Existence of Mild Solutions for Nonlocal Evolution Equations with the Hilfer Derivatives

Abstract

The existence of mild solutions for Hilfer fractional evolution equations with nonlocal conditions in a Banach space is investigated in this manuscript. No assumptions about the compactness of a function or the Lipschitz continuity of a nonlinear function are imposed on the nonlocal item and the nonlinear function, respectively. However, we assumed that the nonlocal item is continuous, the nonlinear term is continuous and satisfies some specified assumptions, and the associated semigroup is compact. Our theorems are proved by means of approximate techniques, semigroup methods, and fixed point theorem. These methods are useful for fixing the noncompactness of operators caused by some specified given assumptions on this paper. The results obtained here improve some known results. Finlay, two examples are presented for illustration of our main results.

The Regional Enlarged Observability for Hilfer Fractional Differential Equations

“…In this section, we will characterize the ReEnOb of System (4) with the output function (5) in the subregion ω of Ω. Let ω be a positive Lebesgue measure, and let us define the restriction mapping (projection mapping) p ω , as follows:…”

“…In recent decades, fractional calculus theory has proven to be a significant tool for the formulation of several problems in science and engineering, where fractional derivatives and integrals can be utilized to describe the characteristics of various real materials in various scientific disciplines; see, e.g., [1][2][3][4][5]. This theory has recently received a large amount of consideration by many academics; we mention Euler, Laplace, Riemann, Liouville, Marchaud, Riesz, and Hilfer; see, e.g., [6][7][8].…”

“…This paper is interested in the concept of ReEnOb for the following sub-diffusion system via Hilfer FDs of order η, type κ and augmented with the OuPuFu (5). We first characterize the ReEnOb of a diffusion system augmented with the OuPuFu in an internal subregion ω of Ω.…”

Mild Solutions for w-Weighted, Φ-Hilfer, Non-Instantaneous, Impulsive, w-Weighted, Fractional, Semilinear Differential Inclusions of Order μ ∈ (1, 2) in Banach Spaces

“…More than thirty years ago, the study of the existence of a mild solution to semi-linear differential Equations and semi-linear differential inclusions containing a fractional differential operator became of interest. Some of these equations contained the Caputo fractional derivative [10][11][12], some involved the Riemann-Liouville fractional differential operator [13,14], some contained the Caputo-Hadamard fractional differential operator [15,16], some included the Hilfer fractional differential operator of order α ∈ (0, 1) in [17][18][19][20][21][22][23][24][25][26], some contained the Katugampola fractional differential operator [27], some contained the Hilfer-Katugampola fractional differential operator of order α ∈ (0, 1) [28][29][30][31][32] and others involved the Hilfer fractional differential operator of order λ ∈ (1, 2) [33].…”

“…Wang et al [20] showed solutions for (3) with the existence of non-instantaneous impulses and where f is a multi-valued function and studied the controllability of the problem. Very recently, Elbukhari et al [23] proved the existence of a mild solution for Problem (3), when Ξ is the infinitesimal generator of a compact C 0 -semigroup and g does not satisfy any assumption such as compactness or Lipschitz continuity, making their findings interesting. Suechoei et al [35] derived the formula of a mild solution for Problem (1) in the particular cases µ ∈ (0, 1),…”