Explicit algorithms for J-fixed points of some non linear mappings in certain Banach spaces

2020 ◽  
Vol 29 (1) ◽  
pp. 27-36
Author(s):  
M. M. GUEYE ◽  
M. SENE ◽  
M. NDIAYE ◽  
N. DJITTE
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Maximal Monotone ◽  
Point Theory ◽  
Duality Mapping ◽  
Monotone Mappings ◽  
Normalized Duality Mapping ◽  
Maximal Monotone Mappings ◽  
Pseudocontractive Mappings

Let E be a real normed linear space and E∗ its dual. In a recent work, Chidume et al. [Chidume, C. E. and Idu, K. O., Approximation of zeros of bounded maximal monotone mappings, solutions of hammerstein integral equations and convex minimizations problems, Fixed Point Theory and Applications, 97 (2016)] introduced the new concepts of J-fixed points and J-pseudocontractive mappings and they shown that a mapping A : E → 2 E∗ is monotone if and only if the map T := (J −A) : E → 2 E∗ is J-pseudocontractive, where J is the normalized duality mapping of E. It is our purpose in this work to introduce an algorithm for approximating J-fixed points of J-pseudocontractive mappings. Our results are applied to approximate zeros of monotone mappings in certain Banach spaces. The results obtained here, extend and unify some recent results in this direction for the class of maximal monotone mappings in uniformly smooth and strictly convex real Banach spaces. Our proof is of independent interest.

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.

Fixed points of multivalued maps under local Lipschitz conditions and applications

2019 ◽  
Vol 150 (3) ◽  
pp. 1467-1494
Author(s):  
Claudio A. Gallegos ◽  
Hernán R. Henríquez
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Point Theory ◽  
Vector Subspace ◽  
Multivalued Maps ◽  
Lipschitz Continuous

AbstractIn this work we are concerned with the existence of fixed points for multivalued maps defined on Banach spaces. Using the Banach spaces scale concept, we establish the existence of a fixed point of a multivalued map in a vector subspace where the map is only locally Lipschitz continuous. We apply our results to the existence of mild solutions and asymptotically almost periodic solutions of an abstract Cauchy problem governed by a first-order differential inclusion. Our results are obtained by using fixed point theory for the measure of noncompactness.

scholarly journals Fixed Points of Kannan Maps in the Variable Exponent Sequence Spaces ℓp(·)

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 76 ◽  
Author(s):  
Afrah Abdou ◽  
Mohamed 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 ℓ p ( · ) . 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 On Fixed Point Theory of Monotone Mappings with Respect to a Partial Order Introduced by a Vector Functional in Cone Metric Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Zhilong Li ◽  
Shujun Jiang
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Partial Order ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Monotone Mappings ◽  
Cone Metric Spaces ◽  
Cone Metric ◽  
Minimal Fixed Point

We presented some maximal and minimal fixed point theorems of set-valued monotone mappings with respect to a partial order introduced by a vector functional in cone metric spaces. In addition, we proved not only the existence of maximal and minimal fixed points but also the existence of the largest and the least fixed points of single-valued increasing mappings. It is worth mentioning that the results on single-valued mappings in this paper are still new even in the case of metric spaces and hence they indeed improve the recent results.

scholarly journals Fixed Points Theorems for Unsaturated and Saturated Classes of Contractive Mappings in Banach Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 713
Author(s):  
Vasile Berinde ◽  
Mădălina Păcurar
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Contractive Mappings ◽  
Contractive Type

Based on the technique of enriching contractive type mappings, a technique that has been used successfully in some recent papers, we introduce the concept of a saturated class of contractive mappings. We show that, from this perspective, the contractive type mappings in the metric fixed point theory can be separated into two distinct classes, unsaturated and saturated, and that, for any unsaturated class of mappings, the technique of enriching contractive type mappings provides genuine new fixed-point results. We illustrate the concept by surveying some significant fixed-point results obtained recently for five remarkable unsaturated classes of contractive mappings. In the second part of the paper, we also identify two important classes of saturated contractive mappings, whose main feature is that they cannot be enlarged by enriching the contractive mappings.

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