Existence uniqueness and stability of mild solutions for semilinear ψ-Caputo fractional evolution equations

Abstract

In this paper, we study the local and global existence, and uniqueness of mild solution to initial value problems for fractional semilinear evolution equations with compact and noncompact semigroup in Banach spaces. In particular, we derive the form of fundamental solution in terms of semigroup induced by resolvent and ψ-function from Caputo fractional derivatives. These results generalize previous work where the classical Caputo fractional derivative is considered. Moreover, we prove the Mittag-Leffler-Ulam-Hyers stability result. Finally, we give examples of time-fractional heat equation to illustrate the result.

Controllability of Impulsive ψ-Caputo Fractional Evolution Equations with Nonlocal Conditions

“…In this section, based on the works in [22,[37][38][39], the existence of a mild solution is obtained for our problems.…”

“…Proof of Lemma 5. By Definition 3.3 in [37], x(t) = S α ψ (t, 0)x 0 + £ t 0 {h(t)} is the mild solution on J 0 .…”

Semilinear Fractional Evolution Inclusion Problem in the Frame of a Generalized Caputo Operator

“…The existence of saturated mild (and global) solutions of Caputo-type fractional semilinear evolution problems with noncompact semigroups has been obtained in Banach spaces by Chen et al [16]. The authors in [17] studied the existence of local (and global) solutions and the uniqueness of a mild solution to fractional semilinear evolution problems with compact (and noncompact) semigroups in Banach spaces. Zhou and Jiao [18] considered a class of nonlocal Cauchy problems for a Caputo-type fractional neutral evolution problem to investigate the existence and uniqueness of mild solutions.…”

“…Motivated by the aforementioned works and inspired by [17], we consider the following fractional evolution inclusion involving ξ-Caputo FD:…”

Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving ψ -Caputo Derivative in Banach and Fréchet Spaces

“…For some recent results of stability analysis by different types of fractional derivative operator, we refer the reader to articles [55][56][57][58][59][60][61][62][63][64], as well as to the recent book by Abbas et al [65] and the references cited therein. More recently, some authors explored another form of stability known as Mittag-Leffler-Ulam-Hyers for the solutions of fractional differential equations [66][67][68][69][70][71][72][73].…”