The G-invariant implicit function theorem in infinite dimensions

1982 ◽  
Vol 92 (1-2) ◽  
pp. 13-30 ◽  
Author(s):  
E. N. Dancer
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Implicit Function Theorem ◽  
Implicit Function ◽  
Infinite Dimensions ◽  
Basic Theorem

SynopsisOur basic theorem is a version of the implicit function theorem in the case of continuous groups of symmetries. The result is sufficiently general to cover a great many applications. It generalizes some earlier work of the author and corrects and improves some work of Vanderbauwhede. We also consider the breaking of symmetries problem and the variational case. Finally, we apply our results to study the periodic solutions of an ordinary differential equation.

The Implicit Function Theorem in the Scalar Case*

1969 ◽  
Vol 12 (6) ◽  
pp. 721-732 ◽  
Author(s):  
H. I. Freedman
Keyword(s):  
Critical Point ◽  
Periodic Solutions ◽  
Implicit Function Theorem ◽  
General System ◽  
Scalar Case ◽  
Implicit Function ◽  
Singular Case ◽  
Third Order ◽  
Systems Of Differential Equations ◽  
Elementary Calculus

The implicit function theorem has applications at all levels of mathematics from elementary calculus (implicit differentiation) to finding periodic solutions of systems of differential equations ([1, Chapter 14] and [4], for example).In 1961 W. S. Loud [3] studied the case of two equations in three unknowns. He considered only cases where up to third order derivatives were involved and only those cases where the derivative of the solutions at the critical point existed. Coddington and Levinson [1] consider a specific singular case involving n equations in n + m unknowns. In general the number of distinct critical cases involving up to third derivatives for such a general system is not known.

Existence of Self-similar Solutions to the Anisotropic Affine Curve-shortening Flow

2018 ◽  
Vol 2020 (23) ◽  
pp. 9440-9470
Author(s):  
Jian Lu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Smooth Function ◽  
Existence Result ◽  
Curve Shortening Flow ◽  
Self Similar ◽  
Curve Shortening ◽  
Similar Solutions

Abstract In this paper the existence of positive $2\pi $-periodic solutions to the ordinary differential equation $$\begin{equation*} u^{\prime\prime}+u=\frac{f}{u^3} \ \textrm{ in } \mathbb{R} \end{equation*}$$is studied, where $f$ is a positive $2\pi $-periodic smooth function. By virtue of a new generalized Blaschke–Santaló inequality, we obtain a new existence result of solutions.

scholarly journals Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Yongxiang Li ◽  
Lanjun Guo
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Existence Result ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Superlinear Growth ◽  
Existence Of Periodic Solutions

This paper is concerned with the existence of periodic solutions for the fully second-order ordinary differential equation u′′(t)=ft,ut,u′t, t∈R, where the nonlinearity f:R3→R is continuous and f(t,x,y) is 2π-periodic in t. Under certain inequality conditions that f(t,x,y) may be superlinear growth on (x,y), an existence result of odd 2π-periodic solutions is obtained via Leray-Schauder fixed point theorem.

The G-invariant implicit function theorem in infinite dimensions II

1986 ◽  
Vol 102 (3-4) ◽  
pp. 211-220 ◽  
Author(s):  
E. N. Dancer
Keyword(s):  
Banach Spaces ◽  
Group Action ◽  
Implicit Function Theorem ◽  
Implicit Function ◽  
Infinite Dimensions ◽  
Regularity Theorem ◽  
Fredholm Maps

SynopsisIn this paper, we study the perturbation of zeros of maps of Banach spaces where the maps are invariant under continuous groups of symmetries. In some cases, we allow the perturbed maps partially to break the symmetries. Our results improve earlier results of the author by removing smoothness conditions on the group action. The key new idea is a regularity theorem for the zeros of invariant Fredholm maps.

scholarly journals Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Saima Akram ◽  
Allah Nawaz ◽  
Humaira Kalsoom ◽  
Muhammad Idrees ◽  
Yu-Ming Chu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
First Order ◽  
Multiple Periodic Solutions ◽  
Fine Focus ◽  
Nonautonomous Equation ◽  
Maximum Multiplicity

In this article, approaches to estimate the number of periodic solutions of ordinary differential equation are considered. Conditions that allow determination of periodic solutions are discussed. We investigated focal values for first-order differential nonautonomous equation by using the method of bifurcation analysis of periodic solutions from a fine focus Z=0. Keeping in focus the second part of Hilbert’s sixteenth problem particularly, we are interested in detecting the maximum number of periodic solution into which a given solution can bifurcate under perturbation of the coefficients. For some classes like C7,7,C8,5,C8,6,C8,7, eight periodic multiplicities have been observed. The new formulas ξ10 and ϰ10 are constructed. We used our new formulas to find the maximum multiplicity for class C9,2. We have succeeded to determine the maximum multiplicity ten for class C9,2 which is the highest known multiplicity among the available literature to date. Another challenge is to check the applicability of the methods discussed which is achieved by presenting some examples. Overall, the results discussed are new, authentic, and novel in its domain of research.

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