Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space

Abstract

In the presented research, the uniqueness and existence of a mild solution for a fractional system of semilinear evolution equations with infinite delay and an infinitesimal generator operator are demonstrated. The generalized Liouville–Caputo derivative of non-integer-order 1α≤2 and the parameter 0ρ1 are used to establish our model. The ρ-Laplace transform and strongly continuous cosine and sine families of uniformly bounded linear operators are adapted to obtain the mild solution. The Leray–Schauder alternative theorem and Banach contraction principle are used to demonstrate the mild solution’s existence and uniqueness in abstract phase space. The results are applied to the fractional wave equation.

Keywords

• generalized Liouville–Caputo fractional derivative
• <i>ρ</i>-Laplace transformation
• infinite time delay
• mild solution
• Leray–Schauder alternative