Extremal Solutions to Periodic Boundary Value Problem of Nabla Integrodifferential Equation of Volterra Type on Time Scales

Abstract

We firstly establish some new theorems on time scales, and then, by employing them together with a new comparison result and the monotone iterative technique, we show the existence of extremal solutions to the following nabla integrodifferential periodic boundary value problem: ,  , where is a time scale.

1. Introduction

In this paper, we are concerned with the existence of extremal solutions to the following nabla integrodifferential periodic boundary value problem: where is a time scale and satisfy . By proving a new comparison result and developing the monotone iterative technique, we show the extremal solutions of the periodic boundary value problem of nabla integrodifferential equations of Volterra type on time scales.

The study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which may be an arbitrary closed subset of the reals. Many results on this issue have been well documented in the monographs [1, 2] written by Bohner and Peterson. Moreover, an integrodifferential equation on time scales (including time scale ) finds many applications in various mathematical problems [3]. And this leads to the extensive study of the existence of extremal solutions to such kind of equations; see Agarwal et al. [4], Franco [5], Guo [6], Z. He and X. He [7], Nieto and Rodríguez-López [8], Song [9], Xu and Nieto [10], Xing et al. [11], and the references therein. However, to the best of the authors' knowledge, most of them are of ordinary integrodifferential equations and delta integrodifferential equations on time scales, while the nabla integrodifferential equations on time scales have rarely been considered up to now; the main reason is that the theory on nabla derivatives on time scales is not complete. So, in order to study PBVP (1) to fill the gap, we need firstly to establish some new theorems on time scales, including the Induction Principle and Mean Value Theorem, which are very important for getting our main results, and this will be shown in Section 2.2.

In addition, monotone iterative technique coupled with the method of upper and lower solutions has been widely used in the treatment of existence results of initial and boundary value problems for nonlinear differential equations in recent years. The basic idea is that using the upper and lower solutions as an initial iteration one can construct monotone sequences from a corresponding linear problem, and these sequences converge monotonically to the minimal and maximal solutions of the nonlinear problem. When the method is applied to nabla differential equations on time scales, it needs a suitable nabla differential inequality as a comparison principle; this will be shown in Section 3.

For some other work on time scales, we refer the readers to Aderson [12], Agarwal et al. [13], Tisdell et al. [14, 15], and the references therein.

We will assume the following throughout: by we mean that , where . And we denote by .

2. Preliminary

2.1. Some Definitions and Lemmas

For convenience, in this subsection, we give some definitions and lemmas on time scales, which can be found in book [1, 2].

Definition 1 (see [1, page 1]). Let be a time scale. For , one defines the forward jump operator by while one defines the backward jump operator by

Definition 2 (see [2, page 47]). If has a right-scattered minimum , define ; otherwise, set . The backward graininess function is defined by

Definition 3 (see [1, page 47]). For and , define the nabla derivative of at , denoted by , to be the number (provided it exists) with the property that, given any , there is a neighborhood of such that for all .

Definition 4 (see [1, page 48]). The function is -regressive if Define the -regressive class of functions on to be If , then we define circle plus addition by

Definition 5 (see [1, page 48]). For , define circle minus by

Definition 6 (see [1, page 49]). For , let Define the -cylinder transformation by where Log is the principle logarithm function. For , we define for all .

Definition 7 (see [1, page 49]). If , then we define the nabla exponential function by where the -cylinder transformation is as in Definition 6.

Remark 8. From Definitions 6 and 7 we know that implies .

Lemma 9 (see [1, page 51]). Let and . Then one has the following: (i) and ;(ii); (iii); (iv); (v); (vi); (vii); (viii).

Lemma 10 (see [1, page 48]). Assume that are nabla differentiable at . Then (i)the sum is nabla differentiable at with (ii)the product is nabla differentiable at , and the product rules (iii)if , then is nabla differentiable at , and we get the quotient rule (iv)if and are continuous, then

2.2. Some New Results on Time Scales

In order to get Theorem 16 which plays an important role in getting our main results, in this subsection, we need firstly to establish Lemmas 11, 14, and 15. The counterpart about delta derivatives can be found in book [1, 2].

Lemma 11 (left-forward induction principle). Let and assume that is a family of statements satisfying the following. (i)The statement is true.(ii)If is left-scattered and is true, then is also true.(iii)If is left-dense and is true, then there is a neighborhood of such that is true for all .(iv)If is right-dense and is true for all , then is true. Then is true for all .

