scholarly journals The fixed point problem for generalised nonexpansive maps

1997 ◽  
Vol 55 (1) ◽  
pp. 45-61 ◽  
Author(s):  
Michael A. Smyth
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Theory ◽  
Normal Structure ◽  
Structure Type ◽  
Point Theory ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Nonexpansive Maps

This paper is concerned with extending the theory of the existence of fixed points for generalised nonexpansive maps as far as possible. This can be seen as a continuation of the work of Maurey on the extension of the fixed point theory for nonexpansive maps beyond the requirement of normal structure type conditions.

On the theory of fixed point theorems for convex contraction mappings

2015 ◽  
Vol 31 (3) ◽  
pp. 365-371
Author(s):  
VIORICA MURESAN ◽  
◽  
ANTON S. MURESAN ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Theorems ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Shadowing Property ◽  
Contraction Mappings ◽  
Limit Shadowing

Based on the concepts and problems introduced in [Rus, I. A., The theory of a metrical fixed point theorem: theoretical and applicative relevances, Fixed Point Theory, 9 (2008), No. 2, 541–559], in the present paper we consider the theory of some fixed point theorems for convex contraction mappings. We give some results on the following aspects: data dependence of fixed points; sequences of operators and fixed points; well-posedness of a fixed point problem; limit shadowing property and Ulam-Hyers stability for fixed point 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.

A fixed point problem with constraint inequalities via a contraction in incomplete metric spaces

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3365-3379 ◽  
Author(s):  
Z. Ahmadi ◽  
R. Lashkaripour ◽  
H. Baghani
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Complete Metric Spaces ◽  
New Class ◽  
Constraint Inequalities ◽  
Weaker Conditions

In the present paper, firstly, we review the notion of the SO-complete metric spaces. This notion let us to consider some fixed point theorems for single-valued mappings in incomplete metric spaces. Secondly, as motivated by the recent work of H. Baghani et al.(A fixed point theorem for a new class of set-valued mappings in R-complete (not necessarily complete) metric spaces, Filomat, 31 (2017), 3875-3884), we obtain the results of Ansari et al. [J. Fixed Point Theory Appl. (2017), 1145-1163] with very much weaker conditions. Also, we provide some examples show that our main theorem is a generalization of previous results. Finally, we give an application to the boundary value system for our results.

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 A new viscosity-type iteration for a finite family of split variational inclusion and fixed point problems between Hilbert and Banach spaces

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
C. Izuchukwu ◽  
F. O. Isiogugu ◽  
C. C. Okeke
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Iteration Process ◽  
Point Theory ◽  
Fixed Point Problem ◽  
Finite Family ◽  
Variational Inclusion ◽  
Inclusion Problem ◽  
Point Problem ◽  
Common Solution

Abstract In this paper, we introduce a new viscosity-type iteration process for approximating a common solution of a finite family of split variational inclusion problem and fixed point problem. We prove that the proposed algorithm converges strongly to a common solution of a finite family of split variational inclusion problems and fixed point problem for a finite family of type-one demicontractive mappings between a Hilbert space and a Banach space. Furthermore, we applied our results to study a finite family of split convex minimization problems, and also considered a numerical experiment of our results to further illustrate its applicability. Our results extend and improve the results of Byrne et al. (J. Nonlinear Convex Anal. 13:759–775, 2012), Kazmi and Rizvi (Optim. Lett. 8(3):1113–1124, 2014), Moudafi (J. Optim. Theory Appl. 150:275–283, 2011), Shehu and Ogbuisi (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 110(2):503–518, 2016), Takahashi and Yao (Fixed Point Theory Appl. 2015:87, 2015), Chidume and Ezeora (Fixed Point Theory Appl. 2014:111, 2014), and a host of other important results in this direction.

scholarly journals A Real-Time Iterative Projection Scheme for Solving the Common Fixed Point Problem and Its Applications

2018 ◽  
Vol 64 (4) ◽  
pp. 616-636
Author(s):  
A Gibali ◽  
D Teller
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Real Time ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Common Fixed Point Problem ◽  
The Common

