Existence of Mild Solutions for a Class of Hilfer Fractional Evolution Equations with Nonlocal Conditions

Abstract

In this paper, we consider a class of evolution equations with Hilfer fractional derivative. By employing the fixed point theorem and the noncompact measure method, we establish a number of new criteria to guarantee the existence and uniqueness of mild solutions when the associated semigroup is compact or not.

Stability of the fractional Volterra integro‐differential equation by means of ψ‐Hilfer operator

“…Fractional differential equations have already proved to be valuable tools in the modeling of several physical phenomena, because their solutions, in general, allows a better evaluation of the results in several fields of science, for example in engineering, physics, and medicine, among others . Therefore, in recent years, many authors have proposed and proved results of existence and uniqueness for the solutions of such equations, using different methods; Kilbas et al, Podlubny, Zhou, Benchohra and Lazreg, Furati and Kassim, Gu and Trujillo, Mâagli et al, Abbas et al, Yang and Wang, Abbas et al, Zhou et al, and so on are just a few of the most classical references on the subject. Furthermore, these works have paved the way for several new lines of study, such as the fractional theory applied to the so‐called impulsive equations …”

Existence and uniqueness of mild solutions to initial value problems for fractional evolution equations

“…Chen et al [19] studied the existence of mild solutions for a nonautonomous fractional evolution equations with delay in Banach space. Yang and Wang [20] established the existence and uniqueness of mild solutions of fractional evolution equations involving the Hilfer derivative by using the noncompact measure method. In [21][22][23], the authors investigated existence, uniqueness and asymptotic behavior of weak solutions of the initial boundary value problems for time fractional diffusion equations by employing the spectral decomposition of the symmetric uniformly elliptic operator.…”

Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations

“…FDEs involving HFD are widely applicable in biomedical research. These equations are successfully employed to model the irregular boundaries of biological cells and microscopic fluctuations of biomedical matters [37].…”