Caristi's fixed point theorem and selections of set-valued contractions
Let \((X,d)\) be a metric space and \(T:X\to X\) a map which need not be continuous but satisfies \(d(x,Tx)\leq\varphi(x)-\varphi(Tx)\) for some lower semicontinuous function \(\varphi:[0,\infty)\to[0,\infty)\). Caristi proved this result using transfinite induction. \textit{W. A. Kirk} [Colloq. Math. 36, 81-86 (1976; Zbl 0353.53041)] defined a partial ordering on \(X\) by \(x\leq_\varphi y\) iff \(d(x,y)\leq\varphi(x)-\varphi(y)\) in order to prove this theorem. His proof uses Zorn's lemma. \textit{F. E. Browder} [in: Fixed point theorem, Appl. Proc. Sem. Halifax 1975, 23-27 (1976; Zbl 0379.54016)] gave a constructive proof using the axiom of choice only for countable families. \textit{R. Mańka} [Rep. Math. Logic 22, 15-19 (1988; Zbl 0687.04003)] then gave a constructive proof based on Zermelo's theorem. The present author gives a simple derivation of Caristi's theorem from Zermelo's theorem in case \(T\) is continuous. On the other hand, the author describes examples of set-valued contractions which admit (not necessarily continuous) selections which satisfy the assumptions of Caristi's theorem. Finally, the author answers a question posed by W. A. Kirk by proving the following result: Let \(\eta:[0,\infty)\to[0,\infty)\) be a function satisfying \(\eta(0)=0\). Then the right hand lower Dini derivative of \(\eta\) at \(0\) (i.e., \(\liminf_{s\to t^+}[\eta(s)-\eta(t)]/[s-t]\)) vanishes if and only if there is a complete metric space \((X,d)\), a continuous and asymptotically regular mapping \(T:X\to X\) which has no fixed points and a continuous function \(\varphi:[0,\infty)\to[0,\infty)\) such that \(\eta(d(x,Tx))\leq\varphi(x)-\varphi(Tx)\) for all \(x\in X\).
scientific article; zbMATH DE number 1236161
scientific article