Proof. Let We want to show . To achieve a contradiction we assume . But since is closed and nonempty, we have We claim that is true. If , then is true from (I). If and , then is true from (IV). Finally, if , then is true from (II). Hence, in any case, Thus, cannot be left-scattered (as ), and (or ). Hence is left-dense. But now (III) leads to a contradiction. The proof is complete.

Next, for convenience, we give a definition.

Definition 12. A continuous function is called pre-nabla-differentiable with (region of differentiation) , provided is countable and contains no left-scattered elements of , and is nabla differentiable at each .

Remark 13. This is an example; let and let be defined by Then is pre-nabla-differentiable with

Lemma 14 (Mean Value Theorem). Let and be real-valued functions defined on and both pre-nabla-differentiable with ; then implies

Proof. Let , with and denote . Let ; we now show by induction that holds for all . Note that, once we have shown this, the claim of the Mean Value Theorem follows. We now check the conditions given in Lemma 11 as follows.
(I) The statement is trivially satisfied.
(II) Let be left-scattered and assume that holds. Then and Therefore is true.
(III) Suppose is true and is left-dense; that is, . We consider two cases; namely, and . First of all, suppose . Then and are differentiable at and hence there exists a neighborhood of with Thus That is, Hence we have for all Thus is true for all .
For the second case, suppose . Then for . Since and are pre-nabla-differentiable, they are continuous and hence there exists a neighborhood of with Therefore That is, and hence Thus again follows for all .
(IV) Now let be right-dense and suppose that is true for . Then implies that is true as both and are continuous at .

Lemma 15. Suppose is pre-nabla-differentiable with and is a compact interval with endpoints ; then

Proof. Suppose is pre-nabla-differentiable with and with . Defining then By Lemma 14, we get so that This completes the proof.

Theorem 16. Suppose is pre-nabla-differentiable with for each . Assume that for each there exists a compact interval neighborhood such that the sequence Then the limit mapping is predifferentiable with and one has

Proof. Let . Without loss of generality we can assume that . Letting , there exists such that Also by Lemma 15 holds for all , and . Since converges uniformly on , there exists such that Hence, for all , and so that, by letting , for all , and . Let Then there exists such that and since is nabla differentiable at , there also exists a neighborhood of with Altogether we have now, for all , which implies that is nabla differentiable at with ; that is, ; the proof is complete.

3. Some Important Lemmas

In this section, we will give some lemmas which are important for the main results.

Lemma 17 (comparison result). Suppose that there is a function and . Assume that there exist a positive function and a nonnegative function on , such that and Then for all provided that , where ,  ,  .

Proof. Denote . By Definition 5 and (52), we know that ; thus, by Lemma 9, we have Next, we try to show that for all . Otherwise, we have one of the following two cases: (i) for all and ;(ii)there exists such that and .
In case , since and , we have ; also , so, by Remark 8, we have . Together with it follows that .
On the other hand, from (53), we have which implies that is nonincreasing and hence . Thus, we have . Therefore, from , we have for all . Since is positive and decreasing on holds. This is a contradiction.
In case (ii), we have two subcases: When holds, suppose that with , and we claim that there exists such that . Otherwise, for all ; it follows that which is a contradiction. Further, Thus, . Then it follows, together with , that , provided that . It is a contradiction and therefore cannot occur.
When holds, we have . Thus Let , and choose such that with . As in , we can similarly prove that there exists a such that . On the other hand, deducing as before, we have Thus it follows that , which contradicts the condition .
To sum up, we have for all . Thus, for all . The proof is completed.

Lemma 18 (comparison result). Suppose that ,  , and satisfy all the conditions in Lemma 17 and satisfy (52); then for all provided that , where are defined as in Lemma 17, and .

Proof. If the conclusion is not true, we have one of the following two cases: (i) for all and ;(ii)there exist such that and .In case (i), we have which implies that is nonincreasing and so . Then it follows from that , where is a constant. Hence . On the other hand, where . This is a contradiction.
In case (ii), we have two subcases: When holds, suppose that with , and we claim that there exists a such that . Otherwise for all ; it therefore follows that which is a contradiction. On the other hand, Then, together with , it follows that It is a contradiction and therefore does not hold. Similarly, we can prove that case is also wrong. Hence the proof is complete.

