A new Caristi-type inequality is considered and Caristi’s fixed point theorem for mappings of complete metric spaces is developed (in both the single- and set-valued cases). On the basis of this development mappings of complete metric spaces which are contractions with respect to a function of two vector arguments are studied. This function is not required to be a metric or even a continuous function. The proved theorems are generalizations of the Banach contraction principle and Nadler’s theorem.

中文翻译：

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Caristi不等式和α-收缩映射

考虑了一个新的Caristi型不等式，并开发了用于完整度量空间映射的Caristi不动点定理（在单值和集值情况下）。在此基础上，研究了完整度量空间的映射，该度量空间是相对于两个向量自变量的函数的收缩。此功能不需要是度量标准，甚至不必是连续功能。证明定理是Banach压缩原理和Nadler定理的推广。