scholarly journals Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Fatma Aydin Akgun
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Nonlinear Eigenvalue Problems ◽  
Nodal Properties ◽  
Nonlinear Eigenvalue

In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

Nonlinear eigenvalue problems for a class of ordinary differential equations

2002 ◽  
Vol 132 (6) ◽  
pp. 1333-1359 ◽  
Author(s):  
Uri Elias ◽  
Allan Pinkus
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problems ◽  
Various Boundary Conditions ◽  
Image Position ◽  
Nonlinear Eigenvalue

We consider the class of nonlinear eigenvalue problems where yp* = |y|p sgn y, pi > 0 and p0p1 … pn−1 = r, with various boundary conditions. We prove the existence of eigenvalues and study the zero properties and structure of the corresponding eigenfunctions.

Nonlinear eigenvalue problems for the whirling of heavy elastic strings

1980 ◽  
Vol 85 (1-2) ◽  
pp. 59-85 ◽  
Author(s):  
Stuart S. Antman
Keyword(s):  
Material Properties ◽  
Bifurcation Theory ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Qualitative Description ◽  
Nonlinear Eigenvalue Problems ◽  
Elastic Strings ◽  
Global Bifurcation Theory ◽  
Nonlinearly Elastic ◽  
Nonlinear Eigenvalue

SynopsisThis paper combines the global bifurcation theory of Rabinowitz with Sturmian theory and careful estimates to obtain a detailed qualitative description of bifurcating branches of solutions to the equations for whirling nonuniform, nonlinearly elastic strings. These results generalize earlier work of Kolodner and Stuart on inextensible strings. It is shown that the location of solution branches for the generalization of Kolodner's problem is especially sensitive to the material properties of the string, whereas that for Stuart's problem is not. The analysis of a third problem illuminates the source of this dichotomy.

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