A Fixed-Point Approach to the Hyers–Ulam Stability of Caputo–Fabrizio Fractional Differential Equations

Abstract

In this paper, we study Hyers–Ulam and Hyers–Ulam–Rassias stability of nonlinear Caputo–Fabrizio fractional differential equations on a noncompact interval. We extend the corresponding uniqueness and stability results on a compact interval. Two examples are given to illustrate our main results.

Averaging Methods for Second-Order Differential Equations and Their Application for Impact Systems

“…Remark 1. The estimates (12) and ( 12) resemble to Hyers-Ulam stability-see [18]-but the purpose of these estimates is to compare solutions of averaged and original differential equations.…”

“…The estimates (18) and (19) show that there exists some t ∈ (τ 0 − A, τ 0 + A) such that v t = α which means that x intersects Γ at some time t . Moreover, the point x τ 0 − A lies in Γ − and x τ 0 + A in Γ + .…”

“…Recall that 0 > 0 is sufficiently small and fixed in (18), while ∈ (0, 0 ] is a parameter. Similarly, we obtain…”

Extremal Solutions for a Class of Tempered Fractional Turbulent Flow Equations in a Porous Medium

“…On the contrary, fractional-order derivative has nonlocal characteristics; based on this property, the fractional differential equation can also interpret many abnormal phenomena that occur in applied science and engineering, such as viscoelastic dynamical phenomena [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], advection-dispersion process in anomalous diffusion [30][31][32][33][34], and bioprocesses with genetic attribute [35,36]. As a powerful tool of modeling the above phenomena, in recent years, the fractional calculus theory has been perfected gradually by many researchers, and various different types of fractional derivatives were studied, such as Riemann-Liouville derivatives [16,, Hadamardtype derivatives [63][64][65][66][67][68][69][70][71], Katugampola-Caputo derivatives [72], conformable derivatives [73][74][75][76], Caputo-Fabrizio derivatives [77,78], Hilfer derivatives …”

Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras

“…for all x ∈ X. A number of mathematicians were attracted to this result and stimulated to investigate the stability problems of various(functional, differential, difference, integral) equations in some spaces [4][5][6][7][8][9][10][11].…”