Abstract
In this paper, we study boundary value problems for sequential fractional differential equations and inclusions involving Hilfer fractional derivatives, supplemented with Riemann–Stieltjes integral multi-strip boundary conditions. Existence and uniqueness results are obtained in the single-valued case by using the classical Banach and Krasnosel’skiĭ fixed point theorems and the Leray–Schauder nonlinear alternative. In the multi-valued case an existence result is proved by using nonlinear alternative for contractive maps. Examples illustrating our results are also presented.
Highlights
Introduction and preliminariesDifferential equations of fractional order describe many real world processes more accurately as compared to classical order differential equations
Fractional differential equations arise in lots of engineering and clinical disciplines which include biology, physics, chemistry, economics, signal and image processing, control theory, and so on; see the monographs [1,2,3,4,5,6,7,8]
2.1 Existence and uniqueness result In our first result we prove the existence of a unique solution for the sequential Hilfer boundary value problem (1.3) via Banach’s contraction mapping principle
Summary
Two existence results are presented in this subsection. The first one is based on the wellknown Krasnosel’skii fixed point theorem [28]. By the nonlinear alternative of Leray–Schauder type [29], we conclude that the operator A has a fixed point x ∈ U This fixed point is a solution of the sequential Hilfer boundary value problem (1.3). The operators A and B satisfy all the conditions of the nonlinear alternative for contractive maps [33, Corollary 3.8]. It implies that either (i) N has a fixed point in [a, b] or (ii) there is a point x ∈ ∂BM = {x ∈ C([a, b], R) : x ≤ M} and θ ∈ (0, 1) with x = θ N (x). The Hilfer boundary value problem (3.6)–(3.7) has at least one solution on interval [1/7, 13/7]
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