The metric projection on the class \(H(\Omega)\) satisfies Lipschitz condition
Let X be a normed space, H a convex set in X, \(p(x,H)=\inf\{\| x- y\|:y\in H\}\) (\(x\in X)\), \(P^ t_ H(x)=\{y\in H:\| x-y\|\leq t+P(x,H)\}\) and \(P^ 0_ H(x)=P_ H(x)\) (\(t\geq 0)\). The mapping \(P_ H:x\to P_ H(x)\) is said to be a metric projection. Let \(X=B(Q)\) be the space of the real-valued bounded functions \(x=x(q)\) defined on the set Q endowed with the norm \(\| x\| =\sup\{| x(q)|:q\in Q\}\), and let \(\Omega\) be a metric on Q. In this paper, it is shown that in the space B(Q), the metric projection on the class of functions \(H=H(\Omega)=\{y\in B(Q):| y(q_ 1)-y(q_ 2)|\leq \Omega (q_ 1,q_ 2)\), \(q_ 1,q_ 2\in Q\}\) satisfies the Lipschitz condition. A property analogous to differentiability is also proved for the mapping \(P^ t_ H:x\to P^ t_ H(x)\). Main results: Theorem 1. Let \(\Omega_ i\) be a metric on Q, \(y_ i\in B(Q)\), \(t_ i\geq 0\), and \(P_ i=P^{t_ i}_{H(\Omega_ i)}(y_ i) (i=1,2)\). Then \[ P_ 1P_ 2\leq 2H(\Omega_ 1)H(\Omega_ 2)+H(\Omega_ 2)H(\Omega_ 1)+2\| y_ 1-y_ 2\| +| t_ 1-t_ 2|\quad and \] \[ h(P_ 1,P_ 2)\leq (3/2)\delta (\Omega_ 1,\Omega_ 2)+2\| y_ 1- y_ 2\| +1| t_ 1-t_ 2|, \] where \(h(A,B)=\max\{AB,BA\}\), \(AB=\sup_{a\in A}\rho (a,B)\) (A,\(B\subset X)\), and \(\delta (\Omega,\Omega_ 2)=\sup\{|\Omega_ 1(q_ 1,q_ 2)-\Omega_ 2(q_ 1,q_ 2)|:q_ 1,q_ 2\in Q\}\). Theorem 2. Assume that a class \(K\subset B(Q)\) satisfies the condition (\(x\in K\), \(-\infty <r<\infty)\Rightarrow (x+r\in K)\) and \(y\in B(Q)\). Then \(P(y,K)=(1/2)\inf\{\| x_ 1-x_ 2\|:x_ 1,x_ 2\in K\), \(x_ 1\leq y\leq x_ 2\}\) and \(P^ t_ K(y)=\{x\in K:Sy-P-t\leq x\leq Ly+P+t\}\), where \(P=\rho (y,K)\) and \(T\geq 0\). If, in addition, Sy,L\(y\in K\), then \(P_ k(y)\neq\emptyset \) and \(P(y,K)=(1/2)\| Sy-Ly\|\). (Here \((Sy)(q)=(sy)(q,K)=\inf\{x(q):x\in K,y\leq x\}\), \((Ly)(q)=(Ly)(q,K)=\sup\{x(q):x\in K,x\leq y\}\), and the inequality \(x\leq y\) means x(q)\(\leq y(q)\) for every \(q\in Q)\).
scientific article; zbMATH DE number 3872001
scientific article