Abstract and Applied Analysis
Volume 2009 (2009), Article ID 512402, 21 pages
doi:10.1155/2009/512402

Copyright © 2009 Qiying Wei et al. 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

By using the well-known Schauder fixed point theorem and upper and lower solution method, we present some existence criteria for positive solution of an m-point singular p-Laplacian dynamic equation on time scales with the sign changing nonlinearity. These results are new even for the corresponding differential (𝕋=ℝ) and difference equations (𝕋=ℤ), as well as in general time scales setting. As an application, an example is given to illustrate the results.

1. Introduction

Initiated by Hilger in his Ph.D. thesis [1] in 1988, the theory of time scales has been improved greatly ever since, especially in the unification of the theory of differential equations in the continuous case and the theory of finite difference equations in the discrete case. For the time being, it remains active and attracts many distinguished researchers' attention. The reason is two sided. On the one hand, the calculus on time scales not only can unify differential and difference equations, but also can provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. On the other hand, it is also widely applied to the research of biology, heat transfer, stock market, wound healing and epidemic models [2–6], and so forth. For instance, Hoffacker et al. have used the theory to model how students suffering from the eating disorder bulimia are influenced by their college friends. With the theory on time scales, they can model how the number of sufferers changes during the continuous college term as well as during long breaks [5]. Hence, the dynamic equations on time scales are worth studying theoretically and practically [3, 5, 7].

Here and hereafter, we denote is -Laplacian operator, that is, for and where We make the blanket assumption that are points in by an interval we always mean Other types of interval are defined similarly.

Recent research results indicate that considerable work has been made in the existence problems of solutions of boundary value problems on time scales, for details, see [8–16] and the references therein. In particular, some of them are considered the existence of positive solutions of -Laplacian boundary value problems on time scales, see [17–22]. The main tools used in these papers are the various fixed point theorems in cones. Very recently, when the nonlinear term is allowed to change sign, Su et al. [23–25] proved the existence of positive solutions to -Laplacian dynamic equations with sign changing nonlinearity on time scales.

Motivated by references [23–25], we consider the following -point singular -Laplacian boundary value problem on time scales of the form (1.1)(1.2) where is continuous and are continuous, nondecreasing and may be nonlinear, The singularity may occur at and and the nonlinearity is allowed to change sign. In particular, the boundary condition (1.2) includes the Dirichlet boundary condition. We obtain some new existence criteria for positive solutions of the boundary value problem (1.1) and (1.2) by using the upper and lower method. Our results are new even for the corresponding differential ( and difference equations (, as well as in general time scales setting. As an application, an example is given to illustrate these results. In particular, our results improve and generalize some known results of Agarwal et al. [26], O'Regan [27] ( and Lü et al. [28] when ; include the results of Lü et al. [29] when ; extend and include the results of Jiang et al. [30] in the case of .

For the convenience of statements, now we present some basic definitions and lemmas concerning the calculus on time scales that one needs to read this manuscript, which can be found in [3, 7]. One of other excellent sources on dynamical systems on time scales is from the book in [31].

Definition 1.1 (see [3, 7]). A time scale is a nonempty closed subset of It follows that the jump operators defined by (1.3) (supplemented by and ) are well defined. The point is left-dense, left-scattered, right-dense, right-scattered if respectively. If has a right-scattered minimum define otherwise, set If has a left-scattered maximum define otherwise, set . The forward graininess is Similarly, the backward graininess is

Definition 1.2 (see [7]). We say that a function is right-increasing at a point provided the following conditions hold.(i)If is right-scattered, then .(ii)If is right-dense, then there is a neighborhood of such that for all with .Similarly, we say that is right-decreasing if above in (i), and (ii), .

Definition 1.3 (see [3]). A function is called predifferentiable with (region of differential) provided the following conditions hold:(i) is continuous on ;(ii)(iii) is countable and contains no right-scattered elements of (iv) is differentiable at each .

Next, we list some lemmas which will be used in the sequel.

Lemma 1.4 (see [3, 7]). Suppose is a function and let , then one has the following: (i) If is differentiable at , then is continuous at . (ii) If is continuous at and is right-scattered, then is differentiable at with (1.4) (iii) If is right-dense, then is differentiable at if and only one the limit (1.5) exists as a finite number. In this case (1.6) (iv) If is differentiable at , then (1.7)

Lemma 1.5 (see [7]). Suppose is differentiable at If assumes its local right-minimum at , then . If assumes its local right-maximum at , then .

Lemma 1.6 ((Mean Value Theorem) [7]). Let be a continuous function on that is differentiable on . Then there exist such that (1.8)

Lemma 1.7 (see [3]). Suppose and are pre-differential with . If is a compact interval with endpoints then

Now, we can obtain the following lemma which is similar to Lemma 1.7. The proofs are similar to the proofs of Lemma 1.7 by a slight modification and we omit the proofs.

Lemma 1.8. Suppose and are predifferential with . If is a compact interval with endpoints then here

Throughout this paper, it is assumed that

(H1) is continuous; (H2) and (H3) are continuous and nondecreasing, here .

