Extended Riemann Integral Equations Involving Generalized k−hypergeometric Function

Abstract

This research proves the existence of the solution for the Fredholm integral equation of the first kind. Initially, k−Riemann integral equation is considered involving the k−hypergeometric function as kernel. k−fractional integration defined by Mubeen and Habibullah [16] is used to investigate the solution of the integral equation Z x 0 (x − t) c k −1 Γk(c) q+1Fq,k à (ai , k),(b, k) (ci , k) ; 1 − x t ! f(t)dt = g(x) where λ, ai , b, ci > 0, i = 1, . . . , q and f ∈ C◦. To prove the existence of solution, necessary and sufficient conditions are defined. Keywords: k−Pochhammer symbol, k−hypergeometric function, k−Fractional Integration, k−Riemann integral equation, Fredholm integral equation