Abstract
Abstract In this paper, we discuss the solution of the inhomogeneous conformable abstract Cauchy problem. The homogeneous problem is also studied. The analysis of conformable fractional calculus and fractional semigroups is also given. Existence, uniqueness and regularity of a mild solution for the conformable abstract Cauchy problem are studied. Applications illustrating our main abstract results are also given.
Highlights
It is well known that fractional differential equations play an important role in describing many phenomena and processes in various fields of science such as physics, chemistry, control systems, electrodynamics and aerodynamics
The homogeneous problem was studied in [22], but in this paper we discuss the solution under weaker conditions
We introduce a new analysis of fractional semigroups, which is very important for solving abstract fractional Cauchy problems
Summary
It is well known that fractional differential equations play an important role in describing many phenomena and processes in various fields of science such as physics, chemistry, control systems, electrodynamics and aerodynamics. Tα is called the conformable fractional derivative of f of order α. The conformable derivative of u of order α ∈ (0, 1], at t > 0 is defined by dαu(t ) dt α lim ε→0 u(t u(t). Based on the conformable fractional derivative, Abdeljawad et al in [22] introduced the so-called C0-αsemigroup (T(t))t≥0, which is a generalization of the classical strongly continuous semigroup. The conformable α-derivative of T(t) at t = 0 is called the α-infinitesimal generator of the fractional α-semigroup T(t), with domain equals:. (ii) Let X = C([0, ∞) : ) be the Banach space of bounded uniformly continuous functions on [0, ∞) with supremum norm. The homogeneous problem was studied in [22], but in this paper we discuss the solution under weaker conditions For this purpose, we use the semigroup technique.
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