Existence of periodic solutions

Abstract

ABSTRACT. An apparatus for proving existence theorems for periodic solutions of equations with discontinuous right-hand side and differential inclusions is developed.KEY WORDS: differential equations and inclusions, topology of the solution space, existence theorems, Cauchy problem, periodic solutions.In this paper, we explain how to extend the applicability area of the approach [1] to the existence of periodic solutions for wider classes of ordinary differential equations and inclusions.We shall use the ideas developed in [2][3][4]. Part of the notation from [2] will be used without additional explanations (see also [3,4]).Earlier, similar results were proved for equations with a continuous right-hand side or equations whose right-hand side satisfies the Carath~odory condition. As a consequence of the Cauchy problem theory that emerges in the context of our general approach (see [2][3][4][5]), our results turn out to be applicable to equations and inclusions with complicated discontinuities in their right-hand sides, including discontinuities in the space variables.w The length of the existence interval for solutions of the Cauchy problemWe assume that the right-hand side of the differential inclusion in question, Stated in this way, the uniqueness of the zero solution turns out to be similar to the following condition:any solution of inclusion (1) whose domain touches the segment [to, tl] can be continued to the entire segment It0, C1].Indeed, under natural restrictions on (1), the violation of this condition implies that inclusion (1) has a solution z(C), C0 _< t < C1, in the domain U0 such that [Iz(c)l[ --* cc as c -, tl or a solution z(t),Therefore, one can apply the considerations used in [6] to study the length of the existence interval for the solution. In order to simplify formulations, we shall not strive for the greatest possible generality.Recall some key notations from [2-4].The symbol Ri(U) denotes the set of all Z C Cs(U) that satisfy the condition if z E Z and a segment I lies in the domain ~r(z) of the function z, then zli E Z.

On acyclicity of the set of solutions to the Cauchy problem for differential equations

“…In [6] it was proved that the closure of the set/L:e,(U) C/'L:(U) lies in/L:eh(U), where/L:,h(U) denotes the set of elements of Rce(U) that possess a property characterizing the acyclicity of the set of solutions to the Cauchy problem (this property will be described later). Note that the set Rceh(U) appeared when the existence conditions for periodic solutions to differential equations were studied.…”