On existence and stability results for nonlinear fractional delay differential equations

Abstract

We establish existence and uniqueness results for fractional order delay differential equations. It is proved that successive approximation method can also be successfully applied to study Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability, generalized Ulam-Hyers-Rassias stability, Eα-Ulam-Hyers stability and generalized Eα-Ulam-Hyers stability of fractional order delay differential equations.

On the Stability of Fractional Differential Equations Involving Generalized Caputo Fractional Derivative

“…is theory helps us getting an efficient and reliable technique for approximately solving fractional differential equations because there exists a close exact solution when the given problem is Ulam stable. More details from historical point of view and recent developments of such stabilities are reported in [9,[17][18][19][20][21][22][23][24][25][26][27][28][29][30] and the references cited therein. So, the motivation for the elaboration of this paper is the investigation of some kinds of the Ulam-Hyers stability for the following problem involving the concept of Caputo-Katugampola fractional derivative with the case of the α ∈ (1, 2):…”

Ulam-Hyers stability for a nonlinear Volterra integro-differential equation

“…After that with the above idea, Huang et al [3] proved the Ulam-Hyers stability for delay differential equations of the first order by using successive approximation method. Kucche et al [9,10] also applied to prove the Ulam-Hyers stability and E α -Ulam-Hyers stability results for nonlinear implicit fractional differential equations.…”

Global Existence and Ulam--Hyers Stability of $Ψ$--Hilfer Fractional Differential Equations

“…Huang et al [13] investigated HU stability of integer order delay differential equations by method of successive approximation. Kucche and Sutar [14] have extended the idea of [13] and investigated the HU stability of nonlinear delay FDEs with Caputo derivative.…”

“…Motivated by the work of [5,6,14], in this paper, we consider the Ψ-Hilfer fractional differential equation (Ψ-Hilfer FDE) of the form:…”