Lemma 19 (existence result). Assume that all the assumptions on , and in Lemma 17 are satisfied. Then, for any , the periodic boundary value problem has a unique solution provided , where

Proof. We will prove the conclusion by Banach contraction Principle. First, define a Banach Space as follows: We define an operator on as For any two functions , there holds where the integral Thus we have This implies by condition that is a contraction operator on and therefore by Banach Contraction Principle there exists exactly one such that ; that is, Next, we will show that is a solution of (67). In fact, Moreover, by Lemma 9(iii), there holds Thus, the proof is complete.

Lemma 20 (compact result). Assume that is a function sequence on satisfying the following conditions: (i) is bounded on ;(ii) is bounded on .Then there is a subsequence of that converges uniformly on .

Proof. From the assumption, there exists a positive number such that for all and . Since is bounded, we can choose such that , where and .
If there exists some , which means and , then we can find Since is a closed subset of , it is clear in this case that . Then we can define We claim that In fact, there are two cases to consider.
Case I (). If , then and by (78) we have If , then and by (78) we have
Case II (). We have in this case Next we will show that there exists a subsequence of convergent on . In fact, since is bounded, it has a convergent subsequence . Similarly, is bounded and therefore we can choose a convergent subsequence . If we repeat this process, we get Then is convergent on if we choose . From the above argument, for any , there exists a constant such that for . For any fixed , there is some such that . Thus if we set then when , there holds It therefore follows that is convergent uniformly on . Thus the proof is complete.

4. Main Results

Denote In this section, we will make use of iterative technique to prove the main theorem.

Definition 21. Functions are said to be an upper and a lower solution of (1), respectively, if

Theorem 22. Assume that are a lower and an upper solution of (1), respectively. Further, suppose that there exist two positive functions such that one of the following conditions holds: ; and . And where , and are defined as in Lemma 17; is as defined in Lemma 18. Then PBVP (1) has a maximal solution and a minimal solution in ; moreover, there exist monotone iterative sequences and such that ,   uniformly for .

Proof. Without loss of generality, we suppose that (H) holds. For any , we consider the following nonhomogeneous linear integrodifferential equations on time scales: where . It follows by Lemma 11 that (89) has a unique solution , and We define an operator and will prove that and is increasing in . To prove , we set ; we have Denoting , then, by Definition 21 and (53), we have Thus by Lemma 17 we know that for all , which implies that ; that is, . Essentially, with the same method, we can show that .
To prove (b), we firstly show that is increasing. Letting , then by (88) we have Thus, is bounded. Next, we show that is increasing in . Setting ,  , then Denote . By (88) and (93), we have It follows by Lemma 17 that , which implies that ; that is, is increasing; thus (b) has been proved.
Let and . By (a) and (b), we have Next we will show that both and have convergent subsequences. Let . It is obvious that is a bounded set; in fact, From the above discussion, we have In view of the properties of , and , we know that is bounded on ; by Lemma 20, we know that there exists a subsequence of which converges uniformly on to some . Since is nonincreasing, we see that itself converges uniformly on to . Then, by the continuity of and , we have By the definition of , we know that converges uniformly on to . Thus, we have which implies that is solution to PBVP (1). Essentially, with the same method, we can prove that converges uniformly on to some , and is also a solution to PBVP (1).
Finally, we try to show that and are maximal solution and minimal solution to PBVP (1), respectively, on .
Suppose that is any solution to PBVP (1) in ; then there exists such that . Denoting , then by (88) we have Thus, by Lemma 17, we obtain that for all , which implies that . Similarly, we can show that for all . Consequently, by induction, we have for all . Then, by taking limits, we get   all  , which implies that and are maximal solution and minimal solution to PBVP (1), respectively. The proof is complete.

As an application, we consider the second order PBVP on time scales: where ,  . We make the following assumptions. ()There exist satisfying and ()There exist two positive functions with such that

Theorem 23. Suppose that conditions () and () are satisfied. Then there exist monotone sequences , such that and ,   uniformly on . Letting then are the minimal and maximal solutions of (102) satisfying .

Proof. Let . Then , and therefore BVP (102) reduces to the following PBVP: where . Obviously, this is a PBVP of type (1) with . Hence, the conclusion of Theorem 23 follows immediately from Theorem 22. The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The study was supported by the Fundamental Research Funds for the Central Universities (no. 2652012141) and Beijing Higher Education Young Elite Teacher Project.