Nonlinear contractive conditions:: A comparison and related problems

被引:117
作者
Jachymski, Jacek [1 ]
Jozwik, Izabela [2 ,3 ]
机构
[1] Tech Univ Lodz, Inst Math, Wolczanska 215, PL-93005 Lodz, Poland
[2] Tech Univ Lodz, Inst Math, PL-90924 Lodz, Poland
[3] Tech Univ Lodz, Ctr Math & Phys, PL-90924 Lodz, Poland
来源
FIXED POINT THEORY AND ITS APPLICATIONS | 2007年 / 77卷
关键词
nonlinear contractive conditions; phi-contractive map; contractive fixed point; approximate fixed point; domain invariance theorem; separation theorem; semicontinuous function; monotonic function; concave function; subadditive function;
D O I
10.4064/bc77-0-10
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We establish five theorems giving lists of nonlinear contractive conditions which turn out to be mutually equivalent. We derive them from some general lemmas concerning subsets of the plane which may be applied both in the single- or set-valued case as well as for a family of mappings. A separation theorem for concave functions is proved as an auxiliary result. Also, we discuss briefly the following problems for several classes of contractions: stability of procedure of successive approximations, existence of approximate fixed points, continuous dependence of fixed points on parameters, existence of invariant sets for iterated function systems. Moreover, James Dugundji's contribution to the metric fixed point theory is presented. Using his notion of contractions, we also establish an extension of a domain invariance theorem for contractive fields.
引用
收藏
页码:123 / +
页数:4
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