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

Naveed Ahmad,Ghaus Ur Rahman,Zeeshan Ali,Kamal Shah,Akbar Zada

doi:10.1155/2018/6423974

Naveed Ahmad, Ghaus Ur Rahman + Show 3 more

Open Access

https://doi.org/10.1155/2018/6423974

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Abstract

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.

Highlights

Over the past few decades, differential equations of fractional order have got considerable attention from the researchers due to their significant applications in various disciplines of science and technology
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
Since fractional order differential equations play important roles in modeling real world problems related to biology, viscoelasticity, physics, chemistry, control theory, economics, signal and image processing phenomenon, bioengineering, and so forth, it is investigated that fractional order differential equations model real world problems more accurately than differential equations of integer order

Summary

Introduction

Over the past few decades, differential equations of fractional order have got considerable attention from the researchers due to their significant applications in various disciplines of science and technology. The area devoted to the study of existence and uniqueness of solutions to initial/boundary value problems for fractional order differential equations has been studied very well and plenty of papers are available on it in the literature. Influenced from the above-mentioned work, we will study the existence, uniqueness, and different types of Ulam–Hyers stability of the following implicit impulsive fractional order differential equations with nonlocal boundary conditions, involving Caputo derivative cDρy (t) = f (t, y (t) , cDρy (t)) , t ∈ I = [0, 1] , t ≠ tj, j = 1, 2, .

Background

Main Results

Ulam Stability Results

Example

Conclusion