Existence results for nonlinear boundary value problems with integral boundary conditions on an infinite interval
© Yoruk and Hamal; licensee Springer. 2012
Received: 29 June 2012
Accepted: 22 October 2012
Published: 5 November 2012
In this paper, by using fixed point theorems in a cone, the existence of one positive solution and three positive solutions for nonlinear boundary value problems with integral boundary conditions on an infinite interval are established.
MSC:34B10, 39A10, 34B18, 45G10.
where , f may be singular at ; are continuous, nondecreasing functions and for , z in a bounded set, , are bounded; is a continuous function with ; with on and ; , for with in which .
Boundary value problems on an infinite interval appear often in applied mathematics and physics. There are many papers concerning the existence of solutions on the half-line for boundary value problems; see [1–5] and the references therein.
At the same time, boundary value problems with integral boundary conditions are of great importance and are an interesting class of problems. They constitute two, three, multi-point, and nonlocal boundary value problems as special cases. For an overview of the literature on integral boundary value problems, see [6–11] and the references therein.
where ; , , , and are symmetric functions; is continuous. The author obtained the existence of symmetric positive solutions by using the fixed point index theory in cones.
Motivated by the above works, we consider the existence of one and three positive solutions for the BVP (1.1), (1.2). However, to our knowledge, although various existence theorems are obtained for Sturm-Liouville boundary value problems with homogeneous boundary conditions, problems with nonhomogeneous boundary conditions, especially integral boundary conditions on an infinite interval have rarely been considered. Therefore, our boundary conditions are more general.
The rest of the paper is organized as follows. In Section 2, we present some necessary lemmas that will be used to prove our main results. In Section 3, we apply the Schauder fixed point theorem to get the existence of at least one positive solution for the nonlinear boundary value problem (1.1) and (1.2). In Section 4, we use the Leggett-Williams fixed point theorem  to get the existence of at least three positive solutions for the nonlinear boundary value problem (1.1) and (1.2).
where and are given in (2.5) and (2.6) respectively.
where is given by (2.7).
is continuous on .
For each , is continuously differentiable on except .
- (4), for , where(2.8)
For each , satisfies the corresponding homogeneous BVP (i.e., in the BVP (2.1)) on except .
- (6)for and
- (7)For any and , we have
It is convenient to list the following conditions which are to be used in our theorems:
(H1) and also, , , where ; and for , x, y in a bounded set, is bounded and is continuous and may be singular at ; and also, there exists such that for .
(H2) are continuous, nondecreasing functions, and for , z in a bounded set, , are bounded.
(H3) is a continuous function with .
(H4) and .
with the norm .
where is given by (2.7).
Lemma 2.2 ()
M is uniformly bounded in ℬ;
The functions belonging to M are equicontinuous on any compact interval of ;
The functions from M are equiconvergent, that is, given , there corresponds a such that for any and .
Definition 2.1 An operator is called completely continuous if it is continuous and maps bounded sets into relatively compact sets.
3 Existence of at least one positive solution
In this section, we will apply the following Schauder fixed point theorem to get an existence of one positive solution.
Theorem 3.1 (Schauder fixed point theorem)
Let ℬ be a Banach space and S be a nonempty bounded, convex, and closed subset of ℬ. Assume is a completely continuous operator. If the operator A leaves the set S invariant, i.e., if , then A has at least one fixed point in S.
where c is defined by (2.8).
Hence, is well defined.
where is a real number such that , N is a natural number set, .
Similarly, we can see that when as , as . This implies that is a continuous operator for each natural number m.
We must show that there exists a positive constant R such that for each , one has .
Therefore, is equiconvergent. Hence, by Lemma 2.2 and the above discussion, we conclude that for each natural number m, is completely continuous.
Therefore, the operator is completely continuous and maps the set into itself. Hence, the Schauder fixed point theorem can be applied to obtain a solution of the BVP (1.1), (1.2). The theorem is proved. □
where , , , , , , , .
Then by Theorem 3.2, the boundary value problem (3.16)-(3.17) has at least one positive solution.
4 Existence of at least three positive solutions
Theorem 4.1 (Leggett-Williams fixed point theorem )
and for all ;
for all ;
for all with .
Theorem 4.2 Assume that (H1)-(H4) are satisfied and there exists such that holds. Then the boundary value problem (1.1), (1.2) has at least three positive solutions if the following conditions hold:
for and , ;
for and , ;
for and , .
Proof The conditions of the Leggett-Williams fixed point theorem will be shown to be satisfied. Define the cone by and the nonnegative, continuous, concave functional by .
Therefore, we get , and this implies that .
Therefore, condition (i) of Theorem 4.1 is satisfied.
Hence, condition (ii) of Theorem 4.1 holds.
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