Nonlocal Sequential Boundary Value Problems for Hilfer Type Fractional Integro-Differential Equations and Inclusions

Abstract

In the present research, we study boundary value problems for fractional integro-differential equations and inclusions involving the Hilfer fractional derivative. Existence and uniqueness results are obtained by using the classical fixed point theorems of Banach, Krasnosel’skiĭ, and Leray–Schauder in the single-valued case, while Martelli’s fixed point theorem, a nonlinear alternative for multivalued maps, and the Covitz–Nadler fixed point theorem are used in the inclusion case. Examples are presented to illustrate our results.

Some novel analysis of two different Caputo-type fractional-order boundary value problems

“…Some of them used Banach, Leray-Schauder, Krasnoselskii fixed point theorems, and the Banach contraction mapping principle. Several researchers worked by applying the Krein-Rutman theorem, fixed point theorem of Darbo-type, and Lyapunov-type inequality, to know more, see the studies [22,23,24,25,26,27] and references therein.…”

A Survey on Existence Results for Boundary Value Problems of Hilfer Fractional Differential Equations and Inclusions

“…Existence and uniqueness of solutions were studied in [25] for the following new class of boundary value problems consisting of fractional-order sequential Hilfer-type differential equations supplemented with nonlocal integro-multipoint boundary conditions of the form…”

On Ψ-Hilfer Fractional Integro-Differential Equations with Non-Instantaneous Impulsive Conditions

“…The reason for this is FDEs accurately describe many real-world phenomena such as biology, physics, chemistry, signal processing, and many more (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13]). Furthermore, it should be remarked that FDEs have interesting applications in solving inverse problems, and in the modeling of heat flow in porous material (see, e.g., [14][15][16][17][18][19][20][21][22][23][24][25]).…”