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
$x(t)= a(t)+ g(x(t)) \int_0^t v(t,s,x(s))\, \quad\text{for all }t\in[0,T],$
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:

 45G10 Other nonlinear integral equations 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.

Zbl 0999.47041
Full Text:

References:

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