Abstract

By using the functional type cone expansion and compression fixed-point theorem in cones, some new and general results on the existence of positive solution for twin singular boundary value problems with damping term are obtained. An example is given to illustrate our results.

1. Introduction

The study of multipoint boundary value problem for linear second-order ordinary differential equation was initiated by I1'in and Moiseev [1] motivated by the work of Bitsadze and Samarskiĭ [24] on nonlocal linear elliptic boundary value problem, which is a new area of still fairly theoretical exploration in mathematics. Several authors have expounded on various aspects of this theory; see the survey paper by Gupta et al. [57] and the references cited therein. Thereamong, the study of singular boundary value problem for ordinary differential equation has led to several important applications in applied mathematics and physical science, such as the Thomas-Fermi problem which appears in determining the electrical potential in an atom. For other application results, we refer to [810]. With regards to this, increasing attention is paid to question of singular boundary value problem and has obtained many excellent results of the existence of the positive solution for two multiple points nonlinear singular boundary value problem [1115]. The main techniques are the upper and lower solutions method [16], the Leray-Schauder continuation theory [17], and the fixed-point theory in cones [18] et al. We would like to mention some results of Ma and O'Regan [13] and Yao [15], which motivated us to consider the singular boundary value problems. In [13], the authors have studied a multipoint boundary value problem where , , , , is a function satisfying Carathéodory's conditions and ; the Leray-Schauder continuation theorem leads to the existence of single solution. The literature [15] has discussed a second-order boundary value problem where is symmetric on and may be singular at both end points and . The author has proved the existence of symmetric positive solutions and established a corresponding iterative scheme, the main tool being the monotone iterative technique.

A powerful tool for proving existence of solution to boundary value problem is the fixed-point theory. In many cases, it is possible to find single, double, or multiple solutions for boundary value problem, and for the same problem, using the various methods, one can obtain different results under some appropriate conditions. To my best knowledge, very little work has been done on the existence of positive solution for boundary value problem by using the functional type cone expansion and compression fixed-point theorem. The aim of this paper is to establish some new and general results on the existence of positive solution to singular boundary value problems with damping term where , , , , , , , and satisfy(H1), , , ,(H2), ,(H3) is continuous function, may be singular at and/or , and ,(H4) satisfies the Carathéodory condition, that is, for each , the mapping is Lebesgue measurable on and for a.e. , the mapping is continuous on , and for each , there exists such that for all and for a.e. .

We will impose some advisable conditions on the nonlinearity to ensure the existence of at least one positive solution for the above problems. In order to obtain our results, we construct special operator which is the base for further discussion and provide two crucial functionals on cones. Applying the functional type cone expansion and compression fixed-point theorem to the operator and functionals, we obtain some new and general results on the existence of at least one positive solution for the twin singular problems (1.4), (1.5) and (1.4), (1.6). Our results improve and generalize those in [15, 19].

Let and be nonnegative continuous functionals on a cone in real Banach space . For positive numbers and , we define the sets We state the functional type cone expansion and compression fixed-point theorem [20].

Lemma 1.1. Let be a cone in a real Banach space , and let and be nonnegative continuous functionals on . Let be a nonempty bounded subset of , is a completely continuous operator with for . If for all , for each , and for each , , , and , the functionals satisfy the properties then has at least one positive fixed-point such that

2. Main Results

Let be the Banach space with the norm , where and , and let a cone in For , define the operator by where .

Lemma 2.1. If is a fixed-point , then is one solution of the problem (1.4), (1.5).

Proof. Suppose that is a fixed-point and , thus we have Further, The boundary condition (1.5) is satisfied due to the relation between , , and .

For , we define the nonnegative continuous functionals and on by

Lemma 2.2. Let . If , then

Proof. For , that is, . Since nondecreasing on , we have so we get (2.7). Furthermore,

Lemma 2.3. Let . If , then

Proof. For , that is to say, . In view of is nondecreasing on , for , we have Hence,

Lemma 2.4. Let ()–() hold, then is completely continuous.

