On Order Covering Set-Valued Mappings and Their Applications to the Investigation of Implicit Differential Inclusions and Dynamic Models of Economic Processes

Abstract

The present work is devoted to investigation of operator inclusions in partially ordered spaces and application of the obtained results to differential inclusions. We consider the inclusion Υ(x,x)∋y with respect to the unknown x∈X,where Υ:X×X⇒Y is a set-valued mapping, X and Y are partially ordered spaces. It is assumed that the mapping Υ is order covering with respect to the first argument and antitone with respect to the second argument. We prove that for any x0 ∈ X, if the set G(x0) contains an element y0 such that y ≼ y0, then there exists a solution to the inclusion under consideration, which satisfies the inequality x ≼ x0. This statement is applied to investigation of a Cauchi problem for the differential inclusion f(t,x,ẋ,ẋ)∋0 with a bound for the derivative of the unknown function ẋ(t)∈B(t) (here f:[a,b]×Rn×Rn×Rn ⇒Rm,B:[a,b]⇒Rn). We obtain conditions of solvability in the space of absolutely continuous functions, conditions of existence of a solution with the least derivative, and derive the solutions estimates. The latter results are applied to the analysis of the dynamic Walrasian-Evans-Samuelson model of economic processes, which can be reduced to a system of implicit differential inclusions. We establish the existence of the equilibrium and obtain estimates of the equilibrium prices.