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Hammerstein equations with an integral over a noncompact domain. (English) Zbl 0919.45004
Applying the Schauder fixed point principle a few theorems on the existence of continuous solutions of the Hammerstein integral equation (1) $x\left(t\right)={\int }_{T}G\left(t,s\right)r\left(s,x\left(s\right)\right)ds$ are established. The author investigates a very general case assuming that $T$ is a noncompact domain (more precisely: $T$ is a metric locally compact space countable at infinity and equipped with a measure on $\sigma$-algebra of Borel subsets of $T$) and the values of the functions involved are situated in a Banach space $X$. Applications to a nonlinear Wiener-Hopf integral equation are also derived.
##### MSC:
 45G10 Nonsingular nonlinear integral equations 47H30 Particular nonlinear operators