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

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( y1 , y2 )=0⇔ y1 = y2 , and is assumed to be neither symmetric nor satisfying the triangle inequality. The question of well-posedness of the equation ψx =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.

well-posedness of equation, continuous dependence on parameter, coincidence point of two mappings, distance, covering mapping