Abstract
This paper is concerned with the solvability of antiperiodic boundary value problems for secondorder nonlinear differential equations. By using topological methods, some sufficient conditions for the existence of solution are obtained, which extend and improve the previous results.
MSC: 34B05.
Keywords:
antiperiodic boundary value problem; existence of solution; nonlinear1 Introduction
In this paper, we will consider the existence of solutions to secondorder differential equations of the type
subject to the antiperiodic boundary conditions
where T is a positive constant and
Antiperiodic problems have been studied extensively in recent years. For example, antiperiodic boundary value problems for ordinary differential equations were considered in [19]. Also, antiperiodic boundary conditions for impulsive differential equations, partial differential equations and abstract differential equations were investigated in [1016]. The methods and techniques employed in these papers involve the use of the LeraySchauder degree theory [7,8], the upper and lower solutions [2,1012], and a fixed point theorem [1]. By using Schauder’s fixed point theorem and lower and upper solutions method, Wang and Shen in [3] considered the antiperiodic boundary value problem (1.1) and (1.2) when a firstorder derivative is not involved explicitly in the nonlinear term f, namely equation (1.1) reduces to
They proved the following theorems.
Theorem 1.1 ([[3], Theorem 2.2])
Assume there exist constants
for
where
Theorem 1.2 ([[3], Theorem 1.2])
Letγbe a positive constant. Assume there exist a continuous and nondecreasing function
for
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 antiperiodic
boundary value problem (1.1) and (1.2). The significant point here is that the righthand
side of (1.1) may depend on
The paper is organized as follows. In Section 2, we reformulate the antiperiodic
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 LeraySchauder 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
2 Preliminaries
If a function
Let
Lemma 2.1xis a solution of (2.1) if and only ifxsatisfies
where
Proof Suppose
Let
then from (2.3), we have
Multiplying both sides of the above equation by
where
Similarly, multiplying the two sides of (2.4) by
By direct computation, we get
Substituting (2.6) into (2.5),
Hence,
Further from (2.7),
and therefore
Taking into account
and
Substituting (2.8) and (2.9) into (2.7), we get
That is,
On the other hand, assume
where
And
Direct computation yields
Hence,
For later use, we present the following estimations:
Remark 2.1 The integral equation (2.2) we obtained is much simpler than that in [3] which needs a double integral.
Combining Lemma 2.1 and equation (1.1), we can easily get
Theorem 2.1The antiperiodic boundary value problem (1.1) and (1.2) is equivalent to the following integral equation:
where
Define an operator
Lemma 2.2
Proof Noting the continuity of f, this follows in a standard stepbystep process and so is omitted. □
In view of Theorem 2.1, we obtain
Theorem 2.2
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.
Firstly, we introduce some basic properties of the LeraySchauder degree. For more detail, we refer an interested reader to [19,20].
Theorem 3.1The LeraySchauder degree has the following properties.
(i) (Homotopy invariance) Let
(ii) (Existence) If
Now, we give the main result of this section.
Theorem 3.2LetM, Nandλbe positive constants inRand
If all potential solutions to (3.1) satisfy
withMandNindependent ofμ, then the antiperiodic 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
Consider the family of problems associated with (2.12), namely
Note that (3.2) is equivalent to the family of antiperiodic boundary value problems (3.1).
Now, let
From Lemma 2.2, we know that
Hence, the following LeraySchauder degrees are defined and the homotopy invariance principle in Theorem 3.1 applies:
since
4 Main results
In this section, some existence theorems are presented.
Theorem 4.1Let
with
and
then the antiperiodic 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
For each
Since
it follows that
Differentiating both sides of (4.1), we get
then
and because of
Therefore,
The rearrangement yields
By substituting (4.3) into (4.2) and rearranging, we obtain
So,
Hence, Theorem 3.2 holds for positive constants
Theorem 4.2Assume there exist nonnegative constants
then the antiperiodic boundary value problem (1.1) and (1.2) has at least one solution.
Proof Suppose
Similarly,
Therefore, Theorem 3.2 holds for positive constants
Example 4.1 Consider the antiperiodic boundary value problem
We claim (4.4) has at least one solution.
Proof Let
and
Note
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
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 antiperiodic boundary value problem given by
We claim (4.5) has at least one solution.
Proof Let
Thus, the conditions of Theorem 4.3 hold and the solvability follows. □
Remark 4.1 The results of [3] do not apply to the above example since
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):
Competing interests
The author declares that they have no competing interests.
Author’s contributions
The author typed, read and approved the final manuscript.
Acknowledgements
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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