Study of the Existence and Uniqueness of Global Solution for the Fractional Second Order Nonlinear Integro â€“Differential Equations with Boundary Conditions

Ammar Isam Edress

doi:10.31642/jokmc/2018/050201

Abstract

In this paper, the existence of the global solution and its uniqueness is studied for the Second Order Nonlinear Integroâ€“Differential Fractional Equations with boundary conditions by utilizing the Picard approximation method which is given by Sturble, 1962. Furthermore, several given results by Butris, 2010 have been extended

Highlights

Enormous of studies have discussed the solution of the differential and integral equation for fractional order such as [1], [2], [3], [4], [5], [6] and [7]
The focus will be on the 2nd order nonlinear integro–differential fractional equations [8], and [9]
Proof: The proof of this theorem has been given in details by [6]

Summary

INTRODUCTION

Enormous of studies have discussed the solution of the differential and integral equation for fractional order such as [1], [2], [3], [4], [5], [6] and [7]. Consider the following fractional second order nonlinear integro–differential equations, which have the form: t x 2 f t, x t , x t , w t, s g s, x s , x s ds ,0 1. Where the function f t, x, x , y is a continuous in t, x, x , y and defined over the domain:. For all t R1 and x, x1, x2 GR, x , x 1, x 2 G1 and y, y1, y2 G2 where M , K1, K2, K3, L1, L2 are positive t constants and y t, x0 w t, s g s, x s , x s ds where the function w t, s is defined and continuous on the domain -.

EXISTENCE OF THE SOLUTION:

T a and E is identity matrix with xo a B b a

THE UNIQUENESS OF THE SOLUTION