Linking Contractive Self-Mappings and Cyclic Meir-Keeler Contractions with Kannan Self-Mappings
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Research Article Linking Contractive Self-Mappings and Cyclic Meir-Keeler Contractions with Kannan Self-Mappings M. De la Sen Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus of Leioa (Bizkaia)—Aptdo, 644-Bilbao, 48080 Bilbao, Spain Correspondence should be addressed to M. De la Sen, [email protected] Received 1 September 2009; Accepted 22 February 2010 Academic Editor: Wataru Takahashi Copyright q 2010 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Some mutual relations between p-cyclic contractive self-mappings, p-cyclic Kannan self-mappings, and Meir-Keeler p-cyclic contractions are stated. On the other hand, related results about the existence of the best proximity points and existence and uniqueness of fixed points are also formulated.
1. Introduction In the last years, important attention is being devoted to extend the Fixed Point Theory by weakening the conditions on both the maps and the sets where those maps operate 1, 2. For instance, every nonexpansive self-mappings on weakly compact subsets of a metric space have fixed points if the weak fixed point property holds 1. Further, increasing research interest relies on the generalization of Fixed Point Theory to more general spaces than the usual metric spaces such as, for instance, ordered or partially ordered spaces see, e.g., 3–5. Also, important fields of application of Fixed Point Theory exist nowadays in the investigation of the stability of complex continuous-time and discrete-time dynamic systems. The theory has been focused, in particular, on systems possessing internal lags, those being described by functional differential equations, those being characterized as hybrid dynamic systems and those being described by coupled continuous-time and discrete-time dynamics, 6–10. On the other hand, Meir-Keeler self-mappings have received important attention in the context of Fixed Point Theory perhaps due to the associated relaxing in the required conditions for the existence of fixed points compared with the usual contractive mappings 11–14. It also turns out from their definition that such self-mappings are less restrictive
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Fixed Point Theory and Applications
than strict contractive self-mappings so that their associated formalism is applicable to a wider class of real-life problems. Another interest of such self-mappings is their usefulness as a formal tool for the study of p≥2-cyclic contractions, even in the eventual case that the involved subsets of the metric space under study do not intersect, 12 so that there is no fixed point. In such a case, the usual role of fixed points is played by the best proximity points between adjacent subsets in the metric space. The underlying idea is that the best proximity points are fixed points if such subsets intersect while they play a close role to fixed points otherwise.
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