Abstract
In this article, we consider the analytic solutions of the uncertain fractional backward difference equations in the sense of Riemann–Liouville fractional operators which are solved by using the Picard successive iteration method. Also, we consider the existence and uniqueness theorem of the solution to an uncertain fractional backward difference equation via the Banach contraction fixed-point theorem under the conditions of Lipschitz constant and linear combination growth. Finally, we point out some examples to confirm the validity of the existence and uniqueness of the solution.
Highlights
Fractional calculus is based on an old idea that has become important and popular in applications only recently. e idea is to generalize integration and differentiation to noninteger orders in order to develop and extend the theory of calculus and to describe a more extensive range of possible doings in reality
Fractional differential equations have been widely employed in many fields: mathematical analysis, optics and thermal systems, control engineering, and robotics, see, for example, [1,2,3,4,5,6,7,8,9]
It is worth mentioning that the uncertainty theory of fractional difference equations is used to make the problems have a unique solution almost surely
Summary
Fractional calculus is based on an old idea that has become important and popular in applications only recently. e idea is to generalize integration and differentiation to noninteger orders in order to develop and extend the theory of calculus and to describe a more extensive range of possible doings in reality. By using modeling techniques with discrete fractional calculus, some researchers established the existence, uniqueness, monotonicity, multiplicity, and qualitative properties of solutions to uncertain fractional difference equations (UFDEs) in the sense of Riemann–Liouville, Caputo, and AB operators; for further details, see [10,11,12,13,14,15,16,17,18,19,20,21,22] and the references cited therein. For any function f: Na ⟶ R, the backward difference operator is defined by. For any function f: Na ⟶ R, the nabla fractional sum of order ] > 0 in the sense of Riemann–Liouville is defined by. For any ] > 0, the fractional Riemann–Liouville-like backward difference for uncertain sequence ξt is defined by.
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