A local implicit function theorem and application to systems of differential equations

1985 ◽  
Vol 18 (1) ◽  
pp. 251-256 ◽  
Author(s):  
Vaclav Dolezal ◽  
Shiva Shankar
Keyword(s):  
Differential Equations ◽  
Implicit Function Theorem ◽  
Implicit Function ◽  
Systems Of Differential Equations

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.

The implicit function theorem and multi-bump solutions of periodic partial differential equations

2006 ◽  
Vol 136 (3) ◽  
pp. 559-583 ◽  
Author(s):  
Robert Magnus
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Implicit Function Theorem ◽  
Superposition Principle ◽  
Implicit Function ◽  
Partial Differential ◽  
Degenerate Solution ◽  
Positivity Of Solutions ◽  
Perturbation Arguments

A modification of the implicit function theorem is advanced for cases where the continuity of the derivative fails. It is applied to a superposition principle for periodicpartial differential equations. The assumption of the principle, that there should exist a non-degenerate solution, is studied and instances of it realized using perturbation arguments and scaling. The positivity of solutions is considered.

Impulsive boundary value problem with parameter

2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Zokha Belattar ◽  
Abdelkader Lakmeche
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Implicit Function Theorem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Second Order ◽  
Implicit Function ◽  
Impulsive Boundary Value Problem ◽  
The Mean

AbstractIn this work, we investigate the existence of solutions for a class of second order impulsive differential equations using either the implicit function theorem or bifurcation techniques by the mean of Krasnosel'ski theorem.

A scaling approach to bumps and multi-bumps for nonlinear partial differential equations

2006 ◽  
Vol 136 (3) ◽  
pp. 585-614 ◽  
Author(s):  
Robert Magnus
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Asymptotic Formula ◽  
Linear Combination ◽  
Implicit Function Theorem ◽  
Nonlinear Partial Differential Equations ◽  
Implicit Function ◽  
Space Operator ◽  
Norm Topology

The problem −Δu + F(V (εx), u) = 0 is considered in Rn. For small ε > 0, solutions are obtained that approach, as ε → 0, a linear combination of specified functions, mutually translated by O(1/ε). These are the so-called multi-bump solutions. The method involves a rescaling of the variables and the use of a modified implicit function theorem. The usual implicit function theorem is inapplicable, owing to lack of convergence of the derivative of the nonlinear Hilbert space operator, obtained after an appropriate rescaling, in the operator-norm topology. An asymptotic formula for the solution for small ε is obtained.

SOLUTION OF SOME SYSTEMS OF DIFFERENTIAL EQUATIONS IN MATHEMATICAL SYSTEM MAPLE

2013 ◽  
Vol 1 (05) ◽  
pp. 58-65
Author(s):  
Yunona Rinatovna Krakhmaleva ◽  
◽  
Gulzhan Kadyrkhanovna Dzhanabayeva ◽  
Keyword(s):  
Differential Equations ◽  
Systems Of Differential Equations ◽  
Mathematical System

On decomposability of countable systems of differential equations

1993 ◽  
Vol 45 (10) ◽  
pp. 1598-1608
Author(s):  
A. M. Samoilenko ◽  
Yu. V. Teplinskii
Keyword(s):  
Differential Equations ◽  
Systems Of Differential Equations
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