scholarly journals Comment on “Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications”

2017 ◽  
Vol 2017 ◽  
pp. 1-3
Author(s):  
Ziyatkhan Aliyev
Keyword(s):  
Recent Work ◽  
Comparison Theorem ◽  
Fourth Order ◽  
Linear Problem ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Nontrivial Solutions ◽  
Nodal Solutions ◽  
Negative Eigenvalues ◽  
Fourth Order Problems

In the recent paper W. Shen and T. He and G. Dai and X. Han established unilateral global bifurcation result for a class of nonlinear fourth-order eigenvalue problems. They show the existence of two families of unbounded continua of nontrivial solutions of these problems bifurcating from the points and intervals of the line trivial solutions, corresponding to the positive or negative eigenvalues of the linear problem. As applications of this result, these authors study the existence of nodal solutions for a class of nonlinear fourth-order eigenvalue problems with sign-changing weight. Moreover, they also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight. In the present comment, we show that these papers of above authors contain serious errors and, therefore, unfortunately, the results of these works are not true. Note also that the authors used the results of the recent work by G. Dai which also contain gaps.

scholarly journals Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Wenguo Shen ◽  
Tao He
Keyword(s):  
Fourth Order ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Connected Components ◽  
Nodal Solutions ◽  
Linear Problems ◽  
Fourth Order Problems

We establish a unilateral global bifurcation result from interval for a class of fourth-order problems with nondifferentiable nonlinearity. By applying the above result, we firstly establish the spectrum for a class of half-linear fourth-order eigenvalue problems. Moreover, we also investigate the existence of nodal solutions for the following half-linear fourth-order problems:x″″=αx++βx-+ratfx,0<t<1,x(0)=x(1)=x″(0)=x″(1)=0, wherer≠0is a parameter,a∈C([0,1],(0,∞)),x+=max⁡{x,0},x-=-min⁡{x,0},α,β∈C[0,1], andf∈C(R,R),sf(s)>0,fors≠0. We give the intervals for the parameterrwhich ensure the existence of nodal solutions for the above fourth-order half-linear problems iff0∈[0,∞)orf∞∈[0,∞],wheref0=lims→0f(s)/sandf∞=lims→+∞f(s)/s. We use the unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

Some global results for nonlinear fourth order eigenvalue problems

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Ziyatkhan Aliyev
Keyword(s):  
Eigenvalue Problem ◽  
Fourth Order ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Nontrivial Solutions ◽  
Fourth Order Eigenvalue Problem

AbstractIn this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems.

Unilateral Global Bifurcation from Infinity in Nonlinearizable One-Dimensional Dirac Problems

2021 ◽  
Vol 31 (01) ◽  
pp. 2150005
Author(s):  
Ziyatkhan S. Aliyev ◽  
Nazim A. Neymatov ◽  
Humay Sh. Rzayeva
Keyword(s):  
Dirac Equation ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Nontrivial Solutions ◽  
One Dimensional ◽  
Bifurcation From Infinity ◽  
The One ◽  
Nodal Properties

In this paper, we study the unilateral global bifurcation from infinity in nonlinearizable eigenvalue problems for the one-dimensional Dirac equation. We show the existence of two families of unbounded continua of the set of nontrivial solutions emanating from asymptotically bifurcation intervals and having the usual nodal properties near these intervals.

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.

scholarly journals Existence of solutions for a fourth order problem at resonance

2014 ◽  
Vol 32 (2) ◽  
pp. 133
Author(s):  
EL Miloud Hssini ◽  
Mohammed Massar ◽  
Mohamed Talbi ◽  
Najib Tsouli
Keyword(s):  
Fourth Order ◽  
Existence Of Solutions ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Order Problem ◽  
The First Eigenvalue ◽  
At Resonance ◽  
Fourth Order Problem ◽  
Problem At Resonance ◽  
Fourth Order Problems

In this work, we are interested at the existence of nontrivial solutions of two fourth order problems governed by the weighted p-biharmonic  operator. The first is the following$$\Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda_1 m(x)|u|^{p-2}u+f(x,u)-h \mbox{ in }\Omega,\,\, u=\Delta u=0 \mbox{ on }\partial\Omega,$$where $\lambda_1$ is the first eigenvalue for the eigenvalue problem$ \Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda m(x) |u|^{p-2}u\mbox{in }\Omega, \,\, u=\Delta u=0 \mbox{ on } \partial\Omega.$ \\In the seconde problem, we replace  $\lambda_1$ by $\lambda$ suchthat $\lambda_1<\lambda<\bar{\lambda}$, where $\bar{\lambda}$ is given bellow.

scholarly journals Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients

2021 ◽  
Author(s):  
Andreas Dedner ◽  
Alice Hodson
Keyword(s):  
Fourth Order ◽  
Perturbation Parameter ◽  
Two Dimensions ◽  
Optimal Error Estimates ◽  
Projection Operators ◽  
Varying Coefficients ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Perturbation Problems ◽  
Fourth Order Problems

Abstract We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

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