This paper is concerned with the solvability of anti-periodic boundary value problems for second-order nonlinear differential equations. By using topological methods, some sufficient conditions for the existence of solution are obtained, which extend and improve the previous results.
Keywords:anti-periodic boundary value problem; existence of solution; nonlinear
In this paper, we will consider the existence of solutions to second-order differential equations of the type
subject to the anti-periodic boundary conditions
where T is a positive constant and is continuous. Equation (1.1) subject to (1.2) is called an anti-periodic boundary value problem.
Anti-periodic problems have been studied extensively in recent years. For example, anti-periodic boundary value problems for ordinary differential equations were considered in [1-9]. Also, anti-periodic boundary conditions for impulsive differential equations, partial differential equations and abstract differential equations were investigated in [10-16]. The methods and techniques employed in these papers involve the use of the Leray-Schauder degree theory [7,8], the upper and lower solutions [2,10-12], and a fixed point theorem . By using Schauder’s fixed point theorem and lower and upper solutions method, Wang and Shen in  considered the anti-periodic boundary value problem (1.1) and (1.2) when a first-order derivative is not involved explicitly in the nonlinear term f, namely equation (1.1) reduces to
They proved the following theorems.
Theorem 1.1 ([, Theorem 2.2])
Assume there exist constants , , and functions such that
for and . Further suppose that
where . Then (1.2) and (1.3) have at least one solution.
Theorem 1.2 ([, Theorem 1.2])
Letγbe a positive constant. Assume there exist a continuous and nondecreasing function and a nonnegative function with
for and . Further suppose that
Then (1.2) and (1.3) have at least one solution.
In this paper, we are interested in the existence of a solution to the anti-periodic boundary value problem (1.1) and (1.2). The significant point here is that the right-hand side of (1.1) may depend on . The dependence of right-hand side on is naturally seen in many physical phenomena, and we refer the readers to [17,18] for some nice examples. If there appears in nonlinear term, the relative boundary value problem will be more complicated. Meanwhile, we note equation (1.4) or (1.5) implies that is at most linear for x, so the problem has not been solved when is super-linear for x. Motivated by the above two aspects, we devote ourselves to studying the anti-periodic boundary value problem (1.1) and (1.2).
The paper is organized as follows. In Section 2, we reformulate the anti-periodic boundary value problem (1.1) and (1.2) as an equivalent integral equation, which is a widely used technique in the theory of boundary value problem. In Section 3, a general existence result is presented for (1.1) and (1.2). The result provides a natural motivation for the obtention of a priori bounds on solutions and greatly minimizes the proofs of the new results in the following section. The main tool used here is the Leray-Schauder topological degree. In Section 4, some new conditions are presented for (1.1) and (1.2). The new conditions involve linear or quadratic growth constraints on in .
If a function satisfies equations (1.1) and (1.2), we call x a solution of (1.1) and (1.2). Let be a Banach space with the norm , where , .
Let , and consider the anti-periodic boundary value problem
Lemma 2.1xis a solution of (2.1) if and only ifxsatisfies
Proof Suppose is a solution of (2.1) and denote , then the first equation of (2.1) can be rewritten as
then from (2.3), we have
Multiplying both sides of the above equation by and integrating from 0 to t yields
Similarly, multiplying the two sides of (2.4) by and integrating from 0 to t yields
By direct computation, we get
Substituting (2.6) into (2.5),
Further from (2.7),
Taking into account , , we obtain
Substituting (2.8) and (2.9) into (2.7), we get
That is, is a solution of (2.2).
On the other hand, assume is a solution of (2.2). Then
Direct computation yields
Hence, is a solution of (2.1). This proof is complete. □
For later use, we present the following estimations:
Remark 2.1 The integral equation (2.2) we obtained is much simpler than that in  which needs a double integral.
Combining Lemma 2.1 and equation (1.1), we can easily get
Theorem 2.1The anti-periodic boundary value problem (1.1) and (1.2) is equivalent to the following integral equation:
where and is defined in Lemma 2.1.
Define an operator by
Lemma 2.2 is completely continuous.
