Existence of Positive Solutions for Multipoint Boundary Value Problem on the Half-Line with Impulses

Abstract

We consider a multi-point boundary value problem on the half-line with impulses. By using a fixed-point theorem due to Avery and Peterson, the existence of at least three positive solutions is obtained.

1. Introduction

Impulsive differential equations are a basic tool to study evolution processes that are subjected to abrupt changes in their state. For instance, many biological, physical, and engineering applications exhibit impulsive effects (see [1–3]). It should be noted that recent progress in the development of the qualitative theory of impulsive differential equations has been stimulated primarily by a number of interesting applied problems [4–24].

In this paper, we consider the existence of multiple positive solutions of the following impulsive boundary value problem (for short BVP) on a half-line:

(11)

where , , , , and , and satisfy

();

(), , and when is bounded, and are bounded on ;

() and is not identically zero on any compact subinterval of . Furthermore satisfies

(12)

where

(13)

Boundary value problems on the half-line arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations and there are many results in this area, see [8, 13, 14, 20, 25–27], for example.

Lian et al. [25] studied the following boundary value problem of second-order differential equation with a -Laplacian operator on a half-line:

(14)

They showed the existence at least three positive solutions for (1.4) by using a fixed point theorem in a cone due to Avery-Peterson [28].

Yan [20], by using Leray-Schauder theorem and fixed point index theory presents some results on the existence for the boundary value problems on the half-line with impulses and infinite delay.

However to the best knowledge of the authors, there is no paper concerned with the existence of three positive solutions to multipoint boundary value problems of impulsive differential equation on infinite interval so far. Motivated by [20, 25], in this paper, we aim to investigate the existence of triple positive solutions for BVP (1.1). The method chosen in this paper is a fixed point technique due to Avery and Peterson [28].

2. Preliminaries

In this section, we give some definitions and results that we will use in the rest of the paper.

Definition 2.1.

Suppose is a cone in a Banach. The map is a nonnegative continuous concave functional on provided is continuous and

(21)

for all , and . Similarly, the map is a nonnegative continuous convex functional on provided is continuous and

(22)

for all , and .

Let be nonnegative, continuous, convex functionals on and be a nonnegative, continuous, concave functionals on , and be a nonnegative continuous functionals on . Then, for positive real numbers , and , we define the convex sets

(23)

and the closed set

(24)

To prove our main results, we need the following fixed point theorem due to Avery and Peterson in [28].

Theorem 2.2.

Let be a cone in a real Banach space . Let and be nonnegative continuous convex functionals on a cone , be a nonnegative continuous concave functional on , and be a nonnegative continuous functional on satisfying for , such that for some positive numbers and

(25)

for all . Suppose

(26)

is completely continuous and there exist positive numbers , and with such that

(i) and for ;

(ii) for with ;

(iii) and for , with

Then has at least three fixed points such that

(27)

3. Some Lemmas

Define is continuous at each , left continuous at , exists, .

By a solution of (1.1) we mean a function in satisfying the relations in (1.1).

Lemma 3.1.

is a solution of (1.1) if and only if is a solution of the following equation:

(31)

where is defined as (1.3).

The proof is similar to Lemma in [9], and here we omit it.

For , let . Then

(32)

It is clear that . Consider the space defined by

(33)

is a Banach space, equipped with the norm . Define the cone by

(34)

Lemma 3.2 (see [20, Theorem ]).

Let . Then is compact in , if the following conditions hold:

(a) is bounded in ;

(b)the functions belonging to are piecewise equicontinuous on any interval of ;

(c)the functions from are equiconvergent, that is, given , there corresponds such that for any and .

Lemma 3.3.

is completely continuous.

Proof.

Firstly, for , from , it is easy to check that is well defined, and for all . For

(35)

so

(36)

which shows .

Now we prove that is continuous and compact, respectively. Let as in . Then there exists such that . By we have is bounded on . Set , and we have

(37)

Therefore by the Lebesgue dominated convergence theorem and continuity of and , one arrives at

(38)

Therefore is continuous.

Let be any bounded subset of . Then there exists such that for all . Set , , then

(39)

So is bounded.

Moreover, for any and , and , then

(310)

So is quasi-equicontinuous on any compact interval of .

Finally, we prove for any , there exists sufficiently large such that

(311)

Since , we can choose such that

(312)

For , it follows that

(313)

That is (3.11) holds. By Lemma 3.2, is relatively compact. In sum, is completely continuous.

4. Existence of Three Positive Solutions

Let the nonnegative continuous concave functional , the nonnegative continuous convex functionals and , and the nonnegative continuous functionals be defined on the cone by

(41)

For notational convenience, we denote by

(42)

The main result of this paper is the following.

Theorem 4.1.

Assume hold. Let , , and suppose that satisfy the following conditions:

()

() for ,

(),

where . Then (1.1) has at least three positive solutions and such that

(43)

Proof.

Step 1.

From the definition , and , we easily show that

(44)

Next we will show that

(45)

In fact, for , then

(46)

From condition , we obtain

(47)

It follows that

(48)

Thus (4.5) holds.

Step 2.

We show that condition (i) in Theorem 2.2 holds. Taking , then and , which shows . Thus for , there is

(49)

Hence by , we have

(410)

Therefore we have

(411)

This shows the condition (i) in Theorem 2.2 is satisfied.

Step 3.

We now prove (ii) in Theorem 2.2 holds. For with , we have

(412)

Hence, condition (ii) in Theorem 2.2 is satisfied.

Step 4.

Finally, we prove (iii) in Theorem 2.2 is satisfied. Since , so . Suppose that with , then

(413)

by the condition of this theorem,

(414)

Thus condition (iii) in Theorem 2.2 holds. Therefore an application of Theorem 2.2 implies the boundary value problem (1.1) has at least three positive solutions such that

(415)

5. An Example

Now we consider the following boundary value problem

(51)

. Choose , , , . If taking , then , and . Consequently, satisfies the following:

(1), , for ;

(2), for ;

(3), , for .

Then all conditions of Theorem 4.1 hold, so by Theorem 4.1, boundary value problem (5.1) has at least three positive solutions.