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18
A quantitative Mean Ergodic Theorem for uniformly convex Banach spaces
, 2008
"... We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of T. Tao of the Mean Ergodic Theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by ..."
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Cited by 17 (10 self)
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We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of T. Tao of the Mean Ergodic Theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by Avigad, Gerhardy and Towsner [1] and T. Tao [10]. 1
Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
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Cited by 14 (3 self)
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This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
Nonexpansive iterations in uniformly convex Whyperbolic spaces
, 2008
"... We propose the class of uniformly convex Whyperbolic spaces with monotone modulus of uniform convexity (UCWhyperbolic spaces for short) as an appropriate setting for the study of nonexpansive iterations. UCWhyperbolic spaces are a natural generalization both of uniformly convex normed spaces and ..."
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Cited by 6 (0 self)
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We propose the class of uniformly convex Whyperbolic spaces with monotone modulus of uniform convexity (UCWhyperbolic spaces for short) as an appropriate setting for the study of nonexpansive iterations. UCWhyperbolic spaces are a natural generalization both of uniformly convex normed spaces and CAT(0)spaces. Furthermore, we apply proof mining techniques to get effective rates of asymptotic regularity for Ishikawa iterations of nonexpansive selfmappings of closed convex subsets in UCWhyperbolic spaces. These effective results are new even for uniformly convex Banach spaces.
Effective rates of convergence for Lipschitzian pseudocontractive mappings in general Banach spaces
, 2011
"... This paper gives an explicit and effective rate of convergence for an asymptotic regularity result ‖T xn −xn‖ → 0 due to Chidume and Zegeye in 2004 where (xn) is a certain pertubated KrasnoselskiMann iteration schema for Lipschitz pseudocontractive selfmappings T of closed and convex subsets of a ..."
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Cited by 3 (3 self)
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This paper gives an explicit and effective rate of convergence for an asymptotic regularity result ‖T xn −xn‖ → 0 due to Chidume and Zegeye in 2004 where (xn) is a certain pertubated KrasnoselskiMann iteration schema for Lipschitz pseudocontractive selfmappings T of closed and convex subsets of a real Banach space. We also give a qualitative strengthening of the theorem by Chidume and Zegeye by weakening the assumption of the existence of a fixed point. For the bounded case, our bound is polynomial in the data involved.
Quantitative results on Fejér monotone sequences
, 2015
"... We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejér monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of soca ..."
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Cited by 2 (2 self)
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We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejér monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of socalled metastability in the sense of T. Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations (xn) for firmly nonexpansive, asymptotically nonexpansive, strictly pseudocontractive and other types of mappings. Many of the results hold in a general metric setting with some convexity structure added (socalled Whyperbolic spaces). Sometimes uniform convexity is assumed still covering the important class of CAT(0)spaces due to Gromov.
RATES OF CONVERGENCE AND METASTABILITY FOR ABSTRACT CAUCHY PROBLEMS GENERATED BY ACCRETIVE OPERATORS
"... Abstract. We extract rates of convergence and rates of metastability (in the sense of Tao) for convergence results regarding abstract Cauchy problems generated by φaccretive at zero operators A: D(A)( ⊆ X) → 2X where X is a real Banach space, proved in [8], by prooftheoretic analysis of the proof ..."
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Cited by 2 (2 self)
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Abstract. We extract rates of convergence and rates of metastability (in the sense of Tao) for convergence results regarding abstract Cauchy problems generated by φaccretive at zero operators A: D(A)( ⊆ X) → 2X where X is a real Banach space, proved in [8], by prooftheoretic analysis of the proofs in [8] and having assumed a new, stronger notion of uniform accretivity at zero, yielding a new notion of modulus of accretivity. We compute the rate of metastability for the convergence of the solution of the abstract Cauchy problem generated by a uniformly accretive at zero operator to the unique zero of A via proof mining based on a result by the first author. Finally, we apply our results to a special class of Cauchy problems considered in [8]. This work is the first application of proof mining to the theory of partial differential equations. 1.
GEODESIC METRIC SPACES AND GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS
, 2013
"... Abstract. In this paper, we present some common fixed point theorems for two generalized nonexpansive multivalued mappings in CAT(0) spaces as well as in UCED Banach spaces. Moreover, we prove the existence of fixed points for generalized nonexpansive multivalued mappings in complete geodesic metri ..."
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Abstract. In this paper, we present some common fixed point theorems for two generalized nonexpansive multivalued mappings in CAT(0) spaces as well as in UCED Banach spaces. Moreover, we prove the existence of fixed points for generalized nonexpansive multivalued mappings in complete geodesic metric spaces with convex metric for which the asymptotic center of a bounded sequence in a bounded closed convex subset is nonempty and singleton. The results obtained in this paper extend and improve some recent results.