Lp- solution of Weighted Cauchy-type Problem of a Diffre-integral Functional Equation

The topic of fractional calculus, (integration and differentiation of fractional-order) is a one of the singular integral and integro-differential operators, is enjoying interest among mathematicians, physicists and engineers ( see (1)-(2), (5)-(10) and (12)-(13) and the references therein ). In this work, we are concerned with a nonlinear weighted Cauchy type problem ( For the earlier work and application see, for example, (6) and (8)) of a diffre-integral functional equation of fractional order. We will prove some local and global existence theorems for this problem, also we will study the uniqueness and stability of its solution.