Multiple Positive Solutions of the Singular Boundary Value Problem for Second-Order Impulsive Differential Equations on the Half-Line
© Jing Xiao et al. 2010
Received: 17 November 2009
Accepted: 21 February 2010
Published: 29 March 2010
This paper uses a fixed point theorem in cones to investigate the multiple positive solutions of a boundary value problem for second-order impulsive singular differential equations on the half-line. The conditions for the existence of multiple positive solutions are established.
Consider the following nonlinear singular Sturm-Liouville boundary value problems for second-order impulsive differential equation on thehalf-line:
The theory of singular impulsive differential equations has been emerging as an important area of investigation in recent years. For the theory and classical results, we refer the monographs to [1, 2] and the papers [3–19] to readers. We point out that in a second-order differential equation , one usually considers impulses in the position and the velocity . However, in the motion of spacecraft one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in position . The impulses only on the velocity occur also in impulsive mechanics .
In recent paper , by using the Krasnoselskii's fixed point theorem, Kaufmann has discussed the existence of solutions for some second-order boundary value problem with impulsive effects on an unbounded domain. In  Sun et al. and  Liu et al., respectively, discussed the existence and multiple positive solutions for singular Sturm-Liouville boundary value problems for second-order differential equation on the half-line. But the Multiple positive solutions of this case with both singularity and impulses are not to be studied. The aim of this paper is to fill up this gap.
The rest of the paper is organized as follows. In Section 2, we give several important lemmas. The main theorems are formulated and proved in Section 3. And in Section 4, we give an example to demonstrate the application of our results.
2. Several Lemmas
Lemma 2.1 (see ).
For the interval , and the corresponding in Remark 2.2, we define = : , and exist, . = exists . , and . It is easy to see that is a Banach space with the norm , and is a positive cone in . For details of the cone theory, see . is called a positive solution of BVP (1.1) if for all and satisfies (1.1).
Lemma 2.3 (see ).
The main tool of this work is a fixed point theorem in cones.
Lemma 2.4 (see ).
Lemma 2.6 (see ).
The proof of this result is based on the properties of the Green function, so we omit it as elementary.
Let us list some conditions as follows.
Given , for any , as the proof of (2.9), we can get that are equicontinuous on . Since is arbitrary, are locally equicontinuous on . By (2.6), , , and the Lebesgue dominated convergence theorem, we have
3. Main Results
For convenience and simplicity in the following discussion, we use the following notations:
Define the open sets:
Using a similar proof of Theorem 3.1, we can get the following conclusions.
Notice that, in the above conclusions, we suppose that the singularity only exist in , that is, as . If we permit as or and as , then the discussion will be much more complex. Now we state the corresponding results.
Let us define the following.
Corresponding to Theorem 3.2 and Corollary 3.3, there are Theorem 3.6 and Corollary 3.7. We just list here without proof.
To illustrate how our main results can be used in practice we present the following example.
This work is supported by the National Nature Science Foundation of P. R.China (10871063) and Scientific Research Fund of Hunan Provincial Education Department (07A038), partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, project no.PGIDIT06PXIB207023PR.
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