Abstract
In this paper, we study a Volterra–Fredholm integro-differential equation. The considered problem involves the fractional Caputo derivatives under some conditions on the order. We prove an existence and uniqueness analytic result by application of the Banach principle. Then, another result that deals with the existence of at least one solution is delivered, and some sufficient conditions for this result are established by means of the fixed point theorem of Schaefer. Ulam stability of the solution is discussed before including an example to illustrate the results of the proposal.
Highlights
Fractional calculus and differential equations of fractional order are of great importance since they can be used in analyzing and modeling real word phenomena [1,2,3]
There has been a very important progress in the study of the theory of differential equations of fractional order. e theory of differential equations of arbitrary order has been recently proved to be an important tool for modeling many physical phenomena
We have considered a coupled Volterra– Fredholm integro-differential equation, and we have used the Caputo derivative operator
Summary
Fractional calculus and differential equations of fractional order are of great importance since they can be used in analyzing and modeling real word phenomena [1,2,3]. There has been a very important progress in the study of the theory of differential equations of fractional order. E theory of differential equations of arbitrary order has been recently proved to be an important tool for modeling many physical phenomena. Wang et al in [21] studied a nonlinear fractional differential equations with Hadamard derivative and Ulam stability in the weighted space of continuous functions. Ahmad et al in [22] discussed the existence of solutions for an initial value problem of nonlinear hybrid differential equations of Hadamard type
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