Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 845107 | 8 pages | https://doi.org/10.1155/2011/845107

First-Order Boundary Value Problem with Nonlinear Boundary Condition on Time Scales

Academic Editor: Wei-Der Chang
Received16 Mar 2011
Accepted02 Jun 2011
Published01 Aug 2011

Abstract

This work is concerned with the following first-order dynamic equation on time scale with the nonlinear boundary condition . By applying monotone iteration method, we not only obtain the existence of positive solutions, but also establish iterative schemes for approximating the solutions.

1. Introduction

The theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in 1988 in order to unify continuous and discrete analysis. The study of dynamic equations on time scales is a fairly new subject and research in this area is rapidly growing. For some basic definitions and relevant results on time scales, see [2, 3].

Recently, first-order boundary value problems (BVPs for short) on time scales have attracted much attention from many authors. For example, for first-order periodic boundary value problem (PBVP for short) on time scales Cabada [4] developed the method of lower and upper solutions coupled with the monotone iterative techniques to derive the existence of extremal solutions. In [5], Cabada and Vivero studied the existence of solutions for the first-order dynamic equation with nonlinear functional boundary value conditions They proved the uniqueness of solutions and developed the monotone iterative technique when and was a continuous and nonincreasing function. In 2007, Sun and Li [6] considered the first-order PBVP on time scales Some existence criteria of at least one solution were established by using novel inequalities and the Schaefer fixed point theorem. In [7], by applying several well-known fixed point theorems, Sun and Li obtained some existence and multiplicity criteria of positive solutions for the first-order PBVP on time scales In 2010, Zhao and Sun [8] investigated the first-order PBVP on time scales Some existence criteria of positive solutions were established, and the method used was the monotone iterative technique. For other related results, one can refer to [914] and the references therein.

Motivated greatly by the above-mentioned works, in this paper, we are interested in the existence and iteration of positive solutions for the following first-order dynamic equation with nonlinear boundary condition on time scales: where is an arbitrary time scale, is fixed, and . For each interval of , we denote by . Throughout this paper, we always assume that is right-dense continuous. Here, a solution of the BVP (1.6) is said to be positive if is nonnegative and nontrivial. By applying monotone iteration method, we not only obtain the existence of positive solutions for the BVP (1.6), but also establish iterative schemes for approximating the solutions. It is worth mentioning that the initial terms of our iterative schemes are constant functions, which implies that the iterative schemes are significant and feasible. In our arguments, the following monotone iteration method [15] is very crucial.

Theorem 1.1. Let be a normal cone of a Banach space and . Suppose that(a1) is completely continuous,(a2) is monotone increasing on ,(a3) is a lower solution of , that is, ,(a4) is an upper solution of ; that is, .Then, the iterative sequences satisfy and converge to, respectively, and , which are fixed points of .

2. Main Results

Theorem 2.1. Assume that and are continuous and . If there exists a constant such that the following conditions are satisfied: (H1), , ,(H2), ,then the BVP (1.6) has positive solutions.

Proof. Let be equipped with the norm Then, is a Banach space. Denote Then, is a normal cone of . Now, if we define an operator by then it is easy to know that fixed points of are nonnegative solutions of the BVP (1.6).
Let and , . Now, we divide our proof into the following steps.
Step 1. We verify that is completely continuous.
First, we will show that is continuous.
Let and . Then,
For any given , since is uniformly continuous on , there exists such that for any with ,
On the other hand, since is continuous at , there exists such that for any with ,
Let . Then, it follows from that there exists a positive integer such that for any ,
In view of (2.6), (2.7), and (2.8), we know that for any , which indicates that . So, is continuous.
Next, we will show that is compact.
Let be a bounded set. Then, there exists a constant such that for any and , . Define and .
On the one hand, for any , we have which shows that is uniformly bounded.
On the other hand, for any and with , we have which implies that is equicontinuous. Consequently, is compact.
Step 2. We assert that is monotone increasing on .
Suppose that and . Then, , . By (H1) and (H2), we have which shows that .
Step 3. We prove that is a lower solution of .
For any , it is obvious that which implies that .
Step 4. We show that is an upper solution of .
In view of (H1) and (H2), we have which indicates that .
Step 5. We claim that the BVP (1.6) has positive solutions.
In fact, if we construct sequences and as follows: then it follows from Theorem 1.1 that and and converge to, respectively, and , which are solutions of the BVP (1.6). Moreover, it follows from that which shows that and are positive solutions of the BVP (1.6).

Example 2.2. Let . We consider the following BVP on :
Since and , if we choose , then all the conditions of Theorem 2.1 are fulfilled. So, it follows from Theorem 2.1 that the BVP (2.19) has positive solutions and . Furthermore, if we construct sequences and as follows: where and , then

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (10801068).

References

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Copyright © 2011 Ya-Hong Zhao. 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.

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