Proof. By ()–, we observe that , is nondecreasing on , and , so . Since may be singular at and/or , we take the arguments to show that is completely continuous.
Assume that . In view of satisfying the Carathéodory condition, it is easy to see that as , where or . Thus, we have Therefore, This means that the operator is continuous.
Choose two sequences , satisfying for any , such that and as , respectively. Define and an operator sequence by Clearly, is a piecewise continuous function, and the operator is well defined. Further, we can see that is completely continuous.
Let , , and , where . We will prove that approach uniformly on . From the absolute continuity of integral, we obtain where . For each , , we have For each , , we have It is easy to see that, for each and , there is as . Similarly, for any , and , , , respectively, we can obtain that From the above argument, we obtain That is to say, the sequence is uniformly approximate on any bounded subset of . Therefore, is completely continuous.

For convenience, we set

We are now ready to apply a functional type cone expansion and compression fixed-point theorem to the operator to give the sufficient conditions for the existence of at least one positive solution to the problem (1.4), (1.5).

Theorem 2.5. Suppose that (H1)–(H4) hold. Assume that there exist positive numbers , , and with such that(A1), ,(A2), ,(A3),then the operator has at least one fixed-point such that and , and the problem (1.4), (1.5) has at least one positive solution such that

Proof. The cone and operator are defined by (2.1) and (2.2), respectively. By the properties of operator , it suffices to show that the conditions of Lemma 1.1 hold with respect to . In view of Lemma 2.1, it is not difficult to prove that a fixed point of is coincident with the solution of the boundary value problem (1.4), (1.5), so we concentrate on the existence of the fixed point of the operator . Set is a nonempty bounded subset of . From Lemma 2.4, it can be shown that is completely continuous by the Arzela-Ascoli lemma. For , the assumption (A1) implies that the hypotheses of (H3) lead to If , then , and so , that is, . It follows that the conditions of Lemma 1.1 hold with respect to . By the definition of functionals and , we can check that the functionals satisfy the properties for and , for and , .
We now prove that , in Lemma 1.1, holds. In fact, if , by the properties of and Lemma 2.2, for each , Hence, by the assumption (A2) and (2.28), there is Finally, we assert that , in Lemma 1.1, also holds. If , by Lemma 2.3, for, The assumption (A3) and (2.30) imply that
To sum up, the hypotheses of Lemma 1.1 are satisfied. Hence, the operator has at least one fixed point, that is, the problem (1.4), (1.5) has at least one positive solution.

Let the cone Evidently, . For , define the operator by where .

We only give the preliminary lemmas and result of the problem (1.4), (1.6), the proofs are similar to the above argument.

Lemma 2.6. If is a fixed-point , then is one solution of the problem (1.4), (1.6).

For , the nonnegative continuous functionals and on are defined by

Lemma 2.7. Let . If , then

Lemma 2.8. Let . If , then

Lemma 2.9. Let (H1)–(H4) hold, then is completely continuous.

For convenience, we set

Theorem 2.10. Suppose that (H1)–(H4) hold. Assume that , then there exist positive numbers , , and with such that(B1), ,(B2), ,(B3), .
Then the operator has at least one fixed-point such that and , and the problem (1.4), (1.6) has at least one positive solution such that

3. Examples

Consider the problems

Let It is easy to check that hypotheses (H1)–(H4) hold. For the problem (3.1), (3.2), by some calculations, we have , , and . Taking , , and , satisfying the following conditions: , , , , , and . Thus, the hypotheses of Lemma 1.1 are fulfilled, and so the operator has at least one fixed point, that is to say, the problem (3.1), (3.2) has at least one positive solution. For the problem (3.1), (3.3), by some calculations, we obtain , , and . Taking , , and , combining with the following conditions: , , , , , and . So the problem (3.1), (3.3) has at least one positive solution.