Positive Solutions for Impulsive Equations of Third Order in Banach Space
© Jingjing Cai. 2010
Received: 4 September 2010
Accepted: 30 November 2010
Published: 15 December 2010
Using the fixed-point theorem, this paper is devoted to study the multiple and single positive solutions of third-order boundary value problems for impulsive differential equations in ordered Banach spaces. The arguments are based on a specially constructed cone. At last, an example is given to illustrate the main results.
Recently, third-order boundary value problems (cf. [1–9]) have attracted many authors attention due to their wide range of applications in applied mathematics, physics, and engineering, especially in the bridge issue. To our knowledge, most papers in literature concern mainly about the existence of positive solutions for the cases in which the spaces are real and the equations have no parameters. And many authors consider nonlinear term have same linearity. In this paper, we consider the existence of solutions when the nonlinear terms have different properties, the space is abstract and the equations have two different parameters.
Inspired by the above work, the aim of this paper is to establish some simple criteria for the existence of nontrivial solutions for BVP (1.1) under some weaker conditions. The new features of this paper mainly include the following aspects. Firstly, we consider the system (1.1) in abstract space while [3, 8] talk about equations in real space ( ). Secondly, we obtained the positive solutions when the two parameters have different ranges. Thirdly, and in system (1.1) may have different properties. Fourthly, in system (1.1) not only contains but also , which is much more complicated. Finally, the main technique used here is the fixed-point theory and a special cone is constructed to study the existence of nontrivial solutions.
We recall some basic facts about ordered Banach spaces . The cone in induces a partial order on , that is, if and only if , is said to be normal if there exists a positive constant such that implies , without loss of generality, suppose, in present paper, the normal constant . denotes the measure of noncompactness (cf. ).
Some preliminaries and a number of lemmas to the derivation of the main results are given in Section 2, then the proofs of the theorems are given in Section 3, followed by an example, in Section 4, to demonstrate the validity of our main results.
2. Preliminaries and Lemmas
For convenience, let us list the following assumption.
The proof is similar to Lemma 2.2 in , we omit it.
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
We list the assumptions:
3. Main Results
4. An Example
BVP(4.1) has at least one positive solution.
The author thanks Professor Liu and Professor Lou for many useful discussions and helpful suggestions. The work was partially supported by NSFC (10971155) and Innovation program of Shanghai Municipal Education Commission (09ZZ33).
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