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-valued 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.