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On some functional-integral equations with linear modification of the argument. (English) Zbl 1026.45012
The technique of the so-called weakly Picard operators is applied in the study of the following functional-integral equation: $$x(t)= g\bigl(t,(hx)(t), x(t),x(0)\bigr)+ \int^t_0K\bigl(t,s,x (rs)\bigr)ds,$$ for $t \in [0,b]$ and $r\in [0,1]$ being a constant. Here $h:C([0,b],X) \to C([0,b],X) $, $g\in C([0,b]\times X^3,X)$ and $K\in C([0,b] \times[0,b]\times X,X)$, where $X$ is a Banach space. Moreover, the functions $g,h$ and $K$ are supposed to satisfy the classical Lipschitz condition, so the operator determined by the right hand side of the equation (1) is a contraction in a suitable Banach space.
45N05Abstract integral equations, integral equations in abstract spaces
45G10Nonsingular nonlinear integral equations