Abstract

We study the existence of positive solutions for second-order nonlinear differential equations with nonseparated boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a nonlinear alternative principle of Leray-Schauder. Recent results in the literature are generalized and significantly improved.

1. Introduction

In this paper, we establish the positive periodic solutions for the following singular differential equation: and boundary conditions where , , the nonlinearity , and denotes the quasi-derivative of . We call boundary conditions (1.2) the periodic boundary conditions which are important representatives of nonseparated boundary conditions. In particular, the nonlinearity may have a repulsive singularity at , which means that Also may take on negative values. Electrostatic or gravitational forces are the most important examples of singular interactions.

During the last few decades, the study of the existence of periodic solutions for singular differential equations have deserved the attention of many researchers [1–8]. Some classical tools have been used to study singular differential equations in the literature, including the degree theory [7–9], the method of upper and lower solutions [5, 10], Schauder’s fixed point theorem [2, 11, 12], some fixed point theorems in cones for completely continuous operators [13–15], and a nonlinear Leray-Schauder alternative principle [16–18].

However, the singular differential equation (1.1), in which the nonlinearity is dependent on the derivative and does not require to be nonnegative, has not attracted much attention in the literature. There are not so many existence results for (1.1) even when the nonlinearity is independent of the derivative. In this paper, we try to fill this gap and establish the existence of positive -periodic solutions of (1.1); proof of the existence of positive solutions is based on an application of a nonlinear alternative of Leray-Schauder, which has been used by many authors [16–18].

The rest of this paper is organized as follows. In Section 2, some preliminary results will be given, including a famous nonlinear alternative of Leray-Schauder type. In Section 3, we will state and prove the main results.

2. Preliminaries

Let us denote and by the solutions of the following homogeneous equations: satisfying the initial conditions and set Throughout this paper, we assume that (1.1) satisfies the following condition (2.4):

Lemma 2.1 (see [19]). For the solution of the boundary value problem the formula holds, where is the Green's function, and the number is defined by (2.3).

Lemma 2.2 (see [19]). Under condition (2.4), the Green’s function of the boundary value problem (2.5) is positive, that is, , for .
One denotes Thus and .

Remark 2.3. If , , then the Green’s function of the boundary value problem (2.5) has the form It is obvious that for ,, and a direct calculation shows that
Let , and we suppose that is a continuous function. Define an operator: for and . It is easy to prove that is continuous and completely continuous.

3. Main Results

In this section, we state and prove the new existence results for (1.1). In order to prove our main results, the following nonlinear alternative of Leray-Schauder is needed, which can be found in [20]. Let us define the function . The usual -norm over is denoted by and the supremum norm of is denoted by .

Lemma 3.1. Assume is a relatively compact subset of a convex set in a normed space . Let be a compact map with . Then one of the following two conclusions holds:(i) has at least one fixed point in ;(ii)there exist and such that .

Now we present our main existence result of positive solution to problem (1.1).

Theorem 3.2. Suppose that (1.1) satisfies (2.4). Furthermore, assume that there exists a constant such that(H₁) there exists a constant such that for all ;(H₂) there exist continuous, nonnegative functions , , and such that where is nonincreasing, is nondecreasing in and is nondecreasing in ;(H₃) there exist a nonincreasing positive continuous function on , and a constant such that for , where satisfies and ;(H₄) the following inequalities hold: where.

Then (1.1) has at least one positive -periodic solution with .

Proof. Since (H₄) holds, let , and we can choose such that and
To show (1.1) has a positive solution, we should only show that has a positive solution satisfying (1.2). If it is right, then is a solution of (1.1) since where is used.
Consider the family of equations: where ,, and Problem (3.6)−(1.2) is equivalent to the following fixed point of the operator equation: where is a continuous and completely continuous operator defined by and we used the fact
Now we show for any fixed point of (3.8). If not, assume that is a fixed point of (3.8) such that . Note that So we have In order to pass the solutions of the truncation equation (3.6) (with ) to that of the original equation (3.4), we need the fact that is bounded. Now we show that for a solution of (3.6).
Integrating (3.6) from 0 to (with ), we obtain Since , there exists such that ; therefore, So, Similarly we have
By (3.12), we obtain . Thus from condition (H₂) Therefore, This is a contradiction, so .
Using Lemma 3.1, we know that has a fixed point, denoted by , that is, equation has a periodic solution with . Using similar procedure to that of the proof of (3.13), we can prove that In the next lemma, we will show that have a uniform positive lower bound, that is, there exists a constant , independent of , such that for all .
The fact and shows that is a bounded and equi-continuous family on . Thus the Arzela–Ascoli Theorem guarantees that has a subsequence, converging uniformly on to a function . is uniformly continuous since satisfies for all . Moreover, satisfies the integral equation Letting , we arrive at Therefore, is a positive periodic solution of (1.1) and satisfies .

Lemma 3.3. There exists a constant such that any solution of (3.6) (with ) satisfies (3.23) for all large enough.

Proof. By condition (H₃), there exist and a continuous function such that for all , where satisfies condition also like in (H₃).
Choose such that , and let . For , let We first show that for all . If not, assume that for some .
If , we obtain from (3.26) and, if , we obtain So we have Integrating (3.6) (with ) from 0 to , we deduce that This is a contradiction. Thus , and we have
To prove (3.23), we first show
Let ; here if , and if . If , it is easy to verify (3.33) is satisfied. We now show (3.33) holds if . If not, suppose there exists with for some . As , by , there exists (without loss of generality, we assume ) such that and for .
From (3.26), we easily show that
Using (3.6) (with ) for , we have, for ,
As , , so for all , and the function is strictly decreasing on . We use to denote the inverse function of restricted to . Thus there exists such that and By using the method of substitution, we obtain By the facts , and are bounded, one can easily obtain that the second term is bounded. The first term is which is also bounded. As a consequence, there exists such that
On the other hand, by (H₃), we can choose large enough such that for all . This is a contradiction. So (3.33) holds.
Finally, we will show that (3.23) is right in . Noticing estimate (3.33) and employing the method of substitution, we obtain Obviously, the right-hand side of the above equality is bounded. On the other hand, by (H₃), if . Thus we know that for some constant ; the proof is completed.

Corollary 3.4. Let the nonlinearity in (1.1) be where , , , , and is a positive parameter,(i)if , then (1.1) has at least one positive periodic solution for each ;(ii)if , then (1.1) has at least one positive periodic solution for each , where is some positive constant.

Proof. We will apply Theorem 3.2. Take Then conditions (H₁)–(H₃) are satisfied and the existence condition (H₄) becomes for some . So (1.1) has at least one positive periodic solution for Note that if and if . We have (i) and (ii).