Existence of Positive Solution to Second-Order Three-Point BVPs on Time Scales

Abstract

We are concerned with the following nonlinear second-order three-point boundary value problem on time scales , , , , where with and . A new representation of Green's function for the corresponding linear boundary value problem is obtained and some existence criteria of at least one positive solution for the above nonlinear boundary value problem are established by using the iterative method.

1. Introduction

Let be a time scale, that is, is an arbitrary nonempty closed subset of . For each interval of we define For more details on time scales, one can refer to [1–5]. Recently, three-point boundary value problems (BVPs for short) for second-order dynamic equations on time scales have received much attention. For example, in 2002, Anderson [6] studied the following second-order three-point BVP on time scales:

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where , , and . Some existence results of at least one positive solution and of at least three positive solutions were established by using the well-known Krasnoselskii and Leggett-Williams fixed point theorems. In 2003, Kaufmann [7] applied the Krasnoselskii fixed point theorem to obtain the existence of multiple positive solutions to the BVP (1.1). For some other related results, one can refer to [8–10] and references therein.

In this paper, we are concerned with the existence of at least one positive solution for the following second-order three-point BVP on time scales:

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Throughout this paper, we always assume that with , , and

It is interesting that the method used in this paper is completely different from that in [6, 7, 9, 10], that is, a new representation of Green's function for the corresponding linear BVP is obtained and some existence criteria of at least one positive solution to the BVP (1.2) are established by using the iterative method.

For the function , we impose the following hypotheses:

(H1) is continuous;

(H2)for fixed , is monotone increasing on ;

(H3)there exists such that

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Remark 1.1.

If (H3) is satisfied, then

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2. Main Results

Lemma 2.1.

The BVP (1.2) is equivalent to the integral equation

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where

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is called the Green's function for the corresponding linear BVP, here

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is the Green's function for the BVP:

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Proof.

Let be a solution of the BVP:

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Then, it is easy to know that

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Now, if is a solution of the BVP (1.2), then it can be expressed by

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which together with the boundary conditions in (1.2) and (2.6) implies that

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On the other hand, if satisfies (2.1), then it is easy to verify that is a solution of the BVP (1.2).

Lemma 2.2.

For any one has

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Proof.

Since it is obvious from the expression of that

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we know that (2.9) is fulfilled.

Our main result is the following theorem.

Theorem 2.3.

Assume that (H1)–(H3) are satisfied. Then, the BVP (1.2) has at least one positive solution . Furthermore, there exist such that

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Proof.

Let

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Define an operator :

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Then it is obvious that fixed points of the operator in are positive solutions of the BVP (1.2).

First, in view of (H2), it is easy to know that is increasing.

Next, we may assert that , which implies that for any , there exist positive constants and such that

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In fact, for any , there exist such that

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which together with (H2), (H3), and Remark 1.1 implies that

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By Lemma 2.2 and (2.16), for any , we have

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If we let

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then it follows from (2.17) and (2.18) that

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which shows that .

Now, for any fixed , we denote

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and let

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where

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Then, it is easy to know from (2.20), (2.21), (2.22), (2.23), (2.24), (H3), and Remark 1.1 that

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Moreover, if we let , then it follows from (2.22), (2.23), (2.24), and (H3) by induction that

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which together with (2.25) implies that for any positive integers and ,

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Therefore, there exists a such that and converge uniformly to on and

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Since is increasing, in view of (2.28), we have

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So,

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which shows that is a positive solution of the BVP (1.2). Furthermore, since , there exist such that

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