Caristi-like condition and the existence of minima of mappings in partially ordered spaces
Let \((X,\preceq)\) and \((Y,\preceq)\) be partially ordered sets; \(Min (Y)\) denote the set of all minimal elements of \(Y\). It is said that a map \(U \colon X \to Y\) satisfies the Caristi-like condition if \((K)\) \(\forall x \in X,\) if \(U(x) \notin Min (Y)\), then there exists \(x^\prime \in X\) such that \(x^\prime \prec x\), \(U(x^\prime) \prec U(x)\). The authors prove the following theorem on the existence of a minimal point of a map. Let a map \(U \colon X \to Y\) satisfy conditions \((K)\) and \((S)\) for arbitrary infinite chain \(S \subset X,\) if the restriction of \(U\) to \(S\) is strictly isotone, then \(S\) has a lower bound \(w \in X\) for which \(U(w) \prec U(x)\) for all \(x \in S\), \(x \neq w\). Then \(U(X) \cap \text{Min} (Y) \neq \emptyset\) and, for each \(x_0 \in X\), there exists \(\bar{x} \in X\), \(\bar{x} \preceq x_0\) such that \(U(\bar{x}) \in \text{Min} (Y)\), \(U(\bar{x}) \preceq U(x_0)\), It is shown that the result on a minimum of a lower semicontinuous function on a metric space (see [the first author, Proc. Steklov Inst. Math. 291, 24--37 (2015; Zbl 1336.49005); translation from Tr. Mat. Inst. Steklova 291, 30--44 (2015)], Theorem 3) follows from the above theorem. It is applied also to obtain the existence of a coincidence point for a pair of maps of partially ordered spaces.
scientific article; zbMATH DE number 7032509