Existence of Positive Solutions for Multipoint Boundary Value Problem on the Half-Line with Impulses
© J. Li and J.J. Nieto 2009
Received: 7 March 2009
Accepted: 25 April 2009
Published: 4 June 2009
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.
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].
where , , , , and , and satisfy
() , , and when is bounded, and are bounded on ;
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.
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 .
Yan , 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 .
In this section, we give some definitions and results that we will use in the rest of the paper.
for all , and .
To prove our main results, we need the following fixed point theorem due to Avery and Peterson in .
is completely continuous and there exist positive numbers , and with such that
(i) and for ;
(ii) for with ;
(iii) and for , with
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).
where is defined as (1.3).
The proof is similar to Lemma in , and here we omit it.
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 .
is completely continuous.
which shows .
Therefore is continuous.
So is bounded.
So is quasi-equicontinuous on any compact interval of .
That is (3.11) holds. By Lemma 3.2, is relatively compact. In sum, is completely continuous.
4. Existence of Three Positive Solutions
The main result of this paper is the following.
Assume hold. Let , , and suppose that satisfy the following conditions:
() for ,
Thus (4.5) holds.
This shows the condition (i) in Theorem 2.2 is satisfied.
Hence, condition (ii) in Theorem 2.2 is satisfied.
5. An Example
. 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.
This work is supported by the NNSF of China (no. 60671066), A project supported by Scientific Research Fund of Hunan Provincial Education Department (07B041) and Program for Young Excellent Talents in Hunan Normal University, The research of J. J. Nieto has been partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, Project PGIDIT06PXIB207023PR.
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