Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 189768 | 12 pages | https://doi.org/10.1155/2009/189768

Existence of Positive Solutions for -Point Boundary Value Problems on Time Scales

Academic Editor: Binggen Zhang
Received27 Aug 2008
Revised24 Nov 2008
Accepted14 Jan 2009
Published08 Mar 2009

Abstract

We study the one-dimensional -Laplacian -point boundary value problem , , , , where is a time scale, , , some new results are obtained for the existence of at least one, two, and three positive solution/solutions of the above problem by using fixed point theorem, new fixed point theorem due to Avery and Henderson, as well as Leggett-Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional -Laplacian -point boundary value problem on time scales has been studied.

1. Introduction

With the development of p-Laplacian dynamic equations and theory of time scales, a few authors focused their interest on the study of boundary value problems for p-Laplacian dynamic equations on time scales. The readers are referred to the paper [17].

In 2005, He [1] considered the following boundary value problems:where is a time scales, The author showed the existence of at least two positive solutions by way of a new double fixed point theorem.

In 2004, Anderson et al. [2] used the virtue of the fixed point theorem of cone and obtained the existence of at least one solution of the boundary value problem:

In 2007, Geng and Zhu [3] used the Avery-Peterson and another fixed theorem of cone and obtained the existence of three positive solutions of the boundary value problem:Also, in 2007, Sun and Li [4] discussed the existence of at least one, two or three positive solutions of the following boundary value problem:

In this paper, we are concerned with the existence of multiple positive solutions to the m-point boundary value problem for the one dimension p-Laplcaian dynamic equation on time scale where is a time scale, and

(H1)(H2)(H3) and there exists such that

In this paper, we have organized the paper as follows. In Section 2, we give some lemmas which are needed later. In Section 3, we apply the Krassnoselskiifs [8] fixed point theorem to prove the existence of at least one positive solution to the MBVP(1.5). In Section 4, conditions for the existence of at least two positive solutions to the MBVP (1.5) are discussed by using Avery and Henderson [9] fixed point theorem. In Section 5, to prove the existence of at least three positive solutions to the MBVP (1.5) are discussed by using Leggett and Williams [10] fixed point theorem.

For completeness, we introduce the following concepts and properties on time scales.

A time scale is a nonempty closed subset of , assume that has the topology that it inherits from the standards topology on

Definition 1.1. Let be a time scale, for , one defines the forward jump operator by , and the backward jump operator by while the graininess function is defined by . If one says that is right-scattered, while if one says that tis left-scattered. Also, if and then is called right-dense, and if and then is called left-dense. One also needs below the set as follows: if has a left-scattered maximum then otherwise For instance, if then

Definition 1.2. Assume is a function and let Then , one defines to be the number (provided it exists) with the property that any given there is a neighborhood of such thatfor all One says that is delta differentiable (or in short: differentiable) on provided exist for all

If then if then

A function .

(i)If is continuous , then is rd-continuous.(ii)The jump operator is rd-continuous.(iii)If is rd-continuous, then so is

A function is called an antidervative of , provided holds for all . One defines the definite integral byFor all . If then is nondecreasing.

2. The Preliminary Lemmas

Lemma 2.1 (see [5, 6]). Assume that (H1)–(H3) hold. Then is a solution of the MBVP (1.5) on if and only ifwhere and

Lemma 2.2. Assume that conditions (H1)–(H3) are satisfied, then the solution of the MBVP (1.5) on satisfies

Lemma 2.3 (see [5]). If the conditions (H1)–(H3) are satisfied, thenwhere

Lemma 2.4 (see [6]).

Let denote the Banach space with the norm

Define the cone by

The solutions of MBVP (1.5) are the points of the operator defined bySo, It is easy to check that is completely continuous.

