Existence of Three Monotone Solutions of Nonhomogeneous Multipoint BVPs for Second-Order Differential Equations
© Xingyuan Liu. 2008
Received: 9 March 2008
Accepted: 7 July 2008
Published: 27 July 2008
where , are given. It was proved that there exists such that BVP(1.2) has at least one positive solution if and no positive solution if . To study the existence of positive solutions of above BVPs, the Green's functions of the corresponding problems are established and play an important role in the proofs of the main results.
in papers [3–5], sufficient conditions are found for the existence of solutions of BVP(1.3) based on the existence of lower and upper solutions with certain relations. Using the obtained results, under some other assumptions, the explicit ranges of values of and are presented with which BVP has a solution, has a positive solution, and has no solution, respectively. Furthermore, it is proved that the whole plane for and can be divided into two disjoint connected regions and such that BVP has a solution for and has no solution for .
The purpose is to establish sufficient conditions for the existence of at least three solutions of BVP(1.5). It is proved that BVP(1.5) has three monotone solutions under the growth conditions imposed on for all . These solutions may not be positive. The proofs of the main results are proved by using fixed point theorem in cones in Banach spaces, Green's functions and the existence of upper and lower solutions are not used in this paper.
The remainder of this paper is organized as follows. The main results are given in Section 2 and an example to show the main results is given in Section 3.
2. Main Results
In this section, we first present some background definitions in Banach spaces and state an important three fixed point theorem. Then the main results are given and proved.
Lemma 2.5 (see ).
The proofs are simple and are omitted.
To apply Lemma 2.5, we prove that all conditions in Lemma 2.5 are satisfied. By the definitions, it is easy to see that are two nonnegative continuous concave functionals on cone , are three nonnegative continuous convex functionals on cone and for all , there exist constants such that for all . Lemma 2.9 implies that is a positive solution of BVP(1.5) if and only if and is a solution of the operator equation and is completely continuous.
This completes Step 2.
This completes Step 3.
This completes Step 4.
This completes Step 5.
The proof is complete.
Now, we present one example, whose three solutions cannot be obtained by theorems in known papers, to illustrate the main results.
The author is grateful to an anonymous referee for detailed reading and constructive comments which make the presentation of the results readable. This work is supported by Science Foundation of Hunan Educational Committee (08C) and the Natural Science Foundation of Hunan Province, China (no.06JJ5008).
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