Strongly Fejer Monotone Mappings Part I : Relaxations
نویسنده
چکیده
We consider the general class of strongly Fejer monotone map-pings and some of their basic properties. These properties are useful for a convergence theory of corresponding iterative methods which are widely used to solve convex problems (see e.g. 3], 6], 7], 10]). In part I we study the relation between these mappings and their relaxations. 1 Strongly Fejer monotone mappings Let H be a (real) Hilbert space. We consider a nonempty, convex and closed subset Q of H (set of feasible elements) and set-valued mappings (multiop-erators) g : Q ! P(Q) , where P(Q) contains all nonempty subsets of Q. For g we introduce sets of weak and strong xed points, namely F ? (g) := fx 2 Q : x 2 g(x)g ; F + (g) := fx 2 Q : fxg = g(x)g ; where F + (g) F ? (g). As usual mappings (operators) g : Q ! Q are integrated as imbeddings. Here both kinds of xed point sets coincide with F(g) := fx 2 Q : x = g(x)g. We exclude the uninteresting special case that g is the identity I (g 6 = I). Now we turn to the basic concepts.
منابع مشابه
A Relaxed Extra Gradient Approximation Method of Two Inverse-Strongly Monotone Mappings for a General System of Variational Inequalities, Fixed Point and Equilibrium Problems
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