THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER

Abstract

In this paper, we consider boundary value problems for the following nonlinear implicit differential equations with complex order D +x(t) = f t,x(t),D +x(t) , ? = m+i?, t ? J := [0,T], ax(0)+bx(T) = c,where D + is the Caputo fractional derivative of order ? ? C. Let ? ? R , 0 < ? < 1, m ? (0,1], and f : J ×R ? R is given continuous function. Here a,b,c are real constants with a+b = 0. We derive the existence and stability of solution for a class of boundary value problem(BVP) for nonlinear fractional implicit differential equations(FIDEs) with complex order. The results are based upon the Banach contraction principle and Schaefer’s fixed point theorem.