The implicit function theorem and analytic differential equations

1975 ◽  
pp. 191-208 ◽  
Author(s):  
K. R. Meyer
Keyword(s):  
Differential Equations ◽  
Implicit Function Theorem ◽  
Implicit Function

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.

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.

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