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.