scholarly journals Concentration analysis in Banach spaces

2016 ◽  
Vol 18 (03) ◽  
pp. 1550038 ◽  
Author(s):  
Sergio Solimini ◽  
Cyril Tintarev
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniformly Convex ◽  
Opial Condition ◽  
Uniformly Convex Banach Spaces ◽  
Weak Star ◽  
Compactness Arguments

The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach–Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of [Formula: see text]-convergence by Lim [Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179–182] instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and [Formula: see text]-spaces, but not in [Formula: see text], [Formula: see text]. [Formula: see text]-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies the connection of [Formula: see text]-convergence with the Brezis–Lieb lemma and gives a version of the latter without an assumption of convergence a.e.

scholarly journals Common fixed point iterations for a class of multi-valued mappings in $$\mathbf {CAT}(0)$$ spaces

2020 ◽  
Vol 9 (3) ◽  
pp. 681-690
Author(s):  
Khairul Saleh ◽  
Hafiz Fukhar-ud-din
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Finite Family ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces ◽  
Fixed Point Iterations

Abstract In this work, we propose an iterative scheme to approach common fixed point(s) of a finite family of generalized multi-valued nonexpansive mappings in a CAT(0) space. We establish and prove convergence theorems for the algorithm. The results are new and interesting in the theory of $$CAT\left( 0\right) $$ C A T 0 spaces and are the analogues of corresponding ones in uniformly convex Banach spaces and Hilbert spaces.

scholarly journals Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 187-196 ◽  
Author(s):  
Kifayat Ullah ◽  
Muhammad Arshad
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Fixed Points ◽  
Fixed Point Theory ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Point Theory ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Uniformly Convex Banach Spaces

In this paper we propose a new three-step iteration process, called M iteration process, for approximation of fixed points. Some weak and strong convergence theorems are proved for Suzuki generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. Numerical example is given to show the efficiency of new iteration process. Our results are the extension, improvement and generalization of many known results in the literature of iterations in fixed point theory.

scholarly journals A renorming in some Banach spaces with applications to fixed point theory

2010 ◽  
Vol 258 (10) ◽  
pp. 3452-3468 ◽  
Author(s):  
Carlos A. Hernandez Linares ◽  
Maria A. Japon
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Point Theory

scholarly journals Nonnegative solutions of two-point boundary value problems for nonlinear second order integrodifferential equations in Banach spaces

1991 ◽  
Vol 4 (1) ◽  
pp. 47-69 ◽  
Author(s):  
Dajun Guo
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Index Theory ◽  
Point Theory ◽  
Second Order ◽  
Integrodifferential Equations ◽  
Third Order ◽  
Nonnegative Solutions ◽  
Fixed Point Index Theory

In this paper, we combine the fixed point theory, fixed point index theory and cone theory to investigate the nonnegative solutions of two-point BVP for nonlinear second order integrodifferential equations in Banach spaces. As application, we get some results for the third order case. Finally, we give several examples for both infinite and finite systems of ordinary nonlinear integrodifferential equations.

scholarly journals Erratum: Abdou, A. A. N. and Khamsi, M.A. Fixed Points of Kannan Maps in the Variable Exponent Sequence Spaces ℓp(·). Mathematics 2020, 8, 76

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 578
Author(s):  
Afrah A. N. Abdou ◽  
Mohamed Amine Khamsi
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Variable Exponent ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Sequence Spaces ◽  
Vector Spaces ◽  
Nonexpansive Maps ◽  
Banach Contraction

Kannan maps have inspired a branch of metric fixed point theory devoted to the extension of the classical Banach contraction principle. The study of these maps in modular vector spaces was attempted timidly and was not successful. In this work, we look at this problem in the variable exponent sequence spaces lp(·). We prove the modular version of most of the known facts about these maps in metric and Banach spaces. In particular, our results for Kannan nonexpansive maps in the modular sense were never attempted before.

scholarly journals Fixed point theory of Mönch type for weakly sequentially upper semicontinuous maps

2000 ◽  
Vol 61 (3) ◽  
pp. 439-449 ◽  
Author(s):  
Donal O'Regan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Banach Spaces ◽  
Weak Topology ◽  
Existence Result ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Upper Semicontinuous

A variety of fixed point results are presented for weakly sequentially upper semicontinuous maps. In addition an existence result is established for differential equations in Banach spaces relative to the weak topology.

scholarly journals Fixed point approximation of Presic nonexpansive mappings in product of CAT (0) spaces

2016 ◽  
Vol 32 (3) ◽  
pp. 315-322
Author(s):  
HAFIZ FUKHAR-UD-DIN ◽  
◽  
VASILE BERINDE ◽  
ABDUL RAHIM KHAN ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Banach Spaces ◽  
Point Theorem ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Control Parameters ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces

We obtain a fixed point theorem for Presiˇ c nonexpansive mappings on the product of ´ CAT (0) spaces and approximate this fixed points through Ishikawa type iterative algorithms under relaxed conditions on the control parameters. Our results are new in the literature and are valid in uniformly convex Banach spaces.

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