Global bifurcation and nodal solutions for fourth-order problems with sign-changing weight

2013 ◽  
Vol 219 (17) ◽  
pp. 9399-9407 ◽  
Author(s):  
Guowei Dai ◽  
Xiaoling Han
Keyword(s):  
Fourth Order ◽  
Global Bifurcation ◽  
Nodal Solutions ◽  
Fourth Order Problems

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.

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.

scholarly journals Existence of Positive Solutions of a Discrete Elastic Beam Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Ruyun Ma ◽  
Jiemei Li ◽  
Chenghua Gao
Keyword(s):  
Positive Solutions ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Global Bifurcation ◽  
Elastic Beam ◽  
Beam Equation ◽  
Nonlinear Boundary Value Problems ◽  
Order Difference ◽  
Bifurcation Theorem ◽  
Existence Of Positive Solutions

LetTbe an integer withT≥5and letT2={2,3,…,T}. We consider the existence of positive solutions of the nonlinear boundary value problems of fourth-order difference equationsΔ4u(t−2)−ra(t)f(u(t))=0,t∈T2,u(1)=u(T+1)=Δ2u(0)=Δ2u(T)=0, whereris a constant,a:T2→(0,∞),  and  f:[0,∞)→[0,∞)is continuous. Our approaches are based on the Krein-Rutman theorem and the global bifurcation theorem.

Sign in / Sign up

Export Citation Format

Share Document