Space ofω-Periodic Limit Functions and Its Applications to an Abstract Cauchy Problem

Abstract

We introduce a new space consisting of what we callω-periodic limit functions. We investigate some properties of the new function space. In particular, we study inclusion relations among asymptotically periodic type function spaces. Finally, we apply theω-periodic limit functions to investigate the existence and uniqueness of asymptoticallyω-periodic mild solutions of an abstract Cauchy problem.

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

“…We call f ω-periodic limit if g(t) = lim n→∞ f (t + nω) is well defined for each t ∈ R + , where n ∈ N. The collection of such functions will be denoted by P ω L(R + , X). [10]) If f, f 1 and f 2 are ω-periodic limit and g(t) = lim n→∞ f (t + nω) is well defined for each t ∈ R + , then the following statements are true:…”

“…(see [10]) Let f ∈ P ω L(R + , X) and g(t) = lim n→∞ f (t + nω) be well defined for each t ∈ R + . If g(t) = lim n→∞ f (t + nω) uniformly on R + , then f ∈ AP ω (R + , X).…”

“…(see [10]) Let f : R + × X → X be ω-periodic limit on t ∈ R + uniformly for x in bounded subsets of X and assume that f satisfies a Lipschitz condition in x uniformly in t ∈ R + :…”

“…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. In [10], Xie and Zhang investigate some properties of ω-periodic limit functions in order to study the existence and uniqueness of asymptotically ω-periodic mild solutions of the following abstract Cauchy problem x ′ (t) = Ax(t) + f (t, x(t))dt, x(0) = c 0 , where A is the infinitesimal generator of an exponentially stable C 0 -semigroup (T (t)) t≥0 . The aim of this work is to investigate properties of ω-periodic limit functions and to study the existence of asymptotically ω-periodic mild solution to the semilinear differential equation.…”