Exp-type Ulam-Hyers stability of fractional differential equations with positive constant coefficient

Abstract

In this paper, we apply asymptotic behavior on Mittag-Leffler functions E α (z) and E α,α (z) for z > 0 to discuss exp-type Ulam-Hyers stability of c D α t x(t) = λx(t) + f (t, x(t)) for the case λ > 0 on a finite time interval [0, 1] and an unbounded interval (1, ∞).

Hyers–Ulam Stability, Exponential Stability, and Relative Controllability of Non‐Singular Delay Difference Equations

“…Obloza [12] used this idea in differential equations, and later Jung [13] and Khan et al [14] used it in the difference equations. is stability was also discussed in fractional differential equation by Gao et al [15], and some results on Ulam-type stability of a first-order non-linear delay dynamic system were discussed by Shah et al in [16]. Recently, the Hyers-Ulam stability of second order differential equations by using Mahgoub transform and generalized Hyers-Ulam stability of a coupled hybrid system of integro-differential equations involving ϕ-caputo fractional operator was studied in [17,18].…”

Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations

“…However for non-integer order differential equations, the area has not yet properly explored and required further exploration. Recently very few articles on Hyers-Ulam stability for the solutions of differential equations of arbitrary order have been published which we refer in [3,4,7,24,25,27,31]. Motivated by the aforesaid work, this article is concerned to investigate existence and stability of solutions for fractional differential equations of arbitrary order which have boundary conditions involving fractional order derivative in the following form…”

Ulam-Hyers stability of fractional impulsive differential equations

“…The Ulam-Hyers stability of fractional differential equations has become one of most active areas, and has attracted many researchers, see [2,3,5,6,10,11,[13][14][15][16][17]. For the stability theory of impulsive dynamical systems and its applications, Wang et al [9] considered Ulam type stability of impulsive ordinary differential equation.…”