Strongly unique best approximations and centers in uniformly convex spaces (Q1821988)

The basic result of the paper is Lemma 2.1: Let X be a uniformly convex space with the modulus of convexity of power type \(q\geq 2\). Then there is a constant \(d>0\) such that \[ \| tx+(1-t)y\|^ q\leq t\| x\|^ q+(1-t)\| y\|^ q-dw_ q(t)\| x-y\|^ q \] for all x,y\(\in X\) and \(t\in (0,1)\), where \(w_ q(t)=t(1-t)^ q+(1-t)t^ q.\) Using this result we prove that if X satisfies the assumptions of Lemma 2.1 and \(M\subset X\) is a closed convex nonempty subset of X then for every \(x\in X\) there exists a unique \(m\in M\) such that \(\| x- m\|^ q\leq \| x-y\|^ q-d\| m-y\|^ q\) for all \(y\in M\), where \(d>0\) is as in Lemma 2.1. In particular we show that it is true when X is the Lebesgue space \(L_ p\) or the Sobolev space \(H^{m,p}\) \((m\geq 0,1<p<\infty)\). We establish similar results for relative centers and asymptotic centers and we apply them to derive a fixed point theorem for uniformly Lipschitzian mappings.