On Ulam Stability of a Generalized Delayed Differential Equation of Fractional Order

Abstract

We investigate Ulam stability of a general delayed differential equation of a fractional order. We provide formulas showing how to generate the exact solutions of the equation using functions that satisfy it only approximately. Namely, the approximate solution $$\phi $$ ϕ generates the exact solution as a pointwise limit of the sequence $$\varLambda ^n\phi $$ Λ n ϕ with some integral (possibly, nonlinear) operator $$\varLambda $$ Λ . We estimate the speed of convergence and the distance between those approximate and exact solutions. Moreover, we provide some exemplary calculations, involving the Chebyshev and Bielecki norms and some semigauges, that could help to obtain reasonable outcomes for such estimations in some particular cases. The main tool is the Diaz–Margolis fixed point alternative.

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“…In fact, Ulam stability theory helps us to arrive at an efficient and reliable technique for approximating fractional differential equations, and when a given problem is stable, it is believed that there is an approximate solution to fractional differential equations. The study of Ulam-Hyers stability is widely used in algebra, functional analysis, calculus and dynamic systems [20][21][22][23][24][25][26]. The main methods include the successive approximation method, fixed-point theorem and the direct analysis method, among which the research on Ulam-Hyers stability and Ulam-Hyers-Rassias stability has become one of the central themes of mathematical analysis.…”

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