- Research Article
- Open Access

# Multiple Positive Solutions for *m*-Point Boundary Value Problem on Time Scales

- Jie Liu
^{1, 2}and - Hong-Rui Sun
^{2}Email author

**2011**:591219

https://doi.org/10.1155/2011/591219

© J. Liu and H.-R. Sun. 2011

**Received:**29 May 2010**Accepted:**6 August 2010**Published:**10 August 2010

## Abstract

The purpose of this article is to establish the existence of multiple positive solutions of the dynamic equation on time scales
, subject to the multi-point boundary condition
, where
is an increasing homeomorphism and satisfies the relation
for
, which generalizes the usually *p*-Laplacian operator. An example applying the result is also presented. The main tool of this paper is a generalization of Leggett-Williams fixed point theorem, and the interesting points are that the nonlinearity *f* contains the first-order derivative explicitly and the operator
is not necessarily odd.

## Keywords

- Boundary Value Problem
- Multiple Positive Solution
- Fixed Point Index
- Multipoint Boundary
- General Dynamic Equation

## 1. Introduction

The study of dynamic equations on time scales goes back to its founder Hilger [1], and is a new area of still fairly theoretical exploration in mathematics. On one hand, the time scales approach not only unifies calculus and difference equations, but also solves other problems that have a mix of stop-start and continuous behavior. On the other hand, the time scales calculus has tremendous potential for application in biological, phytoremediation of metals, wound healing, stock market and epidemic models [2–6].

where is an increasing homeomorphism and for .

Multipoint boundary value problem (BVP) arise in a variety of different areas of applied mathematics and physics, such as the vibrations of a guy wire of a uniform cross section and composed of parts of different densities can be set up as a multipoint boundary value problem [7]. Small size bridges are often designed with two supported points, which leads to a standard two-point boundary value condition. And large size bridges are sometimes contrived with multipoint supports, which corresponds to a multipoint boundary value condition [8]. Especially, if we let denotes the displacement of the bridge from the unloaded position, and we emphasize the position of the bridge at supporting points near , we can obtain the multipoint boundary condition (1.2). The study of multipoint BVPs for linear second-order ordinary differential equations was initiated by Ilin and Moiseev [9], since then many authors studied more general nonlinear multipoint boundary value problems. We refer readers to [8, 10–14] and the references therein.

with Dirichlet boundary condition. They also extended the results to four-point BVP in [12].

When and the nonlinearity is not involved with the first-order derivative , in [27], Sun and Li discussed the existence and multiplicity of positive solutions for problems (1.1) and (1.2). The main tools used are fixed point theorems in cones.

Thanks to the above-mentioned research articles [16, 27], in this paper we consider the existence of multiple positive solutions for the more general dynamic equation on time scales (1.1) with -point boundary condition (1.2). An example is also given to illustrate the main results. The obtained results are even new for the special cases of difference equations and differential equations, as well as in the general time scale setting. The main result extends and generalizes the corresponding results of Liu [18] and Webb [21] ( ), Sun and Li [27] ( ). We also emphasize that in this paper the nonlinear term is involved with the first-order delta derivative , the operator is not necessary odd and have the more generalized form, and the tool is a generalized Leggett-Williams fixed point theorem [16].

The rest of the paper is organized as follows: in Section 2, we give some preliminaries which are needed later. Section 3 is due to develop existence criteria for at least three and arbitrary odd number positive solution of the boundary value problem (1.1) and (1.2). In the final part of this section, we present an example to illustrate the application of the obtained result.

Throughout this paper, the following hypotheses hold:

(H1) , , for and

(H2) exists and such that and is continuous.

## 2. Preliminaries

In this section, we first present some basic definition, then we define an appropriate Banach space, cone, and integral operator, and finally we list the fixed-point theorem which is needed later.

Definition 2.1.

we know that is concave in . Thus and is the maximum value of . In addition, by direct calculation, we get that each fixed point of the operator in is a positive solution of (1.1) and (1.2). Similar as the proof of Lemma in [27], it is easy to see that is completely continuous.

In order to prove our main results, the following fixed point theorem is important in our argument.

Lemma 2.2 (see [16]).

Let be Banach space, a cone, and . Assume that and are nonnegative continuous convex functionals satisfying (2.6) and (2.7), is a nonnegative continuous concave functional on such that for all , and is a completely continuous operator. Suppose

(C1) , for

(C2) , for ;

(C3) for with

## 3. Main Results

In this section, we impose some growth conditions on which allow us to apply Lemma 2.2 to the operator defined in Section 2 to establish the existence of three positive solutions of (1.1) and (1.2). We note that, from the nonnegativity of and , the solution of (1.1) and (1.2) is nonnegative and concave on .

Then, it is easy to see that and (2.6), (2.7) hold.

Theorem 3.1.

Assume for . If there are positive numbers with , such that the following conditions are satisfied

(i) for

(ii) for

(iii) for

Proof.

By the definition of operator and its properties, it suffices to show that the conditions of Lemma 2.2 hold with respect to the operator

Therefore, (3.5) holds.

As in the argument above, we can get that Thus, condition ( ) of Lemma 2.2 holds.

Therefore, we get for , and condition ( ) in Lemma 2.2 is fulfilled.

The proof is complete.

From Theorem 3.1, we see that, when assumptions as (i), (ii), and (iii) are imposed appropriately on we can establish the existence of an arbitrary odd number of positive solutions of (1.1) and (1.2).

Theorem 3.2.

such that the following conditions hold:

(i) for

(ii) for

Then, BVP (1.1) and (1.2) has at least positive solutions.

Proof.

When it is immediate from condition (i) that which means that has at least one fixed point by the Schauder fixed point theorem. When it is clear that the hypothesis of Theorem 3.1 holds. Then we can obtain at least three positive solutions and . Following this way, we finish the proof by induction. The proof is complete.

In the final part of this section, we give an example to illustrate our results.

Example 3.3.

## Declarations

### Acknowledgments

This work was supported by the NNSF of China (10801065) and NSF of Gansu Province of China (0803RJZA096).

## Authors’ Affiliations

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