Abstract
In this article, we investigate the existence and uniqueness of solutions for conformable derivatives in the Caputo setting with four-point integral conditions, applying standard fixed point theorems such as Banach contraction mapping principle, Krasnoselskii’s fixed point theorem, and Leray–Schauder nonlinear alternative. Further, we present Ulam–Hyers stability results by using direct analysis methods. Different types of Ulam stability, such as Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability, are studied. Examples which support our theoretical results are also presented.
Highlights
Fractional calculus extends the theory of differentiation and integration of integer order to real or complex order
There has been shown a great interest in the study of differential equations and inclusions with non-integer order, since fractional order models are more accurate than integer order models
We study the existence, uniqueness, and Ulam–Hyers stability of solutions for conformable derivatives in the Caputo setting with four-point integral conditions:
Summary
Fractional calculus extends the theory of differentiation and integration of integer order to real or complex order. One of the most prominent research areas in the field of fractional differential equations, which has attracted great attention from the researchers, is devoted to the existence theory of solutions. For theoretical development of the topic, we refer the reader to papers [9–23] and the references cited therein Another important and interesting area of research, which has got great attention from the researchers recently, is devoted to the stability analysis of differential equations for classical and fractional order. We study the existence, uniqueness, and Ulam–Hyers stability of solutions for conformable derivatives in the Caputo setting with four-point integral conditions:. Taking the left-fractional conformable integral operator of order β > 0 for (2.14), we have aIβ,ρ x(t) aIα+β,ρ y(t).
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