Pseudo S-asymptotically ω-periodic solution for a differential equation with piecewise constant argument in a Banach space

Abstract

In this paper, we show the existence of function which is not S-asymptotically ω-periodic, but which is S-asymptotically ω-periodic in the Stepanov sense. We give sufficient conditions for the existence and uniqueness of S-asymptotically ω-periodic solutions for a nonautonomous differential equation with piecewise constant argument in a Banach space when ω is an integer. This is done using the Banach fixed point Theorem. An example involving the heat operator is discussed as an illustration of the theory.

Asymptotically $$\omega $$ ω -Periodic Functions in the Stepanov Sense and Its Application for an Advanced Differential Equation with Piecewise Constant Argument in a Banach Space

“…The equation(3) is of the formx (t) = A(t)x(t) + A 0 (t)x([t]) + f (t, x([t])), x(0) = c 0 .By Theorem 4.3, we claim thatTheorem 4.4. If L + |α| < 1 + γ 0 then the equation(3)admits a unique mild solution u(t) ∈ AP ω (R + , X).…”

“…Some concepts generalise asymptotically ω-periodic functions. It is the case of S-asymptotically ω-periodic functions ( [4], [5], [6], [9]), S-asymptotically ωperiodic functions in the Stepanov sense ( [3], [17]) and asymptotically ωperiodic function in the Stepanov sense ( [16]). S-asymptotically ω-periodic functions have properties similar to those of periodic functions, but the theory of S-asymptotically ω-periodic functions has the advantage to easily allowing the consideration of initial distortions to periodicity.…”

“…In [1], the concept of S-asymptotically ω-periodic in the Stepanov sense was introduced and the application to semilinear first-order abstract differential equations were studied. In [3], the authors show the existence of a functions wich is not S-asymptotically ω-periodic, but wich is S-asymptotically ω-periodic in the Stepanov sense. They study the existence and uniqueness of S-asymptotically ω-periodic of the following differential equation with piecewise constant argument…”

“…is studied, when ω is an integer. Motivated by the work presented in [2], [3] and [16], we investigate the existence of asymptotically ω-periodic solutions for the equation (1), when ω is an integer. This paper is organized as follows.…”

A Set of Almost Automorphic Functions and Applications

“…There have been many papers studying DEPCA, see for instance [6]- [11] and the references therein. Some papers deal with the existence of asymptotically ω-periodic solutions (see for instance [12]), S-asymptotically ω-periodic solutions of DEPCA (see [13]). Other articles deal with the existence of almost automorphic solutions of EPCA (see [14] [15]).…”