Journal of Applied MathematicsVolume 2012 (2012), Article ID 516476, 28 pagesdoi:10.1155/2012/516476

Research Article

Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales

Yongkun Li and Chao Wang

Department of Mathematics, Yunnan University, Yunnan, Kunming 650091, China

Received 20 October 2011; Revised 19 December 2011; Accepted 25 December 2011

Academic Editor: Laurent Gosse

Copyright © 2012 Yongkun Li and Chao Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Using two successive reductions: B-equivalence of the system on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation and by Leggett-Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equation on variable time scales with a transition condition between two consecutive parts of the scale , ,, , where and are parameters, is a variable time scale with -property, , and are -periodic functions of , uniformly with respect to .

1. Introduction

In the last several decades, the theory of dynamic equations on time scales (DETS) has been developed very intensively. For the full description of the equations we refer to the nicely written books [1, 2] and papers [3, 4]. The equations have a very special transition condition for adjoint elements of time scales. To enlarge the field of applications of the DETS, Akhmet and Turan proposed to generalize the transition operator [5], correspondingly to investigate differential equations on variable time scales with transition condition (DETC). In [6], Akhmet and Turan proposed some basic theory of dynamic equations on variable time scales; the method of investigation is by means of two successive reductions: -equivalence of the system [7–9] on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation [5, 7]. Consequently, these results are very effective to develop methods of investigation of mechanical models with impacts.

Also, neutral differential equations arise in many areas of applied mathematics, and for this reason these equations have received much attention in the last few decades; they are not only an extension of functional differential equations but also provide good models in many fields including biology, mechanics, and economics. In particular, qualitative analysis such as periodicity and stability of solutions of neutral functional differential equations has been studied extensively by many authors. We refer to [10–19] for some recent work on the subject of periodicity and stability of neutral equations. In [20], the authors discussed a class of neutral functional differential equations with impulses and parameters on nonvariable time scales where , are parameters, is an -periodic nonvariable time scale, and both of them are -periodic functions, are -periodic functions, are nondecreasing with respect to their second arguments and -periodic with respect to their first arguments, respectively; and there exist two positive constants such that for all and is bounded, is a constant.

To the best of authors’ knowledge, there has been no paper published on the existence of solutions to neutral functional differential equations on variable time scales. Our main purpose of this paper is by using theory of dynamic equations on variable time scales to investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equations on variable time scales with a transition condition between two consecutive parts of the scale where and are parameters, is a variable time scale with -property, , , and are -periodic functions of , uniformly with respect to .

For convenience, we introduce the notation

Throughout this paper, we assume the following.(H₁) is a variable time scale with -property, ,, and are -periodic functions of , uniformly with respect to , for each .(H₂) is nondecreasing with respect to and for each .(H₃) and there exist two positive constants , such that for all .(H₄)There exists a number such that .

2. Preliminaries

Let be a real Banach space and be a cone in . A map is said to be a nonnegative continuous concave functional on if is continuous and

For numbers such that and is a nonnegative continuous concave function on , we define the following sets: .

Now, we state the following Leggett-Williams fixed-point theorem, which is critical to the proof of our main results.

Lemma 2.1 (see [21]). Let be completely continuous and nonnegative continuous concave functional on such that for all . Suppose that there exist positive constants with such that (1) and for ;(2) for ;(3) for with .

Then has at least three fixed points satisfying

Let be a periodic time scale, and let be a Banach space with the norm and let be defined by

Lemma 2.2 (see [20]). If and is a Banach space, then has a bounded inverse on , and for all , and .

Definition 2.3 (see [6]). A nonempty closed set in is said to be a variable time scale if for any the projection of on time axis, that is, the set is a time scale in Hilger sense.Fix a sequence such that for all , and as . Denote , and take a sequence of functions . Assume that(C₁) for some positive numbers ;(C₂)there exists , such that for all .Denote We set Following [6], we denote and .

A transition operator , for all , such that where and , and where is a function. One can easily see that is the time coordinate of , the image of under the operator , and is the space coordinate of the image.

Let and be the moments that the graph of intersects the surface and , respectively, where the surfaces are defined previously. Then, we set the nonvariable time scale which is the domain of , and define the -derivative as in the introduction. That is, for , we have for any other , whenever the limit exists.

Consider the system on variable time scales: where is an continuous real-valued matrix function, is an matrix, functions and are continuous, , and .

