Applications of Fixed Point and Approximate AlgorithmsView this Special Issue
Existence of Positive Solutions for Sturm-Liouville Boundary Value Problems with Linear Functional Boundary Conditions on the Half-Line
By using the Leggett-Williams fixed theorem, we establish the existence of multiple positive solutions for second-order nonhomogeneous Sturm-Liouville boundary value problems with linear functional boundary conditions. One explicit example with singularity is presented to demonstrate the application of our main results.
In this paper, we consider the following Sturm-Liouville boundary value problems on the half-line where is a continuous function, on any subinterval of , here is a Lebesgue integrable function and may be singular at with are linear positive functionals on which are called positive if for .
The theory of nonlocal boundary value problems for ordinary differential equations arises in different areas of applied mathematics and physics. There are many studies for nonlocal, including three-point, m-point, and integral boundary value problems on finite interval by applying different methods [1–3]. It is well known that boundary value problems on infinite interval arise in the study of radial solutions of nonlinear elliptic equations and models of gas pressure in a semi-infinite porous medium [4–6]. But the theory of Sturm-Liouville nonhomogeneous boundary value problems on infinite interval is yet rare.
The linear functional boundary conditions cover some nonlocal three-point, m-point, and integral boundary conditions. In , Zhao and Li investigated some nonlinear singular differential equations with linear functional boundary conditions. However, the differential equations were defined only in a finite interval. Recently, Liu et al.  studied multiple positive solutions for Sturm-Liouville boundary value problems on the half-line However, the authors did not consider the case when Sturm-Liouville boundary value problems are nonhomogeneous. Therefore BVP(1.1) is the direct extension of . So it is worthwhile to investigate BVP(1.1).
In this paper, we always assume that the following conditions hold. and . For any constant and
Motivated and inspired by [5–9], we are concerned with the existence of multiple positive solutions for BVP(1.1) by applying Leggett-Williams fixed theorem. The main new features presented in this paper are as follows. Firstly, Sturm-Liouville nonhomogeneous boundary value problems with linear functional boundary conditions are seldom researched, it brings about many difficulties when we imply the integral equations of BVP(1.1). To solve the problem, we use a new method of undetermined coefficient to obtain the integral equations of boundary value problems with nonhomogeneous boundary conditions. Secondly, we discuss the existence of triple positive solutions and positive solutions of BVP(1.1). Finally, the methods used in this paper are different from [1, 6, 7] and the results obtained in this paper generalize and involve some results in .
The rest of paper is organized as follows. In Section 2, we present some preliminaries and lemmas. We state and prove the main results in Section 3. Finally, in Section 4, one example with a singular nonlinearity is presented to demonstrate the application of Theorem 3.1.
In order to discuss the main results, we need the following lemmas.
Lemma 2.1. Under the condition and , the boundary value problem has a unique solution for any . Moreover, this unique solution can be expressed in the form where , and are defined by
Proof. and in (1.3) are two linear independent solutions of the equation , so the general solutions for the equation can be expressed in the form
where , are undetermined constants. Through verifying directly, when and satisfy and separately, in (2.4) is a solution of BVP(2.1).
Now we need to prove that when in (2.4) is a solution of BVP(2.1), and satisfy and separately.
Let be a solution of BVP(2.1), then That is, .
By (2.4), we have then From (2.7), we obtain that and satisfy and separately. The proof is completed.
Remark 2.2. Assume that holds. Then for any and any solution of BVP(2.1) is nonnegative.
Lemma 2.4. For any constant , there exists , such that, for .
Proof. By (1.3), it is obvious that is increasing, and is decreasing on ; therefore, by (2.3), we have We take , then ; this is because that By Lemma 2.3, we have , then Similarly, we can obtain that . The proof is completed.
In this paper, we use the space with the norm , where and , then is a Banach space.
Let Clearly is a cone of .
Lemma 2.5 (see ). Let exists}, then is precompact if the following conditions hold:(a) is bounded in ;(b)the functions belonging to are locally equicontinuous on any interval of ;(c)the functions from are equiconvergent; that is, given , there corresponds such that for any and .
We shall consider nonnegative continuous and concave functional on ; that is, is continuous and satisfies We denote the set by and The key tool in our approach is the following Leggett-Williams fixed point theorem.
Theorem 2.6 (see ). Let be completely continuous and a nonnegative continuous concave functional on with for any . Suppose that there exist such that(), and , for ;(), for ;() for with .Then has at least three fixed points , with
3. Existence Results
Define the operator by Then is a fixed point of operator if and only if is a solution of BVP(1.1).
For convenience, we denote by
Theorem 3.1. Suppose that hold, and assume there exist with , such that, , , ,,, , , .Then BVP(1.1) has at least three positive solutions , , and with
Proof. Firstly we prove that is continuous.
We will show that is well defined and . For all , by , and are nonnegative functions, and we have . From , , we obtain In the same way, we have By Lemma 2.3, , , and , for all , we have Hence, is well defined. By (3.1), , the Lebesgue dominated convergence theorem and the continuity of , for any , we have That is, ; therefore, .
By Lemma 2.4, we have therefore .
We show that is continuous. In fact suppose and , then there exists , such that . By , we have Therefore, by Lemma 2.3, the continuity of and the Lebesgue dominated convergence theorem imply that Thus, . Therefore is continuous.
Secondly we show that is compact operator.
For any bounded set , there exists a constant such that , for all . By Lemma 2.3, , , and , we have Therefore, .
By (3.4) and (3.5), we have so is bounded.
Given , by and Lemma 2.3(1), we have Therefore for any , by (3.1), the Lebesgue dominated convergence theorem and the continuity of , , and , we have By a similar proof as (3.6), we obtain , as . Thus, is equicontinuous on . Since is arbitrary, is locally equicontinuous on .
By Lemma 2.3(2), and the Lebesgue dominated convergence theorem, we obtain By (3.5), we know that , then Therefore, is equiconvergent at . By Lemma 2.5, is completely continuous.
Finally we will show that all conditions of Theorem 2.6 hold.
From the definition of , we can get for all . For all , we have ; therefore , . By (3.4), (3.5), and , we have that is, for . Thus .
Similarly for any , we have , which means that condition of Theorem 2.6 holds.
In order to apply condition of Theorem 2.6, we choose , then ; this is because and , which means that . For all , we have and , thus , that is, . By , we can get Consequently condition of Theorem 2.6 holds.
We will prove that condition of Theorem 2.6 holds. If , and , by , we have Therefore, condition of Theorem 2.6 is satisfied. Then we can complete the proof of this theorem by Leggett-Williams fixed point theorem.
Theorem 3.2. Suppose that , hold, and assume there exist with , such that, , , , ,, , , , .Then BVP(1.1) has at least positive solutions.