scholarly journals Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations

2007 ◽  
Vol 12 (2) ◽  
pp. 253-267 ◽  
Author(s):  
F. Sadyrbaev ◽  
A. Gritsans
Keyword(s):  
Dirichlet Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nontrivial Solution ◽  
Eigenvalue Problems ◽  
Linear Equations ◽  
Normalization Condition ◽  
Explicit Formulas ◽  
Parameter Dependent

Eigenvalue problems of the form x'' = −λf(x) + µg(x)  (i),  x(0) = 0, x(1) = 0  (ii) are considered. We are looking for (λ, µ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fuchik problem for piece-wise linear equations. In our considerations functions f and g may be super-, sub- and quasi-linear in various combinations. The spectra obtained under the normalization condition (otherwise problems may have continuous spectra) structurally are similar to usual Fuchik spectrum for the Dirichlet problem. We provide explicit formulas for Fuchik spectra for super and super, super and sub, sub and super, sub and sub cases, where superlinear and sublinear parts of equations are of the form |x|2αx and |x|1/(2β+1) respectively (α > 0, β > 0.)

Remark on zeros of solutions of second-order linear ordinary differential equations

2016 ◽  
Vol 23 (4) ◽  
pp. 571-577
Author(s):  
Monika Dosoudilová ◽  
Alexander Lomtatidze
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Nontrivial Solution ◽  
Unique Solvability ◽  
Periodic Boundary ◽  
Periodic Boundary Value Problem ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Zeros Of Solutions

AbstractAn efficient condition is established ensuring that on any interval of length ω, any nontrivial solution of the equation ${u^{\prime\prime}=p(t)u}$ has at most one zero. Based on this result, the unique solvability of a periodic boundary value problem is studied.

Oscillation Criteria for Semilinear Equations in General Domains

1976 ◽  
Vol 19 (2) ◽  
pp. 137-144 ◽  
Author(s):  
W. Allegretto
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Linear Equations ◽  
Oscillation Theory ◽  
General Nature ◽  
Nonlinear Ordinary Differential Equations ◽  
Exterior Domains ◽  
Semilinear Equations

Oscillation theory for nonlinear ordinary differential equations has been extensively developed in recent years by several authors. We refer the reader to the recent paper by Noussair and Swanson, [2], where an extensive bibliography may be found. The situation is somewhat different for the case of second order partial differential equations, an area which has recently been virtually untouched, except for the establishment of criteria which depend on a comparison with suitable linear equations and therefore are essentially linear in nature. A bibliography of such results may also be found in [2]. Of a more general nature have been the paper by the author [1], and the more recent work of Noussair and Swanson, [2]. Although the methods employed and the equations considered in [1] and [2] are different, both papers obtained results only for a class of exterior domains of Rn, of which the typical example is the complement of a bounded sphere.

scholarly journals Distributional and entire solutions of ordinary differential and functional differential equations

1983 ◽  
Vol 6 (2) ◽  
pp. 243-269 ◽  
Author(s):  
S. M. Shah ◽  
Joseph Wiener
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Integral Equations ◽  
Linear Equations ◽  
Functional Differential Equations ◽  
Generalized Function ◽  
Entire Solutions ◽  
Polynomial Coefficients ◽  
Functional Differential

A brief survey of recent results on distributional and entire solutions of ordinary differential equations (ODE) and functional differential equations (FDE) is given. Emphasis is made on linear equations with polynomial coefficients. Some work on generalized-function solutions of integral equations is also mentioned.

scholarly journals Some Properties of the Hermite Polynomials and Their Squares and Generating Functions

2016 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Explicit Formulas ◽  
Derivatives Of

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

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.

The interlacing of eigenvalues for periodic multi-parameter problems

1978 ◽  
Vol 80 (3-4) ◽  
pp. 357-362 ◽  
Author(s):  
Patrick J. Browne
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Image Position ◽  
Periodic Equation

SynopsisThis paper studies a linked system of second order ordinary differential equationswhere xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.

NONLINEAR SPECTRA: THE NEUMANN PROBLEM

2009 ◽  
Vol 14 (1) ◽  
pp. 33-42 ◽  
Author(s):  
Armands Gritsans ◽  
Felix Sadyrbaev
Keyword(s):  
Neumann Problem ◽  
Nontrivial Solution ◽  
Eigenvalue Problems ◽  
Linear Equations ◽  
The Neumann Problem

Eigenvalue problems of the form x” = −λf(x+ ) + μg(x− ), x‘(a) = 0, x' (b) = 0 are considered. We are looking for (λ,μ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fučík problem for piece‐wise linear equations. In our considerations functions f and g may be nonlinear. Consequently spectra may differ essentially from those for the Fučík equation.

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