zbMATH — the first resource for mathematics

On a class of integral equations of Urysohn type with strong non-linearity. (English. Russian original) Zbl 1245.45005
Izv. Math. 76, No. 1, 163-189 (2012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 1, 173-200 (2012).
This paper is devoted to a study of homogeneous Urysohn integral equations with strong non-linearity on the positive semi-axis \[ \varphi(x)=\int_{0}^{\infty}K(x,t,\varphi(t))dt, \quad x\in (0,+\infty), \] and their non-homogeneous versions \[ H(x)=g(x)+\int_{0}^{\infty}K(x,t,H(t))dt, \quad x\in (0,+\infty) \] under some restrictions on the kernel \(K(x,t,\tau)\).
It is assumed that some non-linear integral operator of Wiener-Hopf-Hammerstein type is a local minorant of the corresponding Urysohn operator. Using special methods of the linear theory of convolution-type integral equations, the author constructs positive solutions for these classes of Urysohn equations. The asymptotic behaviour of these solutions at infinity is also studied. As an auxiliary fact in the course of the proof of these assertions, the author obtains a one-parameter family of positive solutions for non-linear integral equations of Wiener-Hopf-Hammerstein type whose operator is a minorant for the original Urysohn operator. Particular examples of non-linear integral equations are given for which all the hypotheses of the main theorems hold.

45G10 Other nonlinear integral equations
45M05 Asymptotics of solutions to integral equations
45M20 Positive solutions of integral equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)
Full Text: DOI