We apply the Kurzweil-Henstock integral setting to prove a Fredholm Alternative-type result for the integral equation x (t)− K Z [a,b] α (t, s)x (s) ds = f (t) , t ∈ [a, b] , where x and f are Kurzweil integrable functions (possibly highly oscillating) defined on a compact interval [a, b] of the real line with values on Banach spaces. An application is given.

Abstract We define the $$L^r$$ L r -variational integral and we prove that it is equivalent to $$HK_r$$ H K -integral defined in 2004 by P. Musial Y. Sagher Studia Mathematica paper The $$L^...

In this paper, we have studied the Henstock - Kurzweil integral which is a generalized Riemann means. Hen-stock natural extension of integral. We defined Banach space valued function with respect to bounded variation an real increasing function. investigated elementary properties variation. proved convergence theorems and Saks lemma functions vari-ation. Equi-integrability equi-integrable theor...