- Research Article
- Open Access

# Positive Solutions for Boundary Value Problems of Second-Order Functional Dynamic Equations on Time Scales

- IlkayYaslan Karaca
^{1}Email author

**2009**:829735

https://doi.org/10.1155/2009/829735

© Ilkay Yaslan Karaca. 2009

**Received:**3 February 2009**Accepted:**26 February 2009**Published:**16 March 2009

## Abstract

Criteria are established for existence of least one or three positive solutions for boundary value problems of second-order functional dynamic equations on time scales. By this purpose, we use a fixed-point index theorem in cones and Leggett-Williams fixed-point theorem.

## Keywords

- Banach Space
- Ordinary Differential Equation
- Dynamic Equation
- Green Function
- Difference Equation

## 1. Introduction

Throughout this paper we let be any time scale (nonempty closed subset of ) and be a subset of such that and for is not right scattered and left dense at the same time.

Some preliminary definitions and theorems on time scales can be found in books [3, 4] which are excellent references for calculus of time scales.

We will assume that the following conditions are satisfied.

(H1)

(H2) is continuous with respect to and for , where denotes the set of nonnegative real numbers.

(H4)

if then ; for , where denotes the set of all positively regressive and rd-continuous functions.

(H5) and are defined on and , respectively, where

There have been a number of works concerning of at least one and multiple positive solutions for boundary value problems recent years. Some authors have studied the problem for ordinary differential equations, while others have studied the problem for difference equations, while still others have considered the problem for dynamic equations on a time scale [5–10]. However there are fewer research for the existence of positive solutions of the boundary value problems of functional differential, difference, and dynamic equations [1, 2, 11–13].

Our problem is a dynamic analog of the BVPs (1.1) and (1.2). But it is more general than them. Moreover, conditions for the existence of at least one positive solution were studied for the BVPs (1.1) and (1.2). In this paper, we investigate the conditions for the existence of at least one or three positive solutions for the BVP (1.3). The key tools in our approach are the following fixed-point index theorem [14], and Leggett-Williams fixed-point theorem [15].

Theorem 1.1 (see [14]).

- (i)
If for , then

- (ii)
If for , then

Theorem 1.2 (see [15]).

- (i)
and for all

- (ii)
for

- (iii)
for with

## 2. Preliminaries

First, we give the following definitions of solution and positive solution of BVP (1.3).

Definition 2.1.

- (1)
is nonnegative on .

- (2)
- (3)
- (4)
is -differentiable, is -differentiable on and is continuous.

- (5)
for

Furthermore, a solution of (1.3) is called a positive solution if for

(See [8].)

Lemma 2.2 (see [8]).

where and are given in (2.7) and (2.8), respectively. It is obvious from (2.6), (H1) and (H4), that holds.

Lemma 2.3.

- (i)
for

- (ii)
for and

Proof.

where is as in (2.12).

It is easy to derive that is a positive solution of BVP (1.3) if and only if is a nontrivial fixed point of , where be defined as before.

Lemma 2.4.

Proof.

Thus, .

Lemma 2.5.

is completely continuous.

Lemma 2.6.

for all , then there exist such that , for and , for .

Proof.

Thus, by Theorem 1.1, we conclude that for . The proof is therefore complete.

## 3. Existence of One Positive Solution

In this section, we investigate the conditions for the existence of at least one positive solution of the BVP (1.3).

In the next theorem, we will also assume that the following condition on .

Theorem 3.1.

If (H1)–(H6) are satisfied, then the BVP (1.3) has at least one positive solution.

Proof.

Fix and let for . Then, satisfies (2.27). Define by

Hence, has a fixed point in .

Let . Since for and .

Theorem 3.2.

If (H1)–(H5) and (H7) are satisfied, then the BVP (1.3) has at least one positive solution.

Proof.

Therefore, has a fixed point in . The proof is completed.

Corollary 3.3.

Using the following (H8) or (H9) instead of (H6) or (H7), the conclusions of Theorems 3.1 and 3.2 are true. For ,

## 4. Existence of Three Positive Solutions

In this section, using Theorem 1.2 (the Leggett-Williams fixed-point theorem) we prove the existence of at least three positive solutions to the BVP (1.3).

Theorem 4.1.

Suppose there exists constants such that

(D1) for

(D2) for

- (a)
- (b)
there exists a constant such that for and ,

Proof.

Thus

Thus Consequently, the assumption (D3) holds, then there exist a number such that and

The remaining conditions of Theorem 1.2 will now be shown to be satisfied.

By (D1) and argument above, we can get that Hence, condition (ii) of Theorem 1.2 is satisfied.

As a result, yields

Thus, all conditions of Theorem 1.2 are satisfied. It implies that the TPBVP (1.3) has at least three positive solutions with

## 5. Examples

Example 5.1.

Since , . It is clear that (H1)–(H5) and (H8) are satisfied. Thus, by Corollary 3.3, the BVP (5.1) has at least one positive solution.

Example 5.2.

Clearly, is continuous and increasing . We can also see that . By (2.12), (4.1), and (4.2), we get , and .

from , so that (b) of (D3) is met. Summing up, there exist constants and satisfying

## Authors’ Affiliations

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