Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated . Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given. 1991 M.R. Subject Classification. Primary 47H09, 49M45, 65-02, 65J05, 90C25; Secondary 26B25, 41A65, 46C99, 46N10, 47N10, 52A05, 52A41, 65F10, 65K05, 90C90, 92C55. Key words and phrases. Angle between two subspaces, averaged mapping, Cimmino's method, computerized tomography, convex feasibility problem, convex function, convex inequalities, convex programming, convex set, Fej'er monotone sequence, firmly nonexpansive mapping, H...

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We consider a wide class of iterative methods arising in numerical mathematics and optimization which are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods which makes them strongly convergent without additional assumptions. Several applications are discussed. AMS 1991 subject classification. Primary: 65J15, 47N10; secondary 41A29, 47H05, 47H09, 65K10, 90C25. Key words. Convex feasibility, Fej'er-monotonicity, firmly nonexpansive mapping, fixed point, Haugazeau, maximal monotone operator, projection, proximal point algorithm, resolvent, subgradient algorithm. 1 Introduction Let H be a real Hilbert space with scalar product h\Delta j \Deltai, norm k \Delta k, and distance d. In 1965, Bregman [5] proposed a simple iterative method for finding a common point of m intersecting closed convex sets (S i ) 1im in H. He showed that, given an arbitrary starting point x 0 2 H, the sequence (x n ) n0 gene...

We give a quantitative analysis of a result due to Borwein, Reich and Shafrir on the asymptotic behaviour of the general Krasnoselski-Mann iteration for nonexpansive selfmappings of convex sets in arbitrary normed spaces. Besides providing explicit bounds we also get new qualitative results concerning the independence of the rate of convergence of the norm of that iteration from various input data. In the special case of bounded convex sets, where by well-known results of Ishikawa, Edelstein/O'Brian and Goebel/Kirk the norm of the iteration converges to zero, we obtain uniform bounds which do not depend on the starting point of the iteration and the nonexpansive function, but only depend on the error #, an upper bound on the diameter of C and some very general information on the sequence of scalars # k used in the iteration. Only in the special situation, where # k := # is constant, uniform bounds were known in that bounded case. For the unbounded case, no quantitative information was ...

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In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, Hölder-Lipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.

Maximal monotone operator theory is about to turn (or just has turned) fifty. I intend to briefly survey the history of the subject. I shall try to explain why maximal monotone operators are both interesting and important—culminating with a description of the remarkable progress made during the past decade. 1

Let (x n ) and (x n ) be two vector sequences converging to a common limit. First, we shall define nonlinear hybrid procedures which consist of constructing a new vector sequence (y n ) with better convergence properties than (x n ) and (x n ). Then, this procedure is used for accelerating the convergence of a given sequence and applied to the construction of fixed point methods. New methods are derived. Finally, the connection between fixed point iterations and methods for the numerical integration of differential equations is also exploited. Numerical results are given.