For any we define the oriented interval as

Consider the nonvariable time scale where are defined by (2.5) for the variable time scale , and take a continuation of which is Lipschitzian with the same Lipschitz constant ; furthermore, if is a monotone function, a continuation can also have the same monotony with . Set .(C₃) for arbitrary , where is a Lipschitz constant.

By (C₃), in [6], one can see the following important lemma.

Lemma 2.4 (see [6]). Assume (C₃) is satisfied. Then there are mappings , such that, corresponding to each solution of (2.10), there is a solution of the system such that for all except possibly on and , where and are the moments that meets the surfaces and , respectively. Furthermore, the functions satisfy the inequality uniformly with respect to for all such that and ; here is a bounded function. Under the sense of Lemma 2.4, we say that systems (2.10) and (2.13) are -equivalent.

Proof. Fix . Let be the solution of (2.10) such that , and assume that and are solutions of and , respectively. Let be the solution of the system with the initial condition .We first note that . Moreover, for and for , Thus, we have Substituting (2.18) in (2.13), we see that satisfies the first conclusion of the lemma.Next, we prove (2.14). Let . By employ integrals (2.16) and (2.17), we find that the solutions and determined above satisfy the inequalities and on and, where Let be the solution of (2.10) such that , and assume that and are solutions of and, respectively. Let be the solution of (2.15) with initial condition . Without loss of any generality, we assume that and . Application of the Gronwall-Bellman lemma shows that, for , The equation gives us Thus, we obtain Now condition (C₃) together with (2.23) leads to Hence (2.23) becomes On the other hand, gives us Using the transition operators and (2.25) we get Condition (C₃) and (2.28) imply that From (2.27)–(2.29) we obtain where Solutions and on satisfy the inequality Now subtracting the expression from (2.18) and using (2.20), (2.24), (2.29), and (2.31), we conclude that (2.14) holds. The proof is complete.

A special transformation called -substitution [5], which is change of the independent variable and defined for as where . Setting , we see that this transformation has an inverse given by

Lemma 2.5 (see [5, 6]). if

Proof. Assume that . Then, The assertion for can be proved in the same way. The proof is complete.

Definition 2.6 (see [5]). The time scale is said to have an -property if there exists a number such that whenever .

Definition 2.7 (see [5]). A sequence is said to satisfy an -property if there exist numbers and such that for all .

Definition 2.8 (see [6]). The variable time scale is said to satisfy an -property if is in whenever is. In this case, there exists such that the sequences and satisfy the -property and for all .Suppose now that (2.10) is -periodic; that is, satisfies the -property, and are -periodic functions of , and uniformly with respect to .

Lemma 2.9 (see [6]). If (2.10) is -periodic, then the sequence is -periodic uniformly with respect to .

Proof. Since the variable time scale satisfies an -property, by (2.18), one can easily see that is -periodic uniformly with respect to . The proof is complete.

Lemma 2.10 (see [5]). If has an -property, then the sequence , is -periodic with

Proof. In order to prove this lemma, we only need to verify that for all . Assume that for some and . Then where we have used the fact that All other cases can be verified similarly. The proof is complete.

Lemma 2.11 (see [5, 6]). If satisfies an -property, then . Proof. Assume that . By Lemma 2.10, we have The assertion for can be proved in the same way. The proof is complete.

Lemma 2.12 (see [5, 6]). A function is an -periodic function on , if and only if is an -periodic function on , where .

Proof. By Lemma 2.11, . Then the equality completes the proof.

For any fixed , we set and consider the Banach space Let be defined by

Using the inverse transformation of , we can obtain From the second equation, it is easy to get that is, . Hence

Therefore, we can obtain the other variable time scale plane by the inverse transformation of and

Hence, (1.2) can be changed into the following form: where .

Define a cone in by where .

Lemma 2.13 (see [20]). Suppose that conditions (H₁)–(H₄) hold and and , then where .

(H₅) for arbitrary , where is a Lipschitz constant.

In view of (H₅) by Lemma 2.4, it is easy to get the following lemma.

Lemma 2.14. Assume that (H₅) is satisfied. Then there are mappings such that, corresponding to each solution of (2.48), there is a solution of the system such that for all except possibly on and where and are the moments that meets the surfaces and , respectively. Furthermore, the functions satisfy the inequality uniformly with respect to for all such that and ; here is a bounded function. Under the sense of lemma 2.14, we say that systems (2.48) and (2.52) are -equivalent.

