbÞ

t

hZ s

akðgÞyðgÞDg Ds, then hðqðbÞÞ ¼ 0; hDðtÞ ¼ �

Z t

akðsÞyðsÞDs. Integrating by parts and Using (3.10), we have

hðtÞj j ¼Z qðbÞ

ayðgÞ

Z qðbÞ

gkðgÞDs

" #Dg

����������¼ R qðbÞ

a yðgÞZ qðbÞ

g

Z qðbÞ

nkðnÞDsDn

" #D

Dg

������������

¼ yðgÞZ qðbÞ

g

Z qðbÞ

nkðnÞDsDn

" #g¼qðbÞ

g¼a

�Z qðbÞ

ayDðgÞ

Z qðbÞ

rðgÞ

Z qðbÞ

nkðnÞDsDn

" #Dg

������������

6

Z qðbÞ

ayDðgÞ�� �� Z qðbÞ

rðgÞ

Z qðbÞ

ajkðnÞjDsDn

����������Dg

6

Z qðbÞ

a

Z qðbÞ

rðgÞ

Z qðbÞ

ajkðnÞjDsDn

����������2

Dg

24

35

12 Z qðbÞ

ayDðgÞ�� ��2Dg

" #12

6 ðb�aÞZ qðbÞ

a

Z qðbÞ

rðgÞjkðnÞjDn

����������2

Dg

24

35

12 Z qðbÞ

ayDðgÞ�� ��2Dg

" #12

ð4:19Þ

L. Li, M. Han / Applied Mathematics and Computation 232 (2014) 542–547 547

and

Z qðbÞ

ayrðtÞ

Z t

akðsÞyðsÞDs

� �Dt ¼ �

Z qðbÞ

ayrðtÞhDðtÞDt ¼ �yðqðbÞÞhðqðbÞÞ þ yðaÞhðaÞ þ

Z qðbÞ

ahðtÞyDðtÞDt

6

Z qðbÞ

ajhðtÞj2Dt

" #12 Z qðbÞ

ayDðtÞ�� ��2Dt

" #12

6 ðb� aÞ32

Z qðbÞ

a

Z qðbÞ

rðgÞjkðnÞjDn

����������2

Dg

24

35

12 Z qðbÞ

ayDðtÞ�� ��2Dt; ð4:20Þ

where we have used (4.19).From (4.22), (4.18) and (4.20), we get

Z qðbÞ

ayDðtÞ�� ��2Dt 6 ðb� aÞ

32

Z qðbÞ

a

Z qðbÞ

rðgÞjkðnÞjDn

����������

2

Dg

24

35

12 Z qðbÞ

ayDðtÞ�� ��2Dt ð4:21Þ

and (4.16) follows from (4.21). h

Theorem 4.2. Assume that y is a nontrivial solution of (4.15) which satisfiesyðaÞ ¼ yðbÞ ¼ 0, then,

1 6 ðb� aÞ3Z b

a

Z b

rðgÞjkðnÞjDn

����������2

Dg: ð4:22Þ

Proof. The Proof is similar to the proof of Theorem 4.1 except that all the ‘‘qðbÞ’’ are replaced by ‘‘b’’. h

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