Caristi不动点定理是非线性分析中一个非常重要的结论,曾被评价很可能成为非线性泛函分析进一步发展的强有力的工具.这个结果不仅是Banach压缩原理的推广,而且在不动点理论和变分方法中产生了深远的影响.事实上,Caristi不动点定理等价于Ekeland变分原理.近40年来,Caristi不动点定理在多方面得到讨论与推广,条件的减弱和形式更加一般化使得应用的范围越来越广,特别是减弱条件中关于度量函数的要求更具有重要的意义.应用半序集及极小元定理,在完备度量空间中得到τ函数型Caristi不动点定理需要与下方有界下半连续泛函相关的不等式条件.推广了已有文献中的结果,将原来不等式条件中的度量函数d减弱为τ函数,不等式右端具有上半连续函数的形式,推论中包含了多值映射的结果.%Caristi fixed point theorem is a very important conclusion in nonlinear analysis, and it was remarked that it is likely to be a powerful tool in further development of nonlinear functional analysis.This result is not only a generalization of Banach contraction principle, but also has a profound influence on the fixed point theory and variational methods.In fact, Caristi fixed point theorem is equivalent to Ekeland variational principle.In the past forty years, Caristi fixed point theorem has been discussed and generalized in many aspects.The weakening of the conditions and the more general forms make the application more and more widely, it is especially more important to reduce the requirement of metric function in the condition.By using partially ordered set and minimal element theorem, Caristi fixed point theorem of τ function`s type is obtained in complete metric spaces under the inequality condition concerning lower bounded and lower semi-continuous functional.Some results in references are extended, the metric function d in original inequality condition is reduced to τ function and the right side of the inequality has the form of upper semi-continuous function.The result of multi valued mapping is included in the corollary.
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