scholarly journals Some Logical Metatheorems with Applications in Functional Analysis

2003 ◽  
Vol 10 (21) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Functional Analysis ◽  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Inner Product ◽  
Original Proof ◽  
Uniform Bounds ◽  
Existence Proofs ◽  
Special Cases

In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. `Uniform' here means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, normed linear spaces, uniformly convex spaces as well as inner product spaces.

scholarly journals A Discussion on Random Meir-Keeler Contractions

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 245
Author(s):  
Cheng-Yen Li ◽  
Erdal Karapınar ◽  
Chi-Ming Chen
Keyword(s):  
Functional Analysis ◽  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Random Fixed Point

The aim of this paper is to enrich random fixed point theory, which is one of the cornerstones of probabilistic functional analysis. In this paper, we introduce the notions of random, comparable MT- γ contraction and random, comparable Meir-Keeler contraction in the framework of complete random metric spaces. We investigate the existence of a random fixed point for these contractions. We express illustrative examples to support the presented results.

scholarly journals Some Common Fixed-Point Theorems for Generalized-Contractive-Type Mappings on Complex-Valued Metric Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Chakkrid Klin-eam ◽  
Cholatis Suanoom
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Contractive Type ◽  
Complex Valued ◽  
Common Fixed Point Theorems

Fixed-point theory in complex valued metric spaces has greatly developed in recent times. In this paper, we prove certain common fixed-point theorems for two single-valued mappings in such spaces. The mappings we consider here are assumed to satisfy certain metric inequalities with generalized fixed-point theorems due to Rouzkard and Imdad (2012). This extends and subsumes many results of other authors which were obtained for mappings on complex-valued metric spaces.

scholarly journals Semicompatibility and fixed point theorems in an unboundedD-metric space

2005 ◽  
Vol 2005 (5) ◽  
pp. 789-801
Author(s):  
Bijendra Singh ◽  
Shishir Jain ◽  
Shobha Jain
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Contractive Condition ◽  
Restricted Domain

Rhoades (1996) proved a fixed point theorem in a boundedD-metric space for a contractive self-map with applications. Here we establish a more general fixed point theorem in an unboundedD-metric space, for two self-maps satisfying a general contractive condition with a restricted domain ofxandy. This has been done by using the notion of semicompatible maps inD-metric space. These results generalize and improve the results of Rhoades (1996), Dhage et al. (2000), and Veerapandi and Rao (1996). These results also underline the necessity and importance of semicompatibility in fixed point theory ofD-metric spaces. All the results of this paper are new.

Two open problems in the fixed point theory of contractive type mappings on quasimetric spaces

2017 ◽  
Vol 33 (2) ◽  
pp. 169-180
Author(s):  
MITROFAN M. CHOBAN ◽  
◽  
VASILE BERINDE ◽  
◽  
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Partial Answer ◽  
Open Problems ◽  
Complete Answer ◽  
Contractive Type ◽  
Generalized Distances

Two open problems in the fixed point theory of quasi metric spaces posed in [Berinde, V. and Choban, M. M., Generalized distances and their associate metrics. Impact on fixed point theory, Creat. Math. Inform., 22 (2013), No. 1, 23–32] are considered. We give a complete answer to the first problem, a partial answer to the second one, and also illustrate the complexity and relevance of these problems by means of four very interesting and comprehensive examples.

EXISTENCE OF SOLUTION FOR A VOLTERRA TYPE INTEGRAL EQUATION USING DARBO-TYPE F-CONTRACTION

2021 ◽  
Vol 10 (6) ◽  
pp. 2687-2710
Author(s):  
F. Akutsah ◽  
A. A. Mebawondu ◽  
O. K. Narain
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Existence Of Solution ◽  
Volterra Type ◽  
Complete Metric Spaces ◽  
Type Integral ◽  
New Type

In this paper, we provide some generalizations of the Darbo's fixed point theorem and further develop the notion of $F$-contraction introduced by Wardowski in (\cite{wad}, D. Wardowski, \emph{Fixed points of a new type of contractive mappings in complete metric spaces,} Fixed Point Theory and Appl., 94, (2012)). To achieve this, we introduce the notion of Darbo-type $F$-contraction, cyclic $(\alpha,\beta)$-admissible operator and we also establish some fixed point and common fixed point results for this class of mappings in the framework of Banach spaces. In addition, we apply our fixed point results to establish the existence of solution to a Volterra type integral equation.

scholarly journals A New Approach to the Solution of the Fredholm Integral Equation via a Fixed Point on Extended b-Metric Spaces

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 512 ◽  
Author(s):  
Erdal Karapınar ◽  
Panda Kumari ◽  
Durdana Lateef
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Nash Equilibria ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Real Life ◽  
Point Theory ◽  
New Approach ◽  
Correctness Of Programs

It is very well known that real-life applications of fixed point theory are restricted with the transformation of the problem in the form of f ( x ) = x . (1) The Knaster–Tarski fixed point theorem underlies various approaches of checking the correctness of programs. (2) The Brouwer fixed point theorem is used to prove the existence of Nash equilibria in games. (3) Dlala et al. proposed a solution for magnetic field problems via the fixed point approach.

Fixed Point Theory in Metric Spaces

2018 ◽  
Author(s):  
Praveen Agarwal ◽  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory

scholarly journals Common Fixed Point Theorems for Generalized Cyclic Contraction Pair in Partial B-Metric Spaces

2019 ◽  
Vol 8 (10S) ◽  
pp. 85-89
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Cyclic Contraction ◽  
Common Fixed Point Theorems

In this paper, we introduce the notion of generalized cyclic contraction pair with transitive mapping in partial b-metric spaces. Also, we establish some fixed point theorems for this contraction pair. Our results generalize and improve the result of Oratai Yamaod, Wutiphol Sintunavarat and Yeol Je Cho (Fixed Point Theory App. 2015:164) in partial-b-metric spaces.

scholarly journals On Fuzzy Orders and Metrics

2014 ◽  
Vol 10 ◽  
pp. 48-52
Author(s):  
Vinayak E. Nikumbh
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Ordered Sets

In this paper we present results regarding fixed point theory using notion of fuzzy orders.We propose anapproach based on quasi metrics which in a way unifies the fixed point theory for ordered sets and metric spaces.

Fixed point results for mappings satisfying (ψ φ)-weakly contractive condition in ordered partial metric spaces

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Hemant Nashine
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Contractive Condition ◽  
Partial Metric ◽  
New Class ◽  
Weakly Contractive Condition ◽  
Partial Metric Spaces ◽  
Weakly Contractive

AbstractIn [18], Matthews introduced a new class of metric spaces, that is, the concept of partial metric spaces, or equivalently, weightable quasi-metrics, are investigated to generalize metric spaces (X, d), to develop and to introduce a new fixed point theory. In partial metric spaces, the self-distance for any point need not be equal to zero. In this paper, we study some results for single map satisfying (ψ,φ)-weakly contractive condition in partial metric spaces endowed with partial order. An example is given to support the useability of our results.

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