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.

Consider the following boundary value problem with integral boundary conditions on the half-line 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 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.

Yan Sun et al.[4] studied the existence of positive solutions for singular boundary value problems on the half-line for the following Sturm-Liouville boundary value problem:

where μ is a positive parameter; f is a continuous, non-negative 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, non-negative 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 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 [12] to get the existence of at least three positive solutions for the nonlinear boundary value problem (1.1) and (1.2).

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

whereis given by (2.7).

Furthermore, it is easy to prove the following properties of :

(1) is continuous on .

(2) For each , is continuously differentiable on except .

(3) .

(4) , for , where

(5) 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

where

Obviously, .

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 .

Consider the Banach space

with the norm .

From the above assumptions, we can define an operator by

where is given by (2.7).

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.

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

Let , , then

Hence, by the Lebesgue dominated convergence theorem and the fact that is continuous on t, we have

Also, by (H1) and (H2), we get

So, .

We can show that . Notice that

In addition, we have

Therefore, .

Hence, is well defined.

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

Hence, if , then

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:

where , , , , , , , .

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.

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 (Leggett-Williams 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:

(i) andfor all;

(ii) for all;

(iii) for allwith.

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

forand, ;

(H6) There existand an intervalsuch that

forand, ;

(H7) There exist, , wheresuch that

forand, .

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 .

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.

If , then by (H5) we have

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, for and , we have

Therefore, condition (iii) is also satisfied. Then the Leggett-Williams 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

 □

The authors declare that they have no competing interests.

Both authors typed, read and approved the final manuscript.

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