scholarly journals AN ALTERNATIVE APPROACH TO FRÉCHET DERIVATIVES

2020 ◽  
pp. 1-19
Author(s):  
SHANE ARORA ◽  
HAZEL BROWNE ◽  
DANIEL DANERS
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Banach Spaces ◽  
Short Proof ◽  
Banach Fixed Point Theorem ◽  
Alternative Approach ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
The Schwarz Lemma

We discuss an alternative approach to Fréchet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carathéodory. The approach allows many questions of differentiability to be reduced to questions of continuity. We demonstrate how that simplifies the theory of differentiation, including the rules of differentiation and the Schwarz lemma on the symmetry of second-order derivatives. We also provide a short proof of the differentiable dependence of fixed points in the Banach fixed point theorem.

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.

Fixed point theorems for a system of transformations in generalized metric spaces

2015 ◽  
Vol 08 (04) ◽  
pp. 1550068 ◽  
Author(s):  
Stefan Czerwik ◽  
Krzysztof Król
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Banach Fixed Point Theorem ◽  
Generalized Metric Spaces ◽  
Generalized Metric

In this paper we present the results on the existence of fixed points of system of mappings in generalized metric spaces generalizing the result of Diaz and Margolis. Also the “local fixed point theorems” of a system of such mappings both in generalized and ordinary metric spaces are stated. Banach fixed point theorem and many others are consequences of our results.

scholarly journals Fixed points of holomorphic mappings for domains in Banach spaces

2003 ◽  
Vol 2003 (5) ◽  
pp. 261-274 ◽  
Author(s):  
Lawrence A. Harris
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Holomorphic Function ◽  
Fixed Points ◽  
Banach Spaces ◽  
Point Theorem ◽  
Numerical Range ◽  
Holomorphic Functions ◽  
Holomorphic Mappings

We discuss the Earle-Hamilton fixed-point theorem and show how it can be applied when restrictions are known on the numerical range of a holomorphic function. In particular, we extend the Earle-Hamilton theorem to holomorphic functions with numerical range having real part strictly less than 1. We also extend the Lumer-Phillips theorem estimating resolvents to dissipative holomorphic functions.

Fixed Points-Contractive Functions

2001 ◽  
Author(s):  
Krzysztof A. Sikorski
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Research Area ◽  
Banach Fixed Point Theorem ◽  
Homotopy Methods ◽  
Simplicial Algorithm ◽  
Intensive Research ◽  
Fixed Point Computation

Fixed point computation has been an intensive research area since 1967 when Scarf introduced simplicial algorithm to approximate fixed points. Several algorithms have been invented since then, including restart and homotopy methods. Most of these were designed to approximate fixed points of general maps and used the residual error criterion. In this chapter we consider the absolute and/or relative error criteria for contractive univariate and multivariate functions. The departure of our analysis is the classical Banach fixed point theorem. Namely, we consider a function f : D →D, where D is a closed subset of a Banach space B. We assume that f is contractive with a factor q < 1, i.e., . . . ||f(x) – f(y)|| ≤ q ||x-y||, for all x,y ∈ D. Then, there exists a unique ∝ = ∝ (f) ∈ D such that ∝ is a fixed point of f, ∝ = f (∝)

FIXED POINTS AND APPROXIMATELY C*-TERNARY QUADRATIC HIGHER DERIVATIONS

2013 ◽  
Vol 10 (10) ◽  
pp. 1320017
Author(s):  
M. ESHAGHI GORDJI ◽  
B. ALIZADEH ◽  
M. DE LA SEN ◽  
M. B. GHAEMI
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Point Theorem ◽  
Banach Fixed Point Theorem ◽  
Ulam Stability

In this paper, we prove the generalized Hyers–Ulam stability of C*-ternary quadratic higher derivations of any rank by using the Banach fixed point theorem.

scholarly journals Existence and uniqueness of mild and strong solutions of nonlinear volterra integrodifferential equations in Banach spaces

2010 ◽  
Vol 43 (3) ◽  
Author(s):  
H. L. Tidke ◽  
M. B. Dhakne
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Semigroup Theory ◽  
Strong Solutions ◽  
Banach Fixed Point Theorem ◽  
Volterra Integrodifferential Equation

AbstractIn this paper we prove the existence and uniqueness of mild and strong solutions of a nonlinear Volterra integrodifferential equation with nonlocal condition. Our analysis is based on semigroup theory and Banach fixed point theorem and inequalities are established by Gronwall and B. G. Pachpatte.

scholarly journals Boundary value problems for impulsive fractional differential equations in Banach spaces

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5603-5616 ◽  
Author(s):  
Shuai Yang ◽  
Shuqin Zhang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Banach Spaces ◽  
Point Theorem ◽  
Boundary Value ◽  
Banach Fixed Point Theorem ◽  
Uniqueness Of Solutions ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential

This paper is devoted to studying the existence and uniqueness of solutions to the boundary value problems for a impulsive fractional differential equation in Banach spaces. The arguments are based upon the methods of noncompact measure, Banach fixed point theorem and Krasnoselskii?s fixed point theorem. Some examples are given to demonstrate the application of our main results.

Controllability for some nonlinear impulsive partial functional integrodifferential systems with infinite delay in Banach spaces

2020 ◽  
Vol 4 (2) ◽  
pp. 104-115
Author(s):  
Khalil Ezzinbi ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Resolvent Operator ◽  
Measure Of Noncompactness ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Mönch Fixed Point Theorem ◽  
The Resolvent Operator

This work concerns the study of the controllability for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of the work of K. Balachandran and R. Sakthivel (Journal of Mathematical Analysis and Applications, 255, 447-457, (2001)) and a host of important results in the literature, without assuming the compactness of the resolvent operator. An example is given for illustration.

Weak compactness of Fréchet-derivatives: application to composition operators

1980 ◽  
Vol 29 (4) ◽  
pp. 399-406
Author(s):  
Peter Dierolf ◽  
Jürgen Voigt
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Sobolev Spaces ◽  
Composition Operators ◽  
Weak Compactness ◽  
Nonlinear Integral Equations ◽  
Weakly Compact ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Compact Map

AbstractWe prove a result on compactness properties of Fréchet-derivatives which implies that the Fréchet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.

scholarly journals On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Jing Cui ◽  
Litan Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Banach Fixed Point Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Composition Theorem

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

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