Ulam stability of Caputo q-fractional delay difference equation: q-fractional Gronwall inequality approach

Abstract

In this article, we discuss the existence and uniqueness of solution of a delay Caputo q-fractional difference system. Based on the q-fractional Gronwall inequality, we analyze the Ulam-Hyers stability and the Ulam-Hyers-Rassias stability. An example is provided to support the theoretical results.

On two structures of the fractional q‐sequential integro‐differential boundary value problems

“…In the subsequent pace, Miller 23 introduced a composition of quantum differential equations and Lie theory and obtained some new results in this context. By following this trend in the next decades, numerous researchers generalized this theory and derived useful findings on the q-differential equations and inclusions (for more details, see previous studies [24][25][26][27][28][29][30][31][32] ).…”

An Existence Study on the Fractional Coupled Nonlinear q -Difference Systems via Quantum Operators along with Ulam–Hyers and Ulam–Hyers–Rassias Stability

“…So later, various mathematical q-difference fractional models of IVPs and BVPs have been presented in which different methods like the lower-upper solutions technique, fixedpoint results, and iterative methods have been implemented. For instance, we see q-intego-equation on time scales in [12], q-delay equations in [13], q-integro-equations under the q -integral conditions in [14], singular q-equations in [15], q -sequential symmetric BVPs in [16], q-difference equations having p-Laplacian in [17], four-point q-BVP with different orders in [18], oscillation on q-difference inclusions in [19], etc.…”

q -fractional functional integro-differential equation with q -integral condition

“…q-calculus, This area of research has several applications, see [2,10,11] and references therein. There are several developments and applications of the q-calculus in mathematical physics, the theory of relativity and special functions [1,4]. In several papers [13,15], integro-differential equation with infinitepoint boundary conditions have been studied.…”