- Research Article
- Open Access
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
This paper is concerned with nonhomogeneous multipoint boundary value problems of second-order differential equations with one-dimensional -Laplacian. Sufficient conditions to guarantee the existence of at least three solutions (may be not positive) of these BVPs are established.
- Differential Equation
- Banach Space
- Continuous Function
- Partial Differential Equation
- Ordinary Differential Equation
where , is a nonnegative and continuous function. Under some assumptions, it was proved that there exists a constant such that
(i)BVP(1.1) has at least two positive solutions if ;
(ii)BVP(1.1) has at least one solution if or ;
(iii)BVP(1.1) has no positive solution if .
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 .
where , for all , is continuous and nonnegative, is continuous with , is called -Laplacian, with , its inverse function is denoted by .
is continuous with on each subinterval of for all , where ;
, satisfy and there exists a constant such that .
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.
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.
Let be a semi-ordered real Banach space. The nonempty convex closed subset of is called a cone in if for all and and and imply .
for all and .
An operator is completely continuous if it is continuous and maps bounded sets into relative compact sets.
Lemma 2.5 (see ).
Furthermore, suppose that are constants with . Let be a completely continuous operator. If
for with ;
Choose . We call for if for all , define the norm for . It is easy to see that is a semi-ordered real Banach space.
It is easy to see that are two nonnegative continuous concave functionals on the cone are three nonnegative continuous convex functionals on cone and for all .
It follows that for all . The proof is complete.
Suppose that is a nonnegative continuous function, and hold. If is a solution of BVP(2.13), then is increasing and positive on .
It follows that . Then for since for all . The proof is complete.
Hence . Since is increasing for , we get that there exists unique constant such that (2.17) holds. The proof is completed.
Then is a cone in .
It is easy to see that there exists a constant such that for all .
Suppose that , and hold. It is easy to show that
(ii) for each ;
(iii) is a solution of BVP(1.5) if and only if and is a solution of the operator equation in cone ;
(iv) is completely continuous.
The proofs are simple and are omitted.
for all ;
for all ;
for all ;
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.
Now, we prove that all conditions of Lemma 2.5 hold. One sees that . The remainder is divided into four steps.
Prove that .
It follows that . Then .
and for every .
It follows that .
This completes Step 2.
It follows that .
This completes Step 3.
Prove that for with .
This completes Step 4.
Prove that for each with
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.
Corresponding to BVP(1.5), one sees that , . It is easy to see that , choose , then .
for all ;
for all ;
for all ;
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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