We consider the inclusion $\widetilde y\in F(x)$ with a multivalued mapping acting in spaces with vector-valued metrics whose values are elements of cones in Banach spaces and can be infinite. A statement about the existence of a solution $x \in X$ and an estimate of its deviation from a given element $x_0 \in X$ in a vector-valued metric are obtained. This result extends the known theorems on similar operator equations and inclusions in metric spaces and in spaces with $n$-dimensional metric to a more general case and, applied to particular classes of functional equations and inclusions, allows to get less restrictive, compared to known, solvability conditions as well as more precise estimates of solutions. We apply this result to the integral inclusion
$$
\widetilde{y}(t)\in f\Bigl(t,\int_a^b \varkappa(t,s) x(s)\,ds, x(t) \Bigr), \ \ t \in [a,b],
$$
where the function $\widetilde y$ is measurable, the mapping $f$ satisfies the Carathéodory conditions, and the solution $x$ is required to be only measurable (the integrability of $x$ is not assumed).
Keywords: space with vector-valued metric, multivalued mapping, vector metric regularity, Lipschitz property with operator coefficient, operator inclusion, integral inclusion
Received June 14, 2023
Revised August 18, 2023
Accepted August 21, 2023
Funding Agency: This work was supported by the Russian Science Foundation (project no. 22- 21-00772, https://rscf.ru/project/22-21-00772/).
Elena Aleksandrovna Panasenko, Cand. Sci. ( Phys.-Math.), Functional Analysis Department, Derzhavin Tambov State University, Tambov, 392000 Russia, e-mail: panlena_t@mail.ru
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Cite this article as: E.A.Panasenko. On operator inclusions in spaces with vector-valued metrics. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 3, pp. 106–127; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S222–S242.
