Existence and uniqueness of anti-periodic solutions for a kind of Rayleigh equation with two deviating arguments

https://doi.org/10.1016/j.na.2009.03.032Get rights and content

Abstract

In this paper, we use the Leray–Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a kind of Rayleigh equation with two deviating arguments of the form x+f(t,x(t))+g1(t,x(tτ1(t)))+g2(t,x(tτ2(t)))=e(t).

Introduction

Consider the Rayleigh equation with two deviating arguments of the form x+f(t,x(t))+g1(t,x(tτ1(t)))+g2(t,x(tτ2(t)))=e(t), where τi, e:RR and f,gi:R×RR are continuous functions, τi and e are T-periodic, f and gi are T-periodic in their first arguments, T>0 and i=1,2.

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, f usually represents a damping or friction term, gj represents the restoring force, e is an externally applied force and τj 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 C1 such that |f(t,x1)f(t,x2)|C1|x1x2|, for all t,x1,x2R;(A0˜) assume that there exists a nonnegative constant C2 such that f(t,u)=f(u),C2|x1x2|2(x1x2)(f(x1)f(x2)) for all x1,x2,uR; (A for  all  t,xR,i=1,2,f(t+T2,x)=f(t,x),gi(t+T2,x)=gi(t,x),e(t+T2)=e(t),τi(t+T2)=τi(t).

Section snippets

Preliminary results

For convenience of formulation, we introduce a continuation theorem [9], [21] as follows.

Lemma 2.1

Let Ω be open bounded in a linear normal space X . Suppose that f˜ is a complete continuous field on Ω¯ . Moreover, assume that the Leray–Schauder degreedeg{f˜,Ω,p}0, for pXf˜(Ω).Then equation f˜(x)=p has at least one solution in Ω .

Let u(t):RR be continuous in t. u(t) is said to be anti-periodic on R if, u(t+T)=u(t),u(t+T2)=u(t), for all tR. We will adopt the following notations: CTk{xCk(R,R),x

Main results

Theorem 3.1

Let (Ahold. Assume that either the condition (A1or the condition (A2is satisfied. Then Eq.(1.1)has a unique anti-periodic solution.

Proof

Consider the auxiliary equation of Eq. (1.1) as follows: x(t)=λf(t,x(t))λg1(t,x(tτ1(t)))λg2(t,x(tτ2(t)))+λe(t)=λQ1(t,x(t),x(t)),λ(0,1]. 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 g1(t,x)=g2(t,x)=(1+sin4t)112πsinx. Then the Rayleigh equation x(t)+116x(t)+116sinx(t)+g1(t,x(tsin2t))+g2(t,x(tcos2t))=140cost has a unique anti-periodic solution with periodic 2π.

Proof

By (4.1), we have b1=b2=16π,C1=18, τ1(t)=sin2t, τ2(t)=cos2t and e(t)=140cost. 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 2π. 

Remark 4.1

Since there exist no results for the uniqueness of anti-periodic solutions of the Rayleigh

References (22)

  • D.W. Lai, Z.X. Wang, D.W. Lai, A delay differential equation appeared in the study of overvoltage, Report to the...
  • Cited by (16)

    View all citing articles on Scopus

    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 ).

    View full text