Existence, uniqueness, and controllability for Hilfer differential equations on times scales

Abstract

We introduce a new version of ψ$$ \psi $$‐Hilfer fractional derivative, on an arbitrary time scale. The fundamental properties of the new operator are investigated, and in particular, we prove an integration by parts formula. Using the Laplace transform and the obtained integration by parts formula, we then propose a ψ$$ \psi $$‐Riemann–Liouville fractional integral on times scales. The applicability of the new operators is illustrated by considering a fractional initial value problem on an arbitrary time scale, for which we prove existence, uniqueness, and controllability of solutions in a suitable Banach space. The obtained results are interesting and nontrivial even for the following particular choices: (i) of the time scale, (ii) of the order of differentiation, and/or (iii) function ψ$$ \psi $$, opening new directions of investigation. Finally, we end the article with comments and future work.

The Existence of Solutions for w-Weighted ψ-Hilfer Fractional Differential Inclusions of Order μ ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces

“…For works on differential equations with Riemann-Liouville derivatives, we refer the reader to [25]; for those with Caputo derivatives, we refer the reader to [21]; for those with Katugampola derivatives, we refer the reader to [26]; and for those with Hilfer fractional derivatives, we refer the reader to [27][28][29][30].…”

“…II-Study the topological properties of the set of solutions to Problem (1). III-Extend the recent work conducted in [21,[25][26][27][28][29][30][31][32][33][34][35][36][37] when the considered fractional differential operator is replaced by D σ,v,ψ,w s i ,ϱ and the dimension of the space is infinite. IV-Generalize this work to the case where the right-hand side contains the infinitesimal generator of a strongly continuous cosine family and the nonlinear part.…”

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].…”