Positive Solutions for Integral Boundary Value Problem with ϕ-Laplacian Operator
© Yonghong Ding. 2011
Received: 20 September 2010
Accepted: 19 January 2011
Published: 23 February 2011
We consider the existence, multiplicity of positive solutions for the integral boundary value problem with -Laplacian , , , , where is an odd, increasing homeomorphism from onto . We show that it has at least one, two, or three positive solutions under some assumptions by applying fixed point theorems. The interesting point is that the nonlinear term is involved with the first-order derivative explicitly.
where the nonlinear term does not depend on the first-order derivative and , . They obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity of by applying Krasnoselskii fixed point theorem.
The main tools are the priori estimate method and the Leray-Schauder fixed point theorem. However, there are few papers dealing with the existence of positive solutions when satisfies (H1) and depends on both and . This paper fills this gap in the literature. The aim of this paper is to establish some simple criteria for the existence of positive solutions of BVP(1.1). To get rid of the difficulty of depending on , we will define a special norm in Banach space (in Section 2).
This paper is organized as follows. In Section 2, we present some lemmas that are used to prove our main results. In Section 3, the existence of one or two positive solutions for BVP(1.1) is established by applying the Krasnoselskii fixed point theorem. In Section 4, we give the existence of three positive solutions for BVP(1.1) by using a new fixed point theorem introduced by Avery and Peterson. In Section 5, we give some examples to illustrate our main results.
Lemma 2.1 (see ).
From the fact that , we know that is strictly decreasing. It follows that is also strictly decreasing. Thus, is strictly concave on [0, 1]. Without loss of generality, we assume that . By the concavity of , we know that , . So we get . By , it is obvious that . Hence, , .
By a similar argument in , ; then the proof is completed.
So is monotone decreasing continuous and . Hence, is nonnegative and concave on [0, 1]. By computation, we can get . This shows that . The continuity of is obvious since is continuous. Next, we prove that is compact on .
Hence, is uniformly bounded and equicontinuous. So we have that is compact on . From (2.13), we know for , , such that when , we have . So is compact on ; it follows that is compact on . Therefore, is compact on .
Lemma 2.6 (see ).
To obtain positive solution for BVP(1.1), the following definitions and fixed point theorems in a cone are very useful.
Theorem 2.8 (see ).
Theorem 2.9 (see ).
3. The Existence of One or Two Positive Solutions
Then BVP(1.1) has at least one positive solution.
The proof is similar to the (i) and (ii); here we omit it.
In the following, we present a result for the existence of at least two positive solutions of BVP(1.1).
Assume that (H1) and (H2) hold. In addition, suppose that one of following conditions is satisfied.
Then BVP(1.1) has at least two positive solutions.
4. The Existence of Three Positive Solutions
We will show that all the conditions of Theorem 2.9 are satisfied.
In this section, we give three examples as applications.
Hence, by Theorem 3.1, BVP(5.1) has at least one positive solution.
Hence, by Theorem 3.2, BVP(5.5) has at least two positive solutions.
The research was supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and Project of NWNU-KJCXGC-3-47.
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