Multiple Positive Solutions for Nonlinear First-Order Impulsive Dynamic Equations on Time Scales with Parameter

Abstract

By using the Leggett-Williams fixed point theorem, the existence of three positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales with parameter are obtained. An example is given to illustrate the main results in this paper.

1. Introduction

Let be a time scale, that is, is a nonempty closed subset of . Let be fixed and be points in , an interval denoting time scales interval, that is, Other types of intervals are defined similarly. Some definitions concerning time scales can be found in [1–5].

In this paper, we are concerned with the existence of positive solutions for the following nonlinear first-order periodic boundary value problem on time scales:

(11)

where is a positive parameter, , is right-dense continuous, , and for each and represent the right and left limits of at .

The theory of impulsive differential equations is emerging as an important area of investigation, since it is a lot richer than the corresponding theory of differential equations without impulse effects. Moreover, such equations may exhibit several real world phenomena in physics, biology, engineering, and so forth, (see [6–8]). At the same time, the boundary value problems for impulsive differential equations and impulsive difference equations have received much attention [9–19]. On the other hand, recently, the theory of dynamic equations on time scales has become a new important branch (see, e.g., [1–5]). Naturally, some authors have focused their attention on the boundary value problems of impulsive dynamic equations on time scales [20–27]. In particular, for the first-order impulsive dynamic equations on time scales

(12)

where is a time scale which has at least finitely-many right-dense points, is regressive and right-dense continuous, is given function, . The paper [21] obtained the existence of one solution to problem (1.2) by using the nonlinear alternative of Leray-Schauder type.

In [22], Benchohra et al. considered the following impulsive boundary value problem on time scales

(13)

They proved the existence of one solution to the problem (1.3) by applying Schaefer's fixed point theorem and the nonlinear alternative of Leray-Schauder type.

In [26], Li and Shen studied the problem (1.3). Some existence results to problem (1.3) are established by using a fixed point theorem, which is due to Krasnoselskii and Zabreiko, and the Leggett-Williams fixed point theorem.

In [27], the first author studied the problem (1.1) when . The existence of positive solutions to the problem (1.1) was obtained by means of the well-known Guo-Krasnoselskii fixed point theorem.

Recently, Sun and Li [28] considered the following periodic boundary value problem:

(14)

By using the fixed point index, some existence, multiplicity and nonexistence criteria of positive solutions to the problem (1.4) were obtained for suitable .

Motivated by the results mentioned above, in this paper, we shall show that the problem (1.1) has at least three positive solutions for suitable by using the Leggett-Williams fixed point theorem [29]. We note that for the case and problem (1.1) reduces to the problem studied by [30].

In the remainder of this section, we state the following theorem, which are crucial to our proof.

Let be a real Banach space and be a cone. A function is called a nonnegative continuous concave functional if is continuous and

(15)

for all and .

Let be constants,

Theorem 1.1 (see [29]).

Let be a completely continuous map and be a nonnegative continuous concave functional on such that Suppose there exist with such that

• (i) and

• (ii)

• (iii) with

Then has at least three fixed points in satisfying

(16)

2. Preliminaries

Throughout the rest of this paper, we always assume that the points of impulse are right-dense for each

We define

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where is the restriction of to and

Let

(22)

with the norm Then X is a Banach space.

Definition 2.1.

A function is said to be a solution of the problem (1.1) if and only if satisfies the dynamic equation

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the impulsive conditions

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and the periodic boundary condition

Lemma 2.2.

Suppose is -continuous, then is a solution of

(25)

where

(26)

if and only if is a solution of the boundary value problem

(27)

Proof.

Since the method is similar to that of in [27, Lemma  3.1], we omit it here.

Lemma 2.3.

Let be defined as Lemma 2.2, then

(28)

Proof.

It is obvious, so we omit it here.

Let

(29)

where It is not difficult to verify that is a cone in

We define an operator by

(210)

By [27, Lemmas  3.3 and 3.4], it is easy to see that is completely continuous.

3. Main Result

Notation 1.

Let

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and for we define

Theorem 3.1.

Assume that there exists a number such that the following conditions:

(H₁) for

(H₂) hold. Then the problem (1.1) has at least three positive solutions for

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Proof.

Let it is easy to see that is a nonnegative continuous concave functional on such that

First, we assert that there exists such that is completely continuous.

In fact, by the condition of (H₂), there exist and such that

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Let if then and we have

(34)

Take then the set is a bounded set. According to that is completely continuous, then maps bounded sets into bounded sets and there exists a number such that

(35)

If we deduce that is completely continuous. If then from (3.4), we know that for any and hold. Then we have is completely continuous. Take then and are completely continuous.

Second, we assert that and for all

In fact, take so Moreover, for then and we have

(36)

Third, we assert that there exist such that if

Indeed, by the condition of (H₂), there exist and such that

(37)

Then we get

(38)

Finally, we assert that if and

To do this, if and then

(39)

To sum up, all the hypotheses of Theorem 1.1 are satisfied by taking Hence has at least three fixed points, that is, the problem (1.1) has at least three positive solutions and such that

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Corollary 3.2.

Using (H₃) instead of (H₂) in Theorem 3.1, the conclusion of Theorem 3.1 remains true.

4. Example

Example 4.1.

Let We consider the following problem on

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where is a positive parameter, and

(42)

Taking then by it is easy to see that So, for all we have Obviously, we have

Therefore, together with Corollary 3.2, it follows that the problem (4.1) has at least three positive solutions for .