Periodic solutions for scalar lienard equations

Autor: V. S. H. Rao, Juan J. Nieto
Rok vydání: 1991
Předmět:
Zdroj: Acta Mathematica Hungarica. 57:15-27
ISSN: 1588-2632
0236-5294
Popis: in which the functions g: R ~ R and e: [0, T] ~ R are continuous and c is a real constant. I f e is periodic of period T, then it may be seen that a solution of the periodic boundary value problem (PBVP, for short) (1.1)--(1.2) is a periodic solution of period T for the equation (1.1). We refer the readers to [10, 16] and the references there in for the literature on the existence of periodic solutions for the equation (1.1). The important special case c = 0 (also known as the conservative case) is treated for T=27r in [4, 9], and more recently in [7, 15, 17]. This paper is organized as follows. Section 2 deals with the existence of solutions for the PBVP (1.1)--(1.2). While studying this problem, we distinguish two cases for the nonlinearity namely, the cases when (I) g is decreasing and (II) g is increasing. For the case (I), we use a result of [6]. We show that when g is strictly increasing the method of [6] is not useful, and in the case (II) we employ an abstract existence theorem for problems at resonance [3, 13, 14]. Also, we present existence results based on the techniques of [5]. We note that our methods and techniques are different from those employed in [10, 16], and thus our results extend some of the results in [16]. In Section 3, we study the structure of the set of solutions of the PBVP (1.1)-(1.2). Here also we consider two cases
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