Abstract

Recently, the authors introduced a concept of a pair of Zamfirescu-type multi-valued mappings between metric spaces and proved a theorem on the existence of coincidence points for such pairs of mappings. It was shown that this theorem is a generalization of the fixed point theorem for a multi-valued Zamfirescu mapping by Neammanee and Kaewkhao (2010). In this paper, the main result is the theorem on the preservation of the existence of coincidence points in a given open set of a metric space for a one-parameter family of pairs of Zamfirescu-type multi-valued mappings. It is shown that this result follows from the authors’ theorem on the preservation of zero existence for a parametric family of \((\alpha,\beta)\)-search functionals introduced earlier by T. N. Fomenko. In addition, we consifder the connection of this result with the Frigon–Granas theorem (1994) on the preservation of the existence of fixed points for a contraction family of multi-valued mappings.

中文翻译：

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Zamfirescu 型多值映射对的单参数族的巧合保持

摘要

最近，作者引入了度量空间之间的一对 Zamfirescu 型多值映射的概念，并证明了关于此类映射对重合点存在的定理。表明该定理是 Neammanee 和 Kaewkhao (2010) 对多值 Zamfirescu 映射的不动点定理的推广。在本文中，主要结果是关于在给定的度量空间的开放集合中保持重合点存在的定理，对于 Zamfirescu 型多值映射对的单参数族。结果表明，该结果遵循作者关于\((\alpha,\beta)\)的参数族保持零存在性的定理- TN Fomenko 之前引入的搜索功能。此外，我们考虑了这个结果与 Frigon-Granas 定理 (1994) 关于多值映射收缩族的不动点存在性的保持的联系。