On Solutions of Quadratic Integral Equations in Orlicz Spaces

In this paper we study the quadratic integral equation of the form x(t)=g(t)+λ.G(x)(t).∫abK(t,s)f(s,x(s))ds.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x(t) = g(t) + \lambda . G(x)(t) . \int\nolimits_a^bK(t, s)f(s, x(s)){\rm d}s.$$\end{document}We discuss the existence of solutions for the above equation in different function spaces. We stress on the case when f has non-polynomial growth which leads to solutions in Orlicz spaces. The detailed theory for a wide class of spaces is presented. Some existence theorems for a.e. monotonic solutions in Orlicz spaces are proved either for strongly nonlinear functions f or for rapidly growing kernel K. The presented method allows us to extend the current results as well as to unify the proofs for both quadratic and non-quadratic cases.