On stability and continuous dependence on parameter of the set of coincidence points of two mappings acting in a space with a distance

Abstract

We consider the problem of coincidence points of two mappings ψ,φ, acting from a metric space (X,ρ) into a space (Y,d), in which a distance d has only one of the properties of the metric: d(y_1,y_2)=0⇔y_1=y_2, and is assumed to be neither symmetric nor satisfying the triangle inequality. The question of well-posedness of the equation ψ(x)=φ(x), which determines the coincidence point, is investigated. It is shown that if x=ξ is a solution to this equation, then for any sequence of α_i-covering mappings ψ_i:X→Y and any sequence of β_i-Lipschitz mappings φ_i:X→Y, α_i>β_i≥0, in the case of convergence d(φ_i (ξ),ψ_i (ξ))→0, equation ψ_i (x)=φ_i (x) has, for any i, a solution x=ξ_i such that ρ(ξ_i,ξ)→0. Further in the article, the dependence of the set "Coin"(t) of coincidence points of mappings ψ(•,t),φ(•,t):X→Y on a parameter t, an element of the topological space T, is investigated. Assuming that the first of these mappings is α-covering and the second one is β-Lipschitz, we obtain an assertion on upper semicontinuity, lower semicontinuity, and continuity of the set-valued mapping "Coin":T⇒X.