Periodic Solutions in a Given Set of Differential Systems

Abstract

The existence of a periodic solution in a given set of Caratheodory differentiaĺ systems is studied. The proof of the main statement is based on the Wazewski-typė approach jointly with the degree arguments. An illustrating example is supplied, and a comparison with the analogous results of other authors is indicated. ᮊ 2001 Elsevier Science

On periodic solutions to nonlinear differential equations in Banach spaces

“…where F : D(A n ) → H is a continuous mapping. Equations (1) and (2) are abstract forms of periodic nonlinear differential equations in functional spaces. For example, let H = C[T] be the Banach space of all complex continuous functions on T = e it : −π ≤ t < π with the supremum norm and α(t) is the operator α(s)x(t) = x(t + s).…”

“…, x(t − ϕ m (t))), has form (1), and the following so-called periodic high-order functional Duffing equation Φ( The theorem on integral for the operator A, theorems on existence of periodic solutions of a linear differential equation of nth order with constant coefficients and systems of linear differential equations with constant coefficients in Banach spaces are obtained in [5].In the case of an existence of periodic solutions, evident forms of all periodic solutions of a linear differential equation of nth order with constant coefficients and systems of linear differential equations with constant coefficients in Banach spaces are given in terms of resolvent and quasi-resolvent operators of A. Existence of periodic solutions to equations of forms (1) and (2) in classical Banach spaces have been studied in many works. In particular, equations of the forms (1) and (2) for the Banach space of continuous vector-valued functions were considered in papers [3,4,7,8,14,15,19], [21]- [25] and for the Banach space of all summable vector-valued functions on T were considered in [1,10,18].…”

“…Existence of periodic solutions to equations of forms (1) and (2) in classical Banach spaces have been studied in many works. In particular, equations of the forms (1) and (2) for the Banach space of continuous vector-valued functions were considered in papers [3,4,7,8,14,15,19], [21]- [25] and for the Banach space of all summable vector-valued functions on T were considered in [1,10,18]. Our approach is based on techniques and results of the theory of periodic one-parameter groups of bounded linear operators in Banach spaces [2,6,9,17].…”