Differential Calculus and the Implicit Function Theorem

1986 ◽  
pp. 130-205
Author(s):  
Eberhard Zeidler
Keyword(s):  
Implicit Function Theorem ◽  
Differential Calculus ◽  
Implicit Function

A differentiation in locally convex spaces

1975 ◽  
Vol 12 (2) ◽  
pp. 183-209 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Implicit Function Theorem ◽  
Differential Calculus ◽  
Linear Operators ◽  
Locally Convex Spaces ◽  
Implicit Function ◽  
Fundamental Theory ◽  
Finite Linear ◽  
Locally Convex ◽  
Convex Spaces

The theory of F-finite linear operators developed by Robert T. Moore is used to construct a differential calculus in locally convex spaces. This note contains the fundamental theory up to the implicit function theorem.

The Origin and Bifurcation of Disclinations in the Gauge Field Theory of Dislocation and Disclination Continuum

1998 ◽  
Vol 12 (25) ◽  
pp. 2599-2617 ◽  
Author(s):  
Guo-Hong Yang ◽  
Yishi Duan
Keyword(s):  
Field Theory ◽  
Gauge Field ◽  
Implicit Function Theorem ◽  
Taylor Expansion ◽  
Gauge Field Theory ◽  
Implicit Function ◽  
Quantum Numbers ◽  
Topological Quantum ◽  
Topological Current ◽  
Limit Points

In the 4-dimensional gauge field theory of dislocation and disclination continuum, the topological current structure and the topological quantization of disclinations are approached. Using the implicit function theorem and Taylor expansion, the origin and bifurcation theories of disclinations are detailed in the neighborhoods of limit points and bifurcation points, respectively. The branch solutions at the limit points and the different directions of all branch curves at 1-order and 2-order degenerated points are calculated. It is pointed out that an original disclination point can split into four disclinations at one time at most. Since the disclination current is identically conserved, the total topological quantum numbers of these branched disclinations will remain constant during their origin and bifurcation processes. Furthermore, one can see the fact that the origin and bifurcation of disclinations are not gradual changes but sudden changes. As some applications of the proposal theory, two examples are presented in the paper.

scholarly journals Uniqueness, continuity and the existence of implicit functions in constructive analysis

2011 ◽  
Vol 14 ◽  
pp. 127-136 ◽  
Author(s):  
H. Diener ◽  
P. Schuster
Keyword(s):  
Implicit Function Theorem ◽  
A Priori ◽  
Implicit Function ◽  
Implicit Functions ◽  
Unique Existence ◽  
Constructive Analysis ◽  
Full Proof

AbstractWe extract a quantitative variant of uniqueness from the usual hypotheses of the implicit function theorem. Not only does this lead to an a priori proof of continuity, but also to an alternative, full proof of the implicit function theorem. Additionally, we investigate implicit functions as a case of the unique existence paradigm with parameters.

scholarly journals Implicit Function Theorem. Part II

2019 ◽  
Vol 27 (2) ◽  
pp. 117-131
Author(s):  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Implicit Function Theorem ◽  
Existence And Uniqueness ◽  
Continuous Linear Operator ◽  
Linear Operators ◽  
Implicit Function ◽  
Continuous Linear ◽  
Lipschitz Continuous ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Continuous Linear Operators

Summary In this article, we formalize differentiability of implicit function theorem in the Mizar system [3], [1]. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here. In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in [6] is cited. We referred to [10], [11], and [2] in the formalization.

scholarly journals Existence of reaction-diffusion-convection waves in unbounded strips

2005 ◽  
Vol 2005 (2) ◽  
pp. 169-193 ◽  
Author(s):  
M. Belk ◽  
B. Kazmierczak ◽  
V. Volpert
Keyword(s):  
Implicit Function Theorem ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Fredholm Property ◽  
Implicit Function ◽  
Solvability Conditions ◽  
All Speeds ◽  
Diffusion Convection ◽  
Rayleigh Numbers ◽  
Property Index

Existence of reaction-diffusion-convection waves in unbounded strips is proved in the case of small Rayleigh numbers. In the bistable case the wave is unique, in the monostable case they exist for all speeds greater than the minimal one. The proof uses the implicit function theorem. Its application is based on the Fredholm property, index, and solvability conditions for elliptic problems in unbounded domains.

Existence of Impact Period-1 Orbit After Grazing Bifurcation for a Soft Impacting System

2021 ◽  
Vol 31 (07) ◽  
pp. 2150103
Author(s):  
Shihui Fu
Keyword(s):  
Periodic Orbits ◽  
Implicit Function Theorem ◽  
Theoretical Study ◽  
Numerical Results ◽  
Implicit Function ◽  
Grazing Bifurcation ◽  
Single Impact

In the paper, the theoretical study on some experimental and numerical results of grazing bifurcation for a soft impacting system are analyzed. After the conditions under which nonimpact period-1 orbit and grazing bifurcation exist are given, we prove that an impact period-1 orbit exists by the implicit function theorem. The details of these periodic orbits are also investigated, which also helps us numerically find them. The method in this paper is still efficient for the multiperiodic orbits with single impact.

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