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An existence theorem for nonlinear Volterra integral equation with deviating argument. (English) Zbl 0625.45013

The paper deals with nonlinear Volterra integral equations with deviating argument

$x\left(t\right)=h\left(t\right)+{\int }_{0}^{t}K\left(t,s,x\left(H\left(s\right)\right)\right)ds,\phantom{\rule{1.em}{0ex}}t\ge 0,$

in ${C}_{g}$ spaces: ${C}_{g}\left({R}_{+},{R}^{n}\right)=\left\{x:{R}_{0}\to {R}^{n}$, continuous and $|x\left(t\right)|/g\left(t\right)$ bounded on ${R}_{+}\right\}$. It is assumed that g is a continuous positive function on ${R}_{+}$. Under various conditions on the data it is shown that the equation under discussion has at least one solution in a convenient ${C}_{g}$ space (using Schauder fixed point theorem). The classical case of Volterra equation is covered by the result.

Reviewer: C.Corduaneanu

##### MSC:
 45G10 Nonsingular nonlinear integral equations 45D05 Volterra integral equations
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