scholarly journals Methods of Bilateral Approximations for Nonlinear Eigenvalue Problems

2019 ◽  
pp. 1-20
Author(s):  
Bohdan Podlevskyi
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problems ◽  
Spectral Problems ◽  
New Algorithms ◽  
Nonlinear Eigenvalue

In this article, the research proposed by the author, the approach to the construction of methods and algorithms of bilateral approximations to the eigenvalues of nonlinear spectral problems, is continued. On the basis of Newton's method, some new algorithms of the bilateral approximations to their eigenvalues are constructed and substantiated.

MULTIPLE POSITIVE SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS ASSOCIATED TO A CLASS OF p-LAPLACIAN LIKE OPERATORS

2003 ◽  
Vol 05 (05) ◽  
pp. 737-759 ◽  
Author(s):  
NOBUYOSHI FUKAGAI ◽  
KIMIAKI NARUKAWA
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Eigenvalue Problems ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Nonlinear Eigenvalue Problems ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Nonlinear Eigenvalue

This paper deals with positive solutions of a class of nonlinear eigenvalue problems. For a quasilinear elliptic problem (#) - div (ϕ(|∇u|)∇u) = λf(x,u) in Ω, u = 0 on ∂Ω, we assume asymptotic conditions on ϕ and f such as ϕ(t) ~ tp0-2, f(x,t) ~ tq0-1as t → +0 and ϕ(t) ~ tp1-2, f(x,t) ~ tq1-1as t → ∞. The combined effects of sub-nonlinearity (p0> q0) and super-nonlinearity (p1< q1) with the subcritical term f(x,u) imply the existence of at least two positive solutions of (#) for 0 < λ < Λ.

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