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Existence of monotone solutions for a nonlinear quadratic integral equation of Volterra type. (English) Zbl 1147.45005

A sufficient condition for the existence of monotone solutions of the following nonlinear quadratic integral equation of Volterra type

x(t)=a(t)+g(x(t)) 0 t v(t,s,x(s))forallt[0,T],

is established. The approach is based on Darbo’s fixed point theorem and the measure of noncompactness introduced by J. Banaś and L. Olszowy [in Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 41, 13–23 (2001; Zbl 0999.47041)].

MSC:
45G10Nonsingular nonlinear integral equations
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties