Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations

Authors : Ahmed KAJOUNI, Najat CHEFNAJ, Khalid HİLAL

Pages : 360-371

Doi:10.53006/rna.1124961

View : 8 | Download : 4

Publication Date : 2022-09-30

Article Type : Research Paper

Abstract :In this work, we prove the existence of a solution for the initial value problem of nonlinear fractional differential equation with quadratic perturbations involving the Caputo fractional derivative insert ignore into journalissuearticles values(   c D α 0 + − ρ t   c D β 0 + ); insert ignore into journalissuearticles values( x insert ignore into journalissuearticles values( t ); f insert ignore into journalissuearticles values( t , x insert ignore into journalissuearticles values( t ); ); ); = g insert ignore into journalissuearticles values( t , x insert ignore into journalissuearticles values( t ); ); , t ∈ J = [ 0 , 1 ] , 1 < α < 2 , 0 < β < α insert ignore into journalissuearticles values( cD0+α−ρt cD0+β);insert ignore into journalissuearticles values(xinsert ignore into journalissuearticles values(t);finsert ignore into journalissuearticles values(t,xinsert ignore into journalissuearticles values(t);););=ginsert ignore into journalissuearticles values(t,xinsert ignore into journalissuearticles values(t););,t∈J=[0,1],1<α<2,0<β<α with conditions x 0 = x insert ignore into journalissuearticles values( 0 ); f insert ignore into journalissuearticles values( 0 , x insert ignore into journalissuearticles values( 0 ); ); x0=xinsert ignore into journalissuearticles values(0);finsert ignore into journalissuearticles values(0,xinsert ignore into journalissuearticles values(0);); and \\ x 1 = x insert ignore into journalissuearticles values( 1 ); f insert ignore into journalissuearticles values( 1 , x insert ignore into journalissuearticles values( 1 ); ); x1=xinsert ignore into journalissuearticles values(1);finsert ignore into journalissuearticles values(1,xinsert ignore into journalissuearticles values(1);); . Dhage`s fixed-point the theorem was used to establish this existence. As an application, we have given example to demonstrate the effectiveness of our main result.
Keywords : Fractional differential equation, Quadratic perturbations, Dhage fixed point