RESEARCH PAPER ON FRACTIONAL ORDER DERIVATIVES AND DARBOUX PROBLEM FOR IMPLICIT DIFFERENTIAL EQUATIONS Sa¨ ıd Abbas 1 , Mouffak Benchohra 2 , Aleksandr N. Vityuk 3 Abstract In this paper we prove some relations between the Riemann-Liouville and the Caputo fractional order derivatives, and we investigate the exis- tence and uniqueness of solutions for the initial value problems (IVP for short), for a class of functional hyperbolic differential equations by using some fixed point theorems. MSC 2010 : Primary 26A33; Secondary 35R11, 34K37 Key Words and Phrases: partial hyperbolic differential equation, frac- tional order, left-sided mixed Riemann-Liouville integral, mixed regularized derivative, solution, fixed point 1. Introduction Fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary non-integer order. The subject is as old as the differential calculus since, starting from some speculations of G.W. Leibniz (1697) and L. Euler (1730), it has been developed up to nowadays (see [24]). The idea of fractional calculus and fractional order differential equa- tions and inclusions has been a subject of interest not only among math- ematicians, but also among physicists and engineers. Indeed, we can find c  2012 Diogenes Co., Sofia pp. 168–182 , DOI: 10.2478/s13540-012-0012-5