scholarly journals Strongly Uniform Bounds from Semi-Constructive Proofs

2004 ◽  
Vol 11 (31) ◽  
Author(s):  
Philipp Gerhardy ◽  
Ulrich Kohlenbach
Keyword(s):  
Functional Analysis ◽  
Fixed Point Theory ◽  
Classical Case ◽  
Point Theory ◽  
Prima Facie ◽  
Uniform Bounds ◽  
Linear Spaces ◽  
Compact Spaces ◽  
Constructive Proofs ◽  
First Time

In 2003, the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie non-constructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and normed linear spaces and guarantee the independence of the bounds from parameters raging over metrically bounded (not necessarily compact!) spaces. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semi-intuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain non-constructive principles. Contrary to the classical case, there are practically no restrictions on the logical complexity of theorems for which bounds can be extracted. Again, our metatheorems guarantee very general uniformities, even in cases where the existence of uniform bounds is not obtainable by (ineffective) straightforward functional analytic means. Already in the purely intuitionistic case, where the existence of effective bounds is implicit, the metatheorems allow one to derive uniformities that may not be obvious at all from a given constructive proofs. Finally, we illustrate our main metatheorem by an example from metric fixed point theory.

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 LMI-Based Stability Criterion of Impulsive T-S Fuzzy Dynamic Equations via Fixed Point Theory

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Ruofeng Rao ◽  
Zhilin Pu
Keyword(s):  
Fixed Point ◽  
Stability Criterion ◽  
High Efficiency ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Linear Matrix ◽  
The Matrix ◽  
Matrix Exponential Function ◽  
Takagi Sugeno ◽  
First Time

By formulating a contraction mapping and the matrix exponential function, the authors apply linear matrix inequality (LMI) technique to investigate and obtain the LMI-based stability criterion of a class of time-delay Takagi-Sugeno (T-S) fuzzy differential equations. To the best of our knowledge, it is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. It is worth mentioning that LMI methods have high efficiency and other advantages in largescale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. At the end of this paper, a numerical example is presented to illustrate the effectiveness of the proposed methods.

Localization and multiplicity in the homogenization of nonlinear problems

2019 ◽  
Vol 9 (1) ◽  
pp. 292-304 ◽  
Author(s):  
Renata Bunoiu ◽  
Radu Precup
Keyword(s):  
Functional Analysis ◽  
Fixed Point Theory ◽  
Nonlinear Problems ◽  
Point Theory ◽  
Homogenization Theory ◽  
Infinitely Many Solutions ◽  
Significant Gain ◽  
Periodic Homogenization ◽  
Nonlinear Functional Analysis ◽  
Nonlinear Functional

Abstract We propose a method for the localization of solutions for a class of nonlinear problems arising in the homogenization theory. The method combines concepts and results from the linear theory of PDEs, linear periodic homogenization theory, and nonlinear functional analysis. Particularly, we use the Moser-Harnack inequality, arguments of fixed point theory and Ekeland's variational principle. A significant gain in the homogenization theory of nonlinear problems is that our method makes possible the emergence of finitely or infinitely many solutions.

scholarly journals LMI-Based Stability Criterion for Impulsive CGNNs via Fixed Point Theory

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Xiongrui Wang ◽  
Ruofeng Rao ◽  
Shouming Zhong
Keyword(s):  
Fixed Point ◽  
Contraction Mapping ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Contraction Mapping Theorem ◽  
Related Literature ◽  
Variation Range ◽  
Exponentially Stable ◽  
The Stability ◽  
First Time

Linear matrices inequalities (LMIs) method and the contraction mapping theorem were employed to prove the existence of globally exponentially stable trivial solution for impulsive Cohen-Grossberg neural networks (CGNNs). It is worth mentioning that it is the first time to use the contraction mapping theorem to prove the stability for CGNNs while only the Leray-Schauder fixed point theorem was applied in previous related literature. An example is given to illustrate the effectiveness of the proposed methods due to the large allowable variation range of impulse.

scholarly journals Fixed Point Theory and the Ulam Stability

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Janusz Brzdęk ◽  
Liviu Cădariu ◽  
Krzysztof Ciepliński
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
Fixed Point Theorems ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Stability Of Functional Equations ◽  
Banach’S Fixed Point Theorem ◽  
The Stability ◽  
First Time ◽  
Stability Results

The fixed point method has been applied for the first time, in proving the stability results for functional equations, by Baker (1991); he used a variant of Banach's fixed point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow the approaches involving a theorem of Diaz and Margolis. The main aim of this survey is to present applications of different fixed point theorems to the theory of stability of functional equations, motivated by a problem raised by Ulam in 1940.

Elements of Geometry of Balls in Banach Spaces

2018 ◽  
Author(s):  
Kazimierz Goebel ◽  
Stanislaw Prus
Keyword(s):  
Functional Analysis ◽  
Graduate Students ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Normed Spaces ◽  
The Subject ◽  
Basic Facts ◽  
Weak Topologies

One of the subjects of functional analysis is classification of Banach spaces depending on various properties of the unit ball. The need of such considerations comes from a number of applications to problems of mathematical analysis. The list of subjects contains: differential calculus in normed spaces, approximation theory, weak topologies and reflexivity, general theory of convexity and convex functions, metric fixed point theory, and others. The aim of this book is to present basic facts from this field. It is addressed to advanced undergraduate and graduate students interested in the subject. For some it may result in further interest, a continuation and deepening of their study of the subject. It may be also useful for instructors running courses on functional analysis, supervising diploma theses or essays on various levels.

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 General Logical Metatheorems for Functional Analysis

2005 ◽  
Vol 12 (21) ◽  
Author(s):  
Philipp Gerhardy ◽  
Ulrich Kohlenbach
Keyword(s):  
Functional Analysis ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Inner Product ◽  
Product Spaces ◽  
Nonlinear Functional Analysis ◽  
Nonlinear Functional ◽  
Large Classes ◽  
Global Boundedness ◽  
Uniformly Continuous

In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, Hölder-Lipschitz, uniformly continuous, bounded and weakly quasi-nonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.

scholarly journals Hyperbolic spaces and directional contractions

2019 ◽  
Vol 09 (03) ◽  
pp. 1950021
Author(s):  
W. A. Kirk ◽  
Naseer Shahzad
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Axiomatic Approach ◽  
Hyperbolic Type ◽  
Hyperbolic Spaces ◽  
First Time

The axiomatic approach to metric convexity goes back to the pioneering work of Karl Menger in 1928. This is an overview of this concept and the role it plays in metric fixed point theory especially in conjunction with spaces possessing a “hyperbolic” type structures. These include the CAT(0) spaces, hyperconvex metric spaces, and [Formula: see text]-trees. Much of the discussion involves the existence of “approximate” fixed point sequences for mappings satisfying weak contractive conditions. Applications of a well-known fixed point theorem due to Caristi are also included. These involve fixed and approximate fixed points for mappings satisfying local “directional” contractive and non-expansive conditions. Convexity plays a role in this part of the discussion as well. While the paper is semi-expository in nature, some detailed proofs appear here for the first time. Also the concept of a weak [Formula: see text]-directional contraction introduced in Sec. 8 appears to be new. Several suggestions for further research are also discussed.

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