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.

Approximate solutions of generalized logistic equation

“…The papers [5,6,7,8,9,14,16,22,28,27,30,29,36,49] are the latest studies on Ulam stability for linear differential and difference equations. In addition, research on Ulam stability for fractional differential and difference equations has increased, e.g., [1,11,13,15,20,45,47].…”

“…Al-Saffar and Kim [4] studied the dynamics of logistic models with noises from the perspective of information theory. For (13), the linear term r(τ )Γ represents a positive feedback, and the nonlinear term r(τ )Γ α K represents a negative feedback. Specifically, they investigated different oscillatory modulations in the parameters for positive and negative feedbacks and investigate their effect on the evolution of the system and probability density functions (PDFs), which is given by c | Γ| , where c > 0 is a constant.…”

Stability of Nonlinear Implicit Differential Equations with Caputo–Katugampola Fractional Derivative

“…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.…”

On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology

“…The stability of FDEs with Hadamard fractional derivative [25] was investigated by Wang et al [26] utilizing a new fractional comparison principle. In [27], the authors focused on the Ulam stability of a generalized delayed differential equation of fractional order. A more recent study presented in [28] discussed the Mittag-Leffler stability of FDEs using the new generalized Hattaf fractional (GHF) derivative [29], which includes many fractional derivatives available in the literature such as the Caputo-Fabrizio fractional derivative [30], the Atangana-Baleanu fractional derivative [31], and the weighted Atangana-Baleanu fractional derivative [32].…”