Nonlinear Analysis: Theory, Methods & Applications
Existence and uniqueness of anti-periodic solutions for a kind of Rayleigh equation with two deviating arguments☆
Introduction
Consider the Rayleigh equation with two deviating arguments of the form where , and are continuous functions, and are -periodic, and are -periodic in their first arguments, and .
In applied sciences, some practical problems associated with the Rayleigh equation can be found in many works in the literature. For example, an excess voltage of ferro-resonance known as some kind of nonlinear resonance having long duration arises from the magnetic saturation of inductance in an oscillating circuit of a power system, and a boosted excess voltage can give rise to some problems in relay protection. To probe this mechanism, a mathematical model was proposed in [1], [2], [3], which is a special case of the Rayleigh equation with two delays. This implies that Eq. (1.1) can represent analog voltage transmission. In a mechanical problem, usually represents a damping or friction term, represents the restoring force, is an externally applied force and is the time lag of the restoring force (see [4]). Some other examples in practical problems concerning physics and engineering technique fields can be found in [5], [6], [7]. Moreover, as pointed out in [4], [8], [9], [10], [11], [12], [13], periodic phenomena and anti-periodic phenomena are widespread in nature, and it is well known that the existence of anti-periodic solutions play a key role in characterizing the behavior of nonlinear differential equations (see [14], [15], [16], [17], [18], [19], [20]). Hence, they have been the object of intensive analysis by numerous authors. In particular, there have been extensive results on the existence of periodic solutions of Eq. (1.1) in the literature. Some of these results can be found in [4], [8], [9], [10], [11], [12], [13]. However, to the best of our knowledge, there exist no results for the existence and uniqueness of anti-periodic solutions of Eq. (1.1). This equation can stand for analog voltage transmission, and voltage transmission process is often an anti-periodic process. Thus, it is worth continuing the investigation of the existence and uniqueness of anti-periodic solutions of Eq. (1.1).
The main purpose of this paper is to establish sufficient conditions for the existence and uniqueness of anti-periodic solutions of Eq. (1.1). Our results are different from those of the references listed above. In particular, an example is also given to illustrate the effectiveness of our results.
It is convenient to introduce the following assumptions:
(A0) assume that there exists a nonnegative constant such that assume that there exists a nonnegative constant such that (A∗) ,
Section snippets
Preliminary results
For convenience of formulation, we introduce a continuation theorem [9], [21] as follows. Lemma 2.1 Let be open bounded in a linear normal space . Suppose that is a complete continuous field on . Moreover, assume that the Leray–Schauder degreeThen equation has at least one solution in .
Let be continuous in . is said to be anti-periodic on if, We will adopt the following notations:
Main results
Theorem 3.1 Let (A∗) hold. Assume that either the condition (A1) or the condition (A2) is satisfied. Then Eq.(1.1)has a unique anti-periodic solution.
Proof Consider the auxiliary equation of Eq. (1.1) as follows: By Lemma 2.3, it is easy to see that Eq. (1.1) has at most one anti-periodic solution. Thus, to prove Theorem 3.1, it suffices to show that Eq. (1.1) has at least one anti-periodic solution. To do this, we shall
An example
Example 4.1 Let . Then the Rayleigh equation has a unique anti-periodic solution with periodic . Proof By (4.1), we have , , and . It is obvious that assumptions (A1) and (A∗) hold. Hence, by Theorem 3.1, Eq. (4.1) has a unique anti-periodic solution with periodic . □
Remark 4.1 Since there exist no results for the uniqueness of anti-periodic solutions of the Rayleigh
References (22)
Periodic solutions of some Vector retarded functional differential equations
J. Math. Anal. Appl.
(1974)- et al.
Periodic solutions for a kind of Liénard equations with deviating arguments
J. Math. Anal. Appl.
(2004) - et al.
Some new results on the existence of periodic solutions to a kind of Rayleigh equation with a deviating argument
Nonlinear Anal.
(2004) - et al.
New results on the periodic solutions for a kind of Rayleigh equation with two deviating arguments
Math. Comput. Modelling
(2007) A priori bounds for periodic solutions of a delay Rayleigh equation
Appl. Math. Lett.
(1999)- et al.
On a class of second-order anti-periodic boundary value problems
J. Math. Anal. Appl.
(1992) - et al.
Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities
Nonlinear Anal.
(2001) - et al.
Anti-periodic solutions for fully nonlinear first-order differential equations
Math. Comput. Modelling
(2007) - et al.
On trigonometric and paratrigonometric Hermite interpolation
J. Approx. Theory
(2004) - et al.
On the existence of periodic solutions of overvoltage model in power system
Acta Math. Sci.
(1996)
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This work was supported by grants (06JJ2063 and 07JJ46001) from the Scientific Research Fund of Hunan Provincial Natural Science Foundation of China, and the Scientific Research Fund of Hunan Provincial Education Department of China (08C616 ).