New results on anti-periodic boundary value problems for second-order nonlinear differential equations
© Liang; licensee Springer 2012
Received: 27 March 2012
Accepted: 27 September 2012
Published: 11 October 2012
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.
where T is a positive constant and is continuous. Equation (1.1) subject to (1.2) is called an anti-periodic boundary value problem.
They proved the following theorems.
Theorem 1.1 ([, Theorem 2.2])
where . Then (1.2) and (1.3) have at least one solution.
Theorem 1.2 ([, Theorem 1.2])
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 , .
That is, is a solution of (2.2).
Hence, is a solution of (2.1). This proof is complete. □
Combining Lemma 2.1 and equation (1.1), we can easily get
where and is defined in Lemma 2.1.
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 operator T.
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.
(Homotopy invariance) Let be a bounded open set, and let be compact. If for each , then is independent of t.
(Existence) If , then .
Now, we give the main result of this section.
with M and N independent of μ, 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 .
Note that (3.2) is equivalent to the family of anti-periodic boundary value problems (3.1).
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.
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.
Hence, Theorem 3.2 holds for positive constants and . The solvability of (1.1) and (1.2) now follows. □
then the anti-periodic boundary value problem (1.1) and (1.2) has at least one solution.
Therefore, Theorem 3.2 holds for positive constants and . The solvability of (1.1) and (1.2) now follows. □
We claim (4.4) has at least one solution.
Then the conclusion follows from Theorem 4.2. □
Now, we reconsider the problem (1.2) and (1.3). The following result is obtained.
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.
We claim (4.5) has at least one solution.
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.
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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