Abstract
In this paper we study the quadratic integral equation of the form $$x(t) = g(t) + \lambda . G(x)(t) . \int\nolimits_a^bK(t, s)f(s, x(s)){\rm d}s.$$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.
Highlights
The paper is devoted to study the following quadratic integral equation b x(t) = g(t) + G(x)(t) · λ K(t, s)f (s, x(s))ds. (1.1)Since the quadratic problems are related with the pointwise product of two operators, it is usually solved in the space of continuous functions or in a context of Banach algebras of continuous functions
We will try to choose the domains of operators defined above in such a way to obtain the existence of solutions in a desired Orlicz space Lφ(I)
We stress on conditions allowing us to consider strongly nonlinear operators and simultaneously to cover both quadratic and classical integral equations
Summary
We prefer an approach to this problem allowing us to consider a wide class of integral equations with solutions in some spaces of discontinuous functions (growth conditions are relaxed). We prefer a method, which allows us to unify classical and quadratic integral equations and to consider the same classes of solutions We started such an approach in [13], but it was done only in that case of Banach–Orlicz algebras or in case when an intermediate space (described later) is L∞ ([12]). In a typical case of quadratic problems the spaces W1 and W2 are supposed both to be the space of continuous functions and some properties of this Banach algebra allow to solve the problem We investigate some properties of this topology on considered Orlicz spaces and use the Darbo fixed-point theorem for proving main results
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