2. Existence Results

Define the Banach space with the norm

To demonstrate existence of positive solutions to problem (1.1) and (1.2), we first approximate the singular problem by means of a sequence of nonsingular problems, and by using the lower and upper solution for nonsingular problem together with Schauders fixed point theorem, and then we establish the existence of solutions to each approximating problem. Our results are new even for the corresponding differential ( and difference equations (, as well as in general time scales setting. If we consider the corresponding differential equation ( of problem (1.1) and (1.2) in the method mentioned above, we obtain the same existence results to problem (1.1) and (1.2). In the same way, we consider the corresponding difference equation ( of problem (1.1) and (1.2), we obtain the same existence results to problem (1.1) and (1.2). Here, the two same existence results are obtained in different settings by using the essentially same method. Naturally, it is quite necessary to consider the existence results to problem (1.1) and (1.2) in same setting. In this case, we need to solve the problem with the help of calculus on time scales, because it not only can unify differential and difference equations, but also can provide accurate information of phenomena that manifests themselves partly in continuous time and partly in discrete time. For example, we can consider the problem (1.1) and (1.2) on time scales (2.1) However, if is taken from (2.1), we cannot study the problem (1.1) and (1.2) only in differential case, neither can we study the problem (1.1) and (1.2) only in difference case.

Now we state and prove our main result.

Theorem 2.1. Let be fixed. Assume that (H1)–(H3) hold and the following conditions are satisfied. (A1) For each , there is a constant such that is a strictly monotone decreasing sequence with , and for ; (A2) There exists a function with and ; (A3) There exists a function with and with for , and for . Then the boundary value problem (1.1) and (1.2) has a positive solution with for

Proof. It follows from the condition (A1) that for each That is, is not empty. Without loss of generality, fix . If then we can suppose that let be such that (2.2) If then we can suppose that let be such that (2.2) holds. Define (2.3) We denote and (2.4) Define a sequence and (2.5) Then (2.6)

Consider the -Laplacian boundary value problem (2.7)(2.8) where (2.9)(2.10) and is the radial retraction function defined by (2.11)

Suppose (2.12) We define the mappings be such that (2.13)(2.14) By using the Arzela-Ascoli theorem on time scales [2], we can show that is continuous and compact. By using the (2.7), (2.8), (2.13) and (2.14), we obtain (2.15) that is (2.16) If (2.17) then hence exists and is continuous. So (2.18) It is clear that solving the boundary value problem (2.7) and (2.8) is equivalent to finding a fixed point of where is compact. Schauder's fixed point theorem guarantees that the boundary value problem (2.7) and (2.8) has a solution with .

We first show that (2.19) If (2.19) is not true, the function has a negative minimum for some We consider two cases, namely, and

Case 1. Assume that , then we claim (2.20) Since has a negative minimum for some in view of Definition 1.2, Lemmas 1.4 and 1.5, we have and there exists a with such that Thus (2.21) which leads to (2.22)

If is left-dense, in view of Lemma 1.4(2.23)

If is left-scattered, by Lemma 1.4 and (2.22) we obtain (2.24) Hence, (2.20) is established.

However, by (2.3), (2.9) and we obtain (2.25)

Assume that then for by (A1) and (A2), we have (2.26) which implies a contraction.

Assume that then in view of (A1), (A2) and , we have (2.27) which implies a contraction.Case 2. Assume that That is, by (2.3), (2.8) and (2.10) together with we have the following three subcases.

(a) If then (2.28) this is a contradiction.

(b) If Assume that then (2.29)

Assume that then (2.30)

Assume that there exist sequences and such that and here then (2.31) Hence, by (2.29), (2.30) and (2.31) together with the monotonicity of we have (2.32) this is a contradiction.

(c) If there exist sequences and such that and here Essentially the same reasoning as before we have this is a contradiction.

Thus, Cases 1–2 imply (2.19) is established. In particular, since for , we obtain

Essentially the same reasoning as the proof of inequality (2.19) we obtain

Hence (2.33)

Now, we discuss the boundary value problem (2.34) where (2.35) Schauder's fixed point theorem guarantees that the boundary value problem (2.34) has a solution with .

Essentially the same reasoning as the proof of inequality (2.33), we have (2.36)

If there exists for some satisfying for Then we investigate the boundary value problem (2.37) where (2.38) It follows from Schauder's fixed point theorem that the boundary value problem (2.37) has a solution with .

By using the similar arguments as above, we have (2.39) Hence, for each the mathematical induction implies that (2.40) Denote (2.41) It follows from Lemma 1.6 that there exist satisfy (2.42) From (2.42), we have (2.43)

So there exists a positive number such that By Lemma 1.8, we have (2.44) The Arzela-Ascoli theorem on time scales [2] guarantees the existence of a subsequence of integers and a function with converging uniformly to on as through Similarly (2.45) Thus there is a subsequence of and a function with converging uniformly to on as through Since , we have on Proceed inductively to obtain subsequence of integers and functions with as and .

