Non-Instantaneous Impulsive Fractional Differential Equations with State Dependent Delay and Practical Stability

2021 ◽  
Vol 41 (5) ◽  
pp. 1699-1718
Author(s):  
Ravi Agarwal ◽  
Ricardo Almeida ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Practical Stability ◽  
Impulsive Fractional Differential Equations ◽  
State Dependent ◽  
State Dependent Delay ◽  
Fractional Differential

Ulam type stability for non-instantaneous impulsive Caputo fractional differential equations with finite state dependent delay

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
State Dependent ◽  
State Dependent Delay ◽  
Caputo Fractional Differential Equations ◽  
Finite State ◽  
Fractional Differential ◽  
Ulam Type Stability

AbstractFour Ulam type stability concepts for non-instantaneous impulsive fractional differential equations with state dependent delay are introduced. Two different approaches to the interpretation of solutions are investigated. We study the case of an unchangeable lower bound of the Caputo fractional derivative and the case of a lower bound coinciding with the point of jump for the solution. In both cases we obtain sufficient conditions for Ulam type stability. An example is also provided to illustrate both approaches.

On fractional differential equations with state-dependent delay via Kuratowski measure of noncompactness

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 451-460 ◽  
Author(s):  
Mohammed Belmekki ◽  
Kheira Mekhalfi
Keyword(s):  
Differential Equations ◽  
Measure Of Noncompactness ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Kuratowski Measure Of Noncompactness ◽  
State Dependent ◽  
State Dependent Delay ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Functional Differential

This paper is devoted to study the existence of mild solutions for semilinear functional differential equations with state-dependent delay involving the Riemann-Liouville fractional derivative in a Banach space and resolvent operator. The arguments are based upon M?nch?s fixed point theoremand the technique of measure of noncompactness.

scholarly journals Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Naveed Ahmad ◽  
Zeeshan Ali ◽  
Kamal Shah ◽  
Akbar Zada ◽  
Ghaus ur Rahman
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Fractional Differential Equations ◽  
Nonlocal Boundary ◽  
Dynamical Problem ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential

We study the existence, uniqueness, and various kinds of Ulam–Hyers stability of the solutions to a nonlinear implicit type dynamical problem of impulsive fractional differential equations with nonlocal boundary conditions involving Caputo derivative. We develop conditions for uniqueness and existence by using the classical fixed point theorems such as Banach fixed point theorem and Krasnoselskii’s fixed point theorem. For stability, we utilized classical functional analysis. Also, an example is given to demonstrate our main theoretical results.

Periodic impulsive fractional differential equations

2017 ◽  
Vol 8 (1) ◽  
pp. 482-496 ◽  
Author(s):  
Michal Fečkan ◽  
Jin Rong Wang
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Fractional Differential Equations ◽  
Periodic Boundary ◽  
Lorenz System ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential ◽  
Existence Of Periodic Solutions ◽  
Lower Limits ◽  
Stability Results

Abstract This paper deals with the existence of periodic solutions of fractional differential equations with periodic impulses. The first part of the paper is devoted to the uniqueness, existence and asymptotic stability results for periodic solutions of impulsive fractional differential equations with varying lower limits for standard nonlinear cases as well as for cases of weak nonlinearities, equidistant and periodically shifted impulses. We also apply our result to an impulsive fractional Lorenz system. The second part extends the study to periodic impulsive fractional differential equations with fixed lower limit. We show that in general, there are no solutions with long periodic boundary value conditions for the case of bounded nonlinearities.

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