Proof. For fixed . Let be the solution of (2.48) such that , and assume that and are solutions of and , respectively. Let be the solution of the system with the initial condition .We first note that . Moreover, for , and for , Thus, we set Substituting (2.57) in (2.52), we see that satisfies the first conclusion of the lemma.The rest of the proof is similar to that of Lemma 2.4, and we can use Gronwall-Bellman lemma to show that satisfies (2.53) and it will be omitted here. This completes the proof.

Next, we will use -substitution, reducing (2.52) to an impulsive differential equation. Letting , we obtain, for , hence and for , we get Thus, the second equation in (2.23) leads to where . Hence, is a solution of the impulsive differential equation:

In the following, we set and consider the Banach space with the norm , where . Define a cone in by where .

Let the operator be defined by where By the assumptions, we have

Lemma 2.15. is an -periodic solution of (2.62) if and only if is a fixed point of the operator .

Proof. If is an -periodic solution of (2.62), for any , there exists such that is the first impulsive point after . Hence, for , we have then Again, for , then So we can obtain Repeating the above process for , we obtain Noticing that and , we find that is a fixed point of .Let be a fixed point of . If , we have By Lemmas 2.11 and 2.12, we have , so it is easy to have and . Therefore, we can obtain If , we can get Therefore, is -periodic solution of (2.62). The proof is complete.

Lemma 2.16. Assume that (H₁)–(H₅) hold, then , and is compact and continuous.

Proof. By the definition of , for , we have Thus, . So in view of (2.66), (2.68), for , we have Therefore, . Next, we will show that is continuous and compact. Firstly, we will consider the continuity of . Let and as , then and as for any . By the continuity of , for any and , we have and denote , and it is easy to see that as implies as , thus where is sufficiently large. For , we have Therefore, is continuous on .Next, we prove that is a compact operator. Let be an arbitrary bounded set in , then there exists a number such that for any . We prove that is compact. In fact, from (H₁), one has ; by (2.53), it is easy to see that for any , one has the following: So for any and , we have where , which implies that and are uniformly bounded on. Therefore, there exists a subsequence of which converges uniformly on ; namely, is compact. The proof is complete.

3. Main Results

Our main results of this paper are as follows.

Theorem 3.1. Assume that (H₁)–(H₅) hold, , for a sufficiently small Lipschitz constant ; suppose that the following conditions hold: (H₆). (H₇)There exist positive constants , and with such that where and .Then for all , (1.2) has at least three positive -periodic solutions, where

Proof. First of all, since and , we have , so Furthermore, in view of (3.1).Now, define for each and a mapping by and a function by For , by Lemma 2.13, we have It follows from (2.51), (2.68), (3.6), and (H₂), for all and that By Lemma 2.16, we know that is completely continuous on.We now assert that the condition (2) of Lemma 2.1 holds. Indeed, if , then similar to above argument, by (3.1), we have Hence, holds. Choose a positive constant such that . Next, we show that the condition (1) of Lemma 2.1 holds. Obviously, is a concave continuous function on with for . We notice that if for , then which implies . For , we have which implies, from (2.50), that And it is also clear that is nondecreasing for and , and we can easily have . Hence for all .Finally, we prove that the condition (3) of Lemma 2.1 holds. Let and , then . We notice that (3.4) implies that Thus To sum up, all the hypotheses of Lemma 2.1 are satisfied. Hence has at least three positive fixed points. That is, (1.2) has at least three positive -periodic solutions. This completes the proof.

Corollary 3.2. Suppose (H₁)–(H₆) hold. If where ; then (1.2) has at least three positive -periodic solutions.

Proof. In view of (3.14), we can choose such that the second inequality in (3.1) holds, and in view of (3.15), we can choose such that the first inequality in (3.1) holds. Therefore, the conclusion of Theorem 3.1 holds. This completes the proof.

4. An Example

Let us consider the variable time scale constructed by , where for all , and consider -periodic system: where is a constant, are nonnegative parameters. In this case, and . Obviously, (H₁)–(H₃) are satisfied, and it is easy to see that (3.14) and (3.15) hold.

By the formula of -substitution and , one can find

Clearly, and , so we can find and it is easy to check that . Thus, (H₄) holds. Furthermore, we also have Hence, for any , one can get So (H₅) is satisfied. For a sufficiently small , one can also have since is a bounded function; for a sufficiently small , one can have such that (H₆) holds. Therefore, according to Corollary 3.2, (4.1) has at least three positive -periodic solutions.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of China under Grant no. 10971183.

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