Existence and Approximation of Fixed Points for Set-Valued Mappings
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Research Article Existence and Approximation of Fixed Points for Set-Valued Mappings Simeon Reich and Alexander J. Zaslavski Department of Mathematics, The Technion-Israel Institute of Technology, 32000 Haifa, Israel Correspondence should be addressed to Simeon Reich, [email protected] Received 8 December 2009; Accepted 29 January 2010 Academic Editor: Tomonari Suzuki Copyright q 2010 S. Reich and A. J. Zaslavski. 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. Taking into account possibly inexact data, we study both existence and approximation of fixed points for certain set-valued mappings of contractive type. More precisely, we study the existence of convergent iterations in the presence of computational errors for two classes of set-valued mappings. The first class comprises certain mappings of contractive type, while the second one contains mappings satisfying a Caristi-type condition.
1. Introduction The study of the convergence of iterations of mappings of contractive type has been an important topic in Nonlinear Functional Analysis since Banach’s seminal paper 1 on the existence of a unique fixed point for a strict contraction 2–5. Banach’s celebrated theorem also yields convergence of iterates to the unique fixed point. During the last fifty years or so, many developments have taken place in this area. Interesting results have also been obtained regarding set-valued mappings, where the situation is more difficult and less understood. See, for example, 5–12 and the references cited therein. As already mentioned above, one of the methods used for proving the classical Banach theorem is to show the convergence of Picard iterations, which holds for any initial point. In the case of set-valued mappings, we do not have convergence of all trajectories of the dynamical system induced by the given mapping. Convergent trajectories have to be constructed in a special way. For instance, in 7, if at the moment t 0, 1, . . . we have reached a point xt , then we choose an element of T xt here T is the given mapping such that xt1 approximates the best approximation of xt from T xt . Since our mapping acts on a general complete metric space, we cannot, in general, choose xt1 as the best approximation of xt by elements of T xt . Instead, we require xt1 to approximate the best approximation up to a positive number t , such that the sequence {t }∞ t0 is summable. This method allowed Nadler 7 to obtain the existence of a fixed point of a strictly contractive set-valued mapping and the authors of 6 to obtain more general
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Fixed Point Theory and Applications
results. In view of this state of affairs, it is important to study convergence of the iterates of set-valued mappings in the presence of errors. In this paper, we study the existence of convergent iterations in the presence of computational errors for two classes of
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