scholarly journals Existence of Solutions of a Discrete Fourth-Order Boundary Value Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Ruyun Ma ◽  
Chenghua Gao ◽  
Yongkui Chang
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Linear Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Linear Eigenvalue Problem ◽  
Existence And Multiplicity ◽  
Nonresonance Condition ◽  
Two Parameter

Leta,bbe two integers withb-a≥5and let𝕋2={a+2,a+3,…,b-2}. We show the existence of solutions for nonlinear fourth-order discrete boundary value problemΔ4u(t-2)=f(t,u(t),Δ2u(t-1)),t∈𝕋2,u(a+1)=u(b-1)=Δ2u(a)=Δ2u(b-2)=0under a nonresonance condition involving two-parameter linear eigenvalue problem. We also study the existence and multiplicity of solutions of nonlinear perturbation of a resonant linear problem.

scholarly journals On a Fourth-Order Boundary Value Problem at Resonance

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Man Xu ◽  
Ruyun Ma
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Elastic Beam ◽  
First Eigenvalue ◽  
At Resonance ◽  
Problem At Resonance ◽  
Spectrum Structure

We investigate the spectrum structure of the eigenvalue problem u4x=λux,  x∈0,1;  u0=u1=u′0=u′1=0. As for the application of the spectrum structure, we show the existence of solutions of the fourth-order boundary value problem at resonance -u4x+λ1ux+gx,ux=hx,  x∈0,1;  u0=u1=u′0=u′1=0, which models a statically elastic beam with both end-points being cantilevered or fixed, where λ1 is the first eigenvalue of the corresponding eigenvalue problem and nonlinearity g may be unbounded.

scholarly journals Existence and Multiplicity of Positive Solutions of a Nonlinear Discrete Fourth-Order Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Ruyun Ma ◽  
Yanqiong Lu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Boundary Value ◽  
Global Bifurcation ◽  
Main Tool ◽  
Bifurcation Theorem ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

we show the existence and multiplicity of positive solutions of the nonlinear discrete fourth-order boundary value problemΔ4ut-2=λhtfut,t∈T2,u1=uT+1=Δ2u0=Δ2uT=0, whereλ>0,h:T2→(0,∞)is continuous, andf:R→[0,∞)is continuous,T>4,T2=2,3,…,T. The main tool is the Dancer's global bifurcation theorem.

scholarly journals Boundary Value Problems for Fourth Order Nonlinearp-Laplacian Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Qinqin Zhang
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Mountain Pass ◽  
Mountain Pass Lemma

We consider the boundary value problem for a fourth order nonlinearp-Laplacian difference equation containing both advance and retardation. By using Mountain pass lemma and some established inequalities, sufficient conditions of the existence of solutions of the boundary value problem are obtained. And an illustrative example is given in the last part of the paper.

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