The method of comparison with a model equation in the study of inclusions in vector metric spaces

For a given multivalued mapping $F:X\rightrightarrows Y$ and a given element $\tilde{y}\in Y$, the existence of a solution $x\in X$ to the inclusion $F(x)\ni\tilde{y}$ and its estimates are studied. The sets $X$ and $Y$ are endowed with vector metrics $\mathcal{P}_X^{E_+}$ and $\mathcal{P}_Y^{M_+}$, whose values belong to cones $E_+$ and $M_+$ of a Banach space $E$ and a linear topological space $M$, respectively. The inclusion is compared with a “model” equation $f(t)=0$, where $f:E_+\to M$. It is assumed that $f$ can be written as $f(t)\equiv g(t,t)$, where the mapping $g:{E}_+\times{E}_+\to M$ orderly covers the set $\{0\}\subset M$ with respect to the first argument and is antitone with respect to the second argument and $-g(0,0)\in M_+$. It is shown that in this case the equation $f(t)=0$ has a solution $t^*\in E_+$. Further, conditions on the connection between $f(0)$ and $F(x_0)$ and between the increments of $f(t)$ for $t\in [0,t^*]$ and the increments of $F(x)$ for all $x$ in the ball of radius $t^*$ centered at $x_0$ for some $x_0$ are formulated, and it is shown that the inclusion has a solution in the ball under these conditions. The results on the operator inclusion obtained in the paper are applied to studying an integral inclusion.