Proof Noting the continuity of f, this follows in a standard step-by-step process and so is omitted. □
In view of Theorem 2.1, we obtain
Theorem 2.2 is a solution of the anti-periodic boundary value problem (1.1) and (1.2) if and only if is the fixed point of the operatorT.
3 General existence
In this section, an abstract existence result is presented for (1.1) and (1.2). The obtained result emphasizes the natural search for a priori bounds on solutions to the boundary value problem, which will be conducted in the following section.
Theorem 3.1The Leray-Schauder degree has the following properties.
(i) (Homotopy invariance) Let be a bounded open set, and let be compact. If for each , then is independent oft.
(ii) (Existence) If , then .
Now, we give the main result of this section.
Theorem 3.2LetM, Nandλbe positive constants inRand be continuous. Consider the family of anti-periodic boundary value problems:
If all potential solutions to (3.1) satisfy
withMandNindependent ofμ, then the anti-periodic boundary value problem (1.1) and (1.2) has at least one solution.
Proof In view of Theorem 2.2, we want to show there exists at least one with x satisfying . This solution will then naturally be in .
Consider the family of problems associated with (2.12), namely
Note that (3.2) is equivalent to the family of anti-periodic boundary value problems (3.1).
Now, let with
From Lemma 2.2, we know that is completely continuous. Therefore, is a compact mapping. By the assumption of the theorem, all possible solutions must satisfy , and thus
Hence, the following Leray-Schauder degrees are defined and the homotopy invariance principle in Theorem 3.1 applies:
since . By the existence property of the Leray-Schauder degree, (3.2) has at least one solution in Ω for all . And hence (1.1) and (1.2) has at least one solution. □
4 Main results
In this section, some existence theorems are presented.
Theorem 4.1Let , and be nonnegative constants and . Iffis continuous and satisfies
then the anti-periodic boundary problem (1.1) and (1.2) has at least one solution.
Proof Consider the family (3.1). We want to show the conditions of Theorem 3.2 hold for some positive constants M and N.
Let be a solution to (3.1) and consider the equivalent equation (3.2), that is,
For each , we have
it follows that
Differentiating both sides of (4.1), we get
and because of
The rearrangement yields
By substituting (4.3) into (4.2) and rearranging, we obtain
Hence, Theorem 3.2 holds for positive constants and . The solvability of (1.1) and (1.2) now follows. □
Theorem 4.2Assume there exist nonnegative constants , and such that
then the anti-periodic boundary value problem (1.1) and (1.2) has at least one solution.
Proof Suppose is a solution of (3.1), and in view of (2.10), we have
Therefore, Theorem 3.2 holds for positive constants and . The solvability of (1.1) and (1.2) now follows. □
Example 4.1 Consider the anti-periodic boundary value problem
We claim (4.4) has at least one solution.
Proof Let , and in Theorem 4.2. Choose , we get for that
Note , we have . Thus, for ,
Then the conclusion follows from Theorem 4.2. □
Now, we reconsider the problem (1.2) and (1.3). The following result is obtained.
Theorem 4.3Suppose is continuous. If there exist nonnegative constants , and such that
then (1.2) and (1.3) has at least one solution.
Proof The proof is similar to Theorem 4.2 and here we omit it. □
An example to highlight the Theorem 4.3 is presented.
Example 4.2 Consider the anti-periodic boundary value problem given by
We claim (4.5) has at least one solution.
Proof Let and see that for . For , and λ to be chosen below, see that
Thus, the conditions of Theorem 4.3 hold and the solvability follows. □
Remark 4.1 The results of  do not apply to the above example since grows more than linearly in . Therefore, we improve the previous results.
Finally, in order to illustrate our main results, we use the ‘bvp4c’ package in MATLAB to simulate. As shown in Figure 1(a) and (b), numerical simulations also suggest that Examples 4.1 and 4.2 with the given coefficients admit at least one solution.
Figure 1. Solutions found by numerical stimulations with (a): , in equation (4.5); (b): , in equation (4.4).
The author declares that they have no competing interests.
The author typed, read and approved the final manuscript.
The author would like to thank anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of work. This research was partially supported by the NNSF of China (No. 11001274) and the Postdoctoral Science Foundation of Central South University and China (No. 2011M501280).
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