Abstract
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.
Keywords:
positive solutions; fixed point theorems; integral boundary conditions; infinite interval1 Introduction
Consider the following boundary value problem with integral boundary conditions on the halfline of an infinite interval of the form
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 halfline for boundary value problems; see [15] 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, multipoint, and nonlocal boundary value problems as special cases. For an overview of the literature on integral boundary value problems, see [611] and the references therein.
Yan Sun et al.[4] studied the existence of positive solutions for singular boundary value problems on the halfline for the following SturmLiouville boundary value problem:
where μ is a positive parameter; f is a continuous, nonnegative function and may be singular at ; with on and ; for . Wang et al.[5] investigated the existence theorems for the boundary value problem given by
where f is a continuous, nonnegative function and may be singular at ; with on and ; for . Also, Feng [11] considered the following boundary value problem with integral boundary conditions on a finite interval:
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 SturmLiouville 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 LeggettWilliams fixed point theorem [12] to get the existence of at least three positive solutions for the nonlinear boundary value problem (1.1) and (1.2).
2 Preliminaries
In this section, we will employ several lemmas to prove the main results in this paper. These lemmas are based on the following BVP for :
Define and to be the solutions of the corresponding homogeneous equation
under the initial conditions,
Using the initial conditions (2.4), we can deduce, from equation (2.3) for and , the following equations:
Let be the Green function for (2.1), (2.2) is given by
where and are given in (2.5) and (2.6) respectively.
Lemma 2.1Suppose the conditionsandhold. Then for any, the BVP (2.1), (2.2) has the unique solution
Furthermore, it is easy to prove the following properties of :
(2) For each , is continuously differentiable on except .
(5) For each , satisfies the corresponding homogeneous BVP (i.e., in the BVP (2.1)) on except .
where
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 .
Consider the Banach space
From the above assumptions, we can define an operator by
Lemma 2.2 ([13])
Let ℬ be defined as before and. ThenMis relatively compact in ℬ if the following conditions hold:
(a) Mis uniformly bounded in ℬ;
(b) The functions belonging toMare equicontinuous on any compact interval of;
(c) The functions fromMare equiconvergent, that is, given, there corresponds asuch thatfor anyand.
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 andSbe a nonempty bounded, convex, and closed subset of ℬ. Assumeis a completely continuous operator. If the operatorAleaves the setSinvariant, i.e., if, thenAhas at least one fixed point inS.
For convenience, let us set
and
Theorem 3.2Assume conditions (H1)(H4) are satisfied. In addition, let there exist a numbersuch that
wherecis defined by (2.8).
Then the BVP (1.1), (1.2) has at least one solutionzwith
Proof Let be the operator defined by (2.9). We claim that A is a completely continuous operator. To justify this, we first show that is well defined. Let , then there exists such that and from conditions (H1) and (H2), we have
and
Hence, by the Lebesgue dominated convergence theorem and the fact that is continuous on t, we have
Also, by (H1) and (H2), we get
We can show that . Notice that
In addition, we have
Next, for any positive integer m, we denote the operator by
and prove that is completely continuous for each . Let as . We will show that as in ℬ. We know that
where is a real number such that , N is a natural number set, .
Therefore, for any , there exists a sufficiently large () such that
From the fact that as , we can see that for the above , there exists a sufficiently large natural number such that if , for any , we have
and
On the other hand, by the continuity of , for the above , there exists a , for any , , such that if , , we have
From the fact that as , there exists a natural number such that when , for any , , if , , we have
In addition to this, by the continuity of and on , for the above , there exists a for any , , such that if , we have
From as , there exists a natural number such that when , for any , if , we have
Similarly, we can see that when as , as . This implies that is a continuous operator for each natural number m.
Choose to be a bounded, convex, and closed set by
We must show that there exists a positive constant R such that for each , one has .
Let . Then for each , we have . Since f, , are positive functions, , . Furthermore, for
and
Inequalities (3.10) and (3.11) yield that . Hence, is uniformly bounded. Using the similar proof as (3.2) and (3.3), we can obtain that for any , ,
Thus, is equicontinuous. It follows from
and
Therefore, is equiconvergent. Hence, by Lemma 2.2 and the above discussion, we conclude that for each natural number m, is completely continuous.
Finally, observe that
and
Hence, inequalities (3.14) and (3.15) imply that and . Then by the assumption (H4) and the absolute continuity of the integral, we get
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. □
Example 3.1 Consider the following boundary value problem:
It is clear that is continuous and singular at . Set and , it follows from a direct calculation that , , and there exists such that the following inequality holds:
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
Definition 4.1 Let ℬ be a Banach space, be a cone in ℬ. By a concave nonnegative continuous functional φ on , we mean a continuous functional with
For being constants with and φ as above, let
and
Theorem 4.1 (LeggettWilliams fixed point theorem [12])
Let ℬ be a Banach space, be a cone of ℬ, andbe a constant. Supposeis a completely continuous operator andφis a nonnegative, continuous, concave functional onwithfor all. If there existr, L, andKwithsuch that the following conditions hold:
ThenAhas at least three positive solutions, , andinsatisfying
and
Theorem 4.2Assume that (H1)(H4) are satisfied and there existssuch thatholds. Then the boundary value problem (1.1), (1.2) has at least three positive solutions if the following conditions hold:
(H5) There exists a constantsuch that
(H6) There existand an intervalsuch that
(H7) There exist, , wheresuch that
Proof The conditions of the LeggettWilliams fixed point theorem will be shown to be satisfied. Define the cone by and the nonnegative, continuous, concave functional by .
Then we have for all . If , then and from (H7) we have
Furthermore,
Therefore, we get , and this implies that .
Now we show that condition (i) of Theorem 4.1 is satisfied. Let for . By the definition of , . Then . If , then by (H6) we get
Therefore, condition (i) of Theorem 4.1 is satisfied.
In a similar way as (4.1), we can see that for each ,
Hence, condition (ii) of Theorem 4.1 holds.
Finally, we show that condition (iii) of Theorem 4.1 is also satisfied. If , we get
and
Hence, we have
Therefore, condition (iii) is also satisfied. Then the LeggettWilliams fixed point theorem implies that A has at least three positive solutions , , and which are solutions to the problem (1.1)(1.2). Furthermore, we have
and
□
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors typed, read and approved the final manuscript.
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