In this paper, we are concerned with the Common Fixed Point Problem (CFPP) with demicontractive operators and its special instance, the Convex Feasibility Problem (CFP) in real Hilbert spaces. Motivated by the recent result of Ordon˜ ez et al. [35] and in general, the field of online/real-time algorithms, e.g., [20, 21, 30], in which the entire input is not available from the beginning and given piece-by-piece, we propose an online/real-time iterative scheme for solving CFPPs and CFPs in which the involved operators/sets emerge along time. This scheme is capable of operating on any block, for any finite number of iterations, before moving, in a serial way, to the next block. The scheme is based on the recent novel result of Reich and Zalas [37] known as the Modular String Averaging (MSA) procedure. The convergence of the scheme follows [37] and other classical results in the fields of fixed point theory and variational inequalities, such as [34]. Numerical experiments for linear and non-linear feasibility problems with applications to image recovery are presented and demonstrate the validity and potential applicability of our scheme, e.g., to online/real-time scenarios.

scholarly journals Cyclic Mappings and Further Results in B-Metric-Like Spaces

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Shengquan Weng ◽  
Quanxin Zhu ◽  
Baoying Du ◽  
Kaibo Shi
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Research Work ◽  
Point Theory ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Generalized Metric Space ◽  
Generalized Metric ◽  
Formula Type

Fixed point problem of many mappings has been widely studied in the research work of fixed point theory. The generalized metric space is one of the research objects of fixed point theory. B-metric-like space is one of the generalized metric spaces; in fact, the research work in B-metric-like spaces is attractive. The intention of this paper is to introduce the concept of other cyclic mappings, named as L β -type cyclic mappings in the setting of B-metric-like space, study the existence and uniqueness of fixed point problem of L β -type cyclic mapping, and obtain some new results in B-metric-like spaces. Furthermore, the main results in this paper are illustrated by a concrete example. The work of this paper extend and promote the previous results in B-metric-like spaces.

scholarly journals Basic problems of the metric fixed point theory and the relevance of a metric fixed point theorem

2013 ◽  
Vol 29 (2) ◽  
pp. 239-258
Author(s):  
IOAN A. RUS ◽  
◽  
MARCEL-ADRIAN SERBAN ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Shadowing Property ◽  
Limit Shadowing ◽  
The Impact

In this paper we present some basic problems of the metric fixed point theory (existence, uniqueness, settheoretic aspects (Bessaga, Janos, Rus, ...), order-theoretic aspects (Ekeland, Bronsted, Caristi, Kirk, Jachymski, ...), convergence of the succesive approximations, data dependence (general estimation, Ulam problem, dependence on the parameters, ...), well-posedness of the fixed point problem, limit shadowing property, stability, Gronwall lemmas, comparison lemmas, retractibility, ...). Following [I. A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevances, Fixed Point Theory, 9 (2008), No. 2, 541–559] we define the relevance of a metrical fixed point theorem by the impact of the theorem on these basic problems. Some case studies are presented.

scholarly journals A New Approach for the Approximations of Solutions to a Common Fixed Point Problem in Metric Fixed Point Theory

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Ishak Altun ◽  
Nassir Al Arifi ◽  
Mohamed Jleli ◽  
Aref Lashin ◽  
Bessem Samet
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Fixed Point Problem ◽  
Point Problem ◽  
New Approach ◽  
Common Fixed Point Problem ◽  
Unique Common Fixed Point

We provide sufficient conditions for the existence of a unique common fixed point for a pair of mappings T,S:X→X, where X is a nonempty set endowed with a certain metric. Moreover, a numerical algorithm is presented in order to approximate such solution. Our approach is different to the usual used methods in the literature.

Perturbation of the fixed point problem for continuous mappings

2020 ◽  
pp. 241-249
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Continuous Mapping ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Bounded Subset ◽  
Completely Continuous ◽  
Continuous Map ◽  
Continuous Mappings

We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.

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