3. Existence of at least One Positive Solutions

Theorem 3.1 (see [8]). Let be a Banach space, and let be a cone. Assume and are open boundary subsets of with and let be a completely continuous operator such that either
(i) for for or(ii) for for hold. Then A has a fixed point in

Theorem 3.2. Assume conditions (H1)–(H3) are satisfied. In addition, suppose there exist numbers such that if and if where

Then the MBVP (1.5) has at least one positive solution.

Proof. Define the cone as in (2.5), define a completely continuous integral operator byFrom (H1)–(H3), Lemmas 2.1 and 2.2, . If with then we get This implies that So, if we set then for .
Let us now set
Then for with by Lemma 2.4 we have Therefore, we haveHence, for Thus by the Theorem 3.1, has a fixed point in Therefore, the MBVP (1.5) has at least one positive solution.

4. Existence of at least Two Positive Solutions

In this section, we apply the Avery-Henderson fixed point theorem [9] to prove the existence of at least two positive solutions to the nonlinear MBVP (1.5) .

Theorem 4.1 (see Avery and Henderson [9]). Let be a cone in a real Banach space Set

If and are increasing, nonnegative continuous functionals on let be a nonnegative continuous functional on with such that, for some positive constants and and for all . Suppose that there exist positive numbers such that for all and

If is a completely continuous operator satisfying

(i) for all ;(ii) for all ;(iii) and for all then has at least two fixed points and such that with and with

Let and Define the increasing, nonnegative and continuous functionals and on by and

From Lemma 2.4, for each

In addition, for each Lemma 2.3 implies

Thus,

We also see that and for all and .

Theorem 4.2. Assume (H1)–(H3) hold, suppose there exist positive numbers such that the function satisfies the following conditions:
(B1) for and ;(B2) for and (B3) for and

Then the MBVP (1.5) has at least two positive solutions and such that with and with

Proof. We now verify that all of the conditions of Theorem 4.1 are satisfied.
Define the cone as (2.5), define a completely continuous integral operator by
and as in (3.1). To verify that condition (i) of Theorem 4.1 holds, we choose , then This implies . Note that We have for As a consequence of (B3), for Since we have from Lemma 2.2, Then condition (i) of Theorem 4.1 holds.
Let . Then This implies for From (B2), we haveHence condition (ii) of Theorem 4.1 holds.
If we first define for , then So
Now, let then This mean that From (B1) and Lemma 2.4, we getThen condition (iii) of Theorem 4.1 holds.
Since all conditions of Theorem 4.1 are satisfied, the MBVP (1.5) has at least two positive solutions and such that with and with

5. Existence of at least Three Positive Solutions

We will use the Leggett-Williams fixed point theorem [10] to prove the existence of at least three positive solutions to the nonlinear MBVP (1.5).

Theorem 5.1 (see Leggett and Williams [10]). Let be a cone in the real Banach space Set

Suppose be a completely continuous operator and be a nonnegative continuous concave functional on with for all . If there exists such that the following condition hold:

(i) and for all (ii) for (iii) for with then has at least three fixed points and in satisfying with

Theorem 5.2. Assume (H1)–(H3) hold . Suppose that there exist constants such that
(C1) for and (C2) for and (C3) for and

Then the MBVP (1.5) has at least three positive solutions and such that with

Proof. The conditions of Theorem 5.1 will be shown to be satisfied. Define the nonnegative continuous concave functional to be the cone as in (2.5), and as in (3.1). We have for all If , then and from assumption (C1) , then we have
This implies that Thus, we have . Since and For we have Using assumption (C2), we obtainHence, condition (i) of Theorem 5.1 holds.
If from assumption (C3), we obtainThis implies that
Consequently, condition (ii) of Theorem 5.1 holds.
We suppose that with Then we getHence, condition (iii) of Theorem 5.1 holds.
Because all of the hypotheses of the Leggett-Williams fixed point theorem are satisfied, the nonlinear MBVP (1.5) has at least three positive solutions and such that and with

Acknowledgment

This work is supported by the Research and Development Foundation of College of Shanxi Province (no. 200811043).

References

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Copyright © 2009 Ying Zhang and ShiDong Qiao. 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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