Criteria of asymptotic ω-periodicity and their applications in a class of fractional differential equations

Abstract

In this paper we first characterize the (Stepanov) asymptotically ω-periodic functions.Then we apply the criteria obtained to investigate the existence and uniqueness of asymptotically ω-periodic mild solutions to semilinear fractional integro-differential equations with Stepanov asymptotically ω-periodic coefficients. MSC: 34C25; 34G20

ASYMPTOTICALLY $\omega$-PERIODIC SOLUTION FOR AN EVOLUTION DIFFERENTIAL EQUATION VIA $\omega$-PERIODIC LIMIT FUNCTIONS

“…(see [9]) A function f ∈ BS p (R + , X) is called asymptotically ω-periodic in the Stepanov sense if it can be expressed as f = g + h, where g ∈ S p P ω (R + , X) and h ∈ BS p 0 (R + , X). The collection of such functions will be denoted by S p AP ω (R + , X).…”

“…The existence of periodic solutions and its various extensions to abstract Cauchy problem has been studied in Several works ([6]- [9]). Recently, the concept of ω-periodic limit functions has been introduced by Xie and Zhang in [10].…”

“…Recently, the concept of ω-periodic limit functions has been introduced by Xie and Zhang in [10]. The class of ω-periodic limit functions generalize asymptotically ω-periodic func-tions ( [11], [12]) in a different way from S-asymptotically ω-periodic functions ( [1]- [5]) and have some relationship with asymptotically ω-periodic functions in the Stepanov sense ( [6], [9]). The concept of S-asymptotically ω-periodic have many applications in functional differential equations, integro-differential equations, fractionnal differential equation.…”

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

“…In [2], the authors prove the existence and uniqueness of mild solution to some functional differential equations with infinite delay in Banach spaces which approach almost automorphic function ( [6], [11]) at infinity and discuss also the existence of S-asymptotically ω-periodic mild solu-tions. In [20], the author discussed about the existence of S-asymptotically ω-periodic mild solution of semilinear fractionnal integro-differential equations in Banach space, where the nonlinear pertubation is S-asymptotically ω-periodic or S-asymptotically ωperiodic in the Stepanov sense ( [10], [20], [21]). The reader may also consult [3], [4], [5], [7], [12] in order to obtain more knowledge about S-asymptotically ω-periodic functions.…”

Asymptotic periodicity for a class of fractional integro--differential equations

“…In [16] the authors proved existence and uniqueness of S-asymptotically ω-periodic mild solutions for a class of linear and semilinear fractional order differential equations by using a generalization of the semigroup theory of linear operators. In [46] the authors investigated the existence and uniqueness of asymptotically ω-periodic mild solutions to semilinear fractional integro-differential equations with Stepanov asymptotically ω-periodic coefficients.…”