Solutions of a system of integral equations in orlicz spaces

Abstract

We consider the following system of integral equations u(i)(t) = integral(1)(0) g(i)(t, s)f(i) (s, u(1)(s), u(2)(s),...,u(n)(s)) ds, a.e. t is an element of [0,1]. 1 <= i <= n. Our aim is to establish criteria such that the above system has a solution (u(1), u(2),..., u(n)) where u(i) is an element of L(phi) (Orlicz space), 1 <= i <= n. We further investigate the system u(i)(t) = integral(1)(0) g(i)(t, s)H(s, u(1)(s), u(2)(s),...,u(n)(s))ds, a.e. t is an element of[0, 1], 1 <= i <= nand establish the existence of constant-sign solutions in Orlicz spaces, i.e., for each 1 <= i <= n, theta u(i) >= 0 and u(i) is an element of L(phi), where theta is an element of {1, -1} is fixed.