Noncommutativity of mappings in hybrid fixed point results
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RESEARCH
Open Access
Noncommutativity of mappings in hybrid fixed point results Hemant Kumar Pathak1 and Rosana Rodríguez-López2* *
Correspondence: [email protected] 2 Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain Full list of author information is available at the end of the article
Abstract In this note, some coincidence and common fixed points of nonlinear hybrid mappings have been obtained under certain noncommutativity conditions of mappings. Our results improve several known results in the field of hybrid fixed point theory. MSC: 54H25; 47H10; 54C60 Keywords: coincidence point; fixed point; occasionally coincidentally idempotent; multi-valued mappings
Introduction As a generalization of the Banach fixed point theorem, Nadler’s contraction principle has lead to an excellent fixed point result in the area of nonlinear analysis. Some other works focused on fixed point results for multi-valued mappings are, for instance, [–]. Coincidence and common fixed points of nonlinear hybrid contractions (i.e., contractions involving single-valued and multi-valued mappings) have been recently studied by many authors. To mention some of the achievements, we cite, for example, [–]. The concept of commutativity of single-valued mappings [] was extended in [] to the setting of a single-valued mapping and a multi-valued mapping on a metric space. This concept of commutativity has been further generalized by different authors, viz weakly commuting [], compatible [], weakly compatible []. It is interesting to note that in all the results obtained so far concerning common fixed points of hybrid mappings the (single-valued and multi-valued) mappings under consideration satisfy either the commutativity condition or one of its generalizations (see, for instance, [–]). In this note, we show the existence of fixed points of hybrid contractions which do not satisfy any of the commutativity conditions or its above-mentioned generalizations. Our result extends and improves several well-known results in the field of hybrid fixed point theory. Some other recent related references are [, ], where common fixed point theorems for hybrid mappings on a symmetric space are proved under the assumptions of weak compatibility and occasional weak compatibility. Some analogous results for the case of contractivity conditions of integral type are presented in [–] and generalized contractive hybrid pairs are considered in []. Finally, in [], fixed point results are proved in topological vector space valued cone metric spaces (with nonnormal cones). © 2013 Pathak and Rodríguez-López; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Pathak and Rodríguez-López Boundary Value Problems 2013, 2013:145
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