Multi-valued nonlinear contraction mappings

Two concepts of nonlinear contractions for multi-valued mappings in complete metric spaces are introduced and three fixed point theorems are proved. Presented theorems are generalizations of very recent fixed point theorems due to Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for s...

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Bibliographic Details
Published inNonlinear analysis Vol. 71; no. 7; pp. 2716 - 2723
Main Author CIRIC, Ljubomir
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Ltd 01.10.2009
Elsevier
Subjects
Complete metric space
Exact sciences and technology
Fixed point
General topology
Global analysis, analysis on manifolds
Hausdorff metric
Mathematical analysis
Mathematics
Multi-valued nonlinear contraction
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
secondary
Complete metric space
Hausdorff metric
Multi-valued nonlinear contraction
primary
Fixed point
Fix point
Metric space
Nonlinear analysis
primary 54H25
Mathematical model
secondary 54C60
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Summary:Two concepts of nonlinear contractions for multi-valued mappings in complete metric spaces are introduced and three fixed point theorems are proved. Presented theorems are generalizations of very recent fixed point theorems due to Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132–139] and Feng and Liu [Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103–112], as well as a fixed point theorem of Mizoguchi and Takahashi [N. Mizoguchi, W. Takahash, Fixed point theorems for multi-valued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177–188] and several others. An example is given to show that presented results are genuine generalizations. Our results are also some contribution to an open problem raised by Reich [S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 5 (1972), 26-42; S. Reich, Some fixed point problems, Atti Acad. Naz. Lincei 57 (1974), 194-198; S. Reich, Some problems and results in fixed point theory, Contemp. Math. 21 (1983), 179–187].
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ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2009.01.116