Now, we define a function with on and Notice, is well defined and for Nextly fix and let be such that let we have (2.46) Hence, for we have which is the positive solution of the following boundary value problem (2.47) Let through we have that satisfies (2.48) It remains to show that is continuous at Now by there exists with Since there exists with for By the monotonicity of for each we have which means for . So is continuous at 0.

If we replace with the singularity occurs at and .

If we replace with the singularity occurs at and .

If we replace with the singularity occurs at

So it is easily obtain the analogue of Theorem 2.1 in this section. See the following remark.

Remark 2.2. If (A3) is appropriately adjusted, we can replace in (A1) by (2.49)(2.50) or (2.51) For example, if (2.49) occurs, (A3) is replaced by

There exists a function such that for for and .

Assume that (H1)–(H3), (A1) and (A2) hold, and in addition suppose the following conditions are satisfied:

(A4) for (2.52)

(A5) There exists a function such that for for and for

(A6)

Then the result in Theorem 2.1 is also true. This follows immediately from Theorem 2.1 if we show (A3) holds. That is to say, if we show for then the result holds Assume it is not true, in view of (A6) we obtain has a negative minimum for some , so ( and essentially the same reasoning as the proof of inequality (2.20), we have However, by (A4), (A5) and we obtain Hence which implies a contradiction.

Corollary 2.3. Let be fixed, suppose (H1)–(H3), (A1), (A2) and (A4)–(A6) hold, then the boundary value problem (1.1) and (1.2) has a solution with for

3. Construction of and

In this section, we consider how to construct a lower solution and an upper solution in certain circumstances. In this section, we assume that (3.1)

Lemma 3.1. Assume that there exists a nonincreasing positive sequence with , then there exist a function satisfying (i) for and (ii) and for

Proof. Let Assume that be such that for and for Let Suppose satisfy and It is easy to show that are continuous and increasing. Denote (3.2) here (3.3) Hence, for and is nondecreasing. Define (3.4) We can easily prove and Thus, we have and with Now since for and for we have for On the other hand, (3.5) by the monotonicity of on respectively, we have (3.6) Consequently,

Without loss of generality, . We have (3.7)

Now we discuss how to construct a lower solution in (A2) and (A4).

(A7) For each there exist a constant and a strictly monotone decreasing sequence with and for ;

(A8) There exists a function such that for and for .

Theorem 3.2. Let be fixed. If (H1)–(H3), (3.1) and (A7)-(A8) hold, then boundary value problem (1.1) and (1.2) has a solution with and for

Proof. By Corollary 2.3, we need only show that conditions (A1), (A2), (A4)–(A6) are satisfied. Without loss of generality, suppose (3.8) by (A7), (A8) and (3.8), we obtain that (A1) and ( A5) hold.

From Lemma 3.1 there exists a function satisfying(i) for and (ii) and for

Assume Let for Then with for Without loss of generality, we have For arbitrary there exists such that We have (3.9) Thus (A4) holds and (A2) is also true if . Also since we have then (A6) is fulfilled. By Corollary 2.3, the boundary value problem (1.1) and (1.2) has a solution with for

We can replace with or So it is easily obtain (see Remark 2.2) the analogue of Theorem 3.2 in this section.

Looking at Theorem 3.2, it is difficulty for us to discuss examples in constructing in (A8). The following theorem removes (A8) and replaces it with an easy verified condition.

Theorem 3.3. Let be fixed. If (H1)–(H3), (A1) and (A2) hold, in addition suppose that the following conditions are satisfied: (3.10) (3.11) Then boundary value problem (1.1) and (1.2) has a solution with and for

Proof. Denote for then with and for (3.12) with (3.13) then (A3) holds. By Theorem 2.1 the result holds.

From Theorems 3.2 and 3.3 we have the following theorem.

Theorem 3.4. Let be fixed. If (H1)–(H3), (3.1) and (A7) hold, in addition suppose there exist constants such that (3.11) and (3.5) are true. Then the problem (1.1) and (1.2) has a solution with and for

Proof. Without loss of generality suppose by (A7) we have (A1) which holds and (3.14) By the similar way as the proof of the Theorem 3.2, there exists a function with for such that for and This together with (3.14) we have Thus all the conditions of the Theorem 3.3 are fulfilled.

4. An Example

In this section, we present an example to illustrate our results. Let (4.1) Consider the following boundary value problem (4.2) It is obvious that . Denote here is constant. Let and We have Note that (H1)–(H3) and (3.1) hold. For and we have (4.3) which implies (A7) is satisfied.

Now we show that (A8) holds with .

Notice that if then (4.4)

If then and (4.5)

If then we have (4.6) by induction, one gets (4.7) Thus, for we have (4.8) Now (4.9) Hence, all conditions of the Theorem 3.2 are satisfied. As a result, the problem (4.2) has a positive solution.

Acknowledgments

This paper is supported by XZIT under Grant XKY2008311 and DEGP under Grant 0709-03.

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