Abstract:
An inclusion with multi-valued mapping acting in spaces with vector-valued metrics is under discussion. It is shown that, if a multi-valued mapping $F$ can be written as $F(x)=\Upsilon(x,x),$ where the mapping $\Upsilon$ is closed and metrically regular with some operator coefficient $K$ with respect to one argument, Lipschitz with operator coefficient $Q$ with respect to the other argument, and the spectral radius of the operator $KQ$ is less than one, then the inclusion $F(x)\ni y$ is solvable. The estimations of the vector-valued distance from a solution $x$ of the inclusion to a given element $x_0$ are derived. In the second part of the paper, these results are used to investigate an integral inclusion of the implicit type with respect to the unknown integrable function.
Keywords:space with vector-valued metric, multi-valued mapping, metrically regular mapping, implicit type integral inclusion.
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