scholarly journals Positive solutions for nonlinear elastic beam models

2001 ◽  
Vol 27 (6) ◽  
pp. 365-375 ◽  
Author(s):  
Bendong Lou
Keyword(s):  
Positive Solutions ◽  
Elastic Beam ◽  
Negative Answer ◽  
Existence Of Positive Solutions ◽  
Nonlinear Elastic ◽  
Positive Results

We give a negative answer to a conjecture of Korman on nonlinear elastic beam models. Moreover, by modifying the main conditions in the conjecture (generalizing the original ones at some points), we get positive results, that is, we obtain the existence of positive solutions for the models.

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.

scholarly journals Positive solutions of beam equations under nonlocal boundary value conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shenglin Wang ◽  
Jialong Chai ◽  
Guowei Zhang
Keyword(s):  
Positive Solutions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Elastic Beam ◽  
Mixed Boundary Conditions ◽  
Order Problem ◽  
Beam Equations ◽  
Existence Of Positive Solutions ◽  
Fourth Order Problem

AbstractIn this article, we study the fourth-order problem with the first and second derivatives in nonlinearity under nonlocal boundary value conditions $$\begin{aligned}& \left \{ \textstyle\begin{array}{l}u^{(4)}(t)=h(t)f(t,u(t),u'(t),u''(t)),\quad t\in(0,1),\\ u(0)=u(1)=\beta_{1}[u],\qquad u''(0)+\beta_{2}[u]=0,\qquad u''(1)+\beta_{3}[u]=0, \end{array}\displaystyle \right . \end{aligned}$$ {u(4)(t)=h(t)f(t,u(t),u′(t),u″(t)),t∈(0,1),u(0)=u(1)=β1[u],u″(0)+β2[u]=0,u″(1)+β3[u]=0, where $f: [0,1]\times\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}_{-}\to \mathbb{R}_{+}$f:[0,1]×R+×R×R−→R+ is continuous, $h\in L^{1}(0,1)$h∈L1(0,1) and $\beta_{i}[u]$βi[u] is Stieltjes integral ($i=1,2,3$i=1,2,3). This equation describes the deflection of an elastic beam. Some inequality conditions on nonlinearity f are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on a special cone in $C^{2}[0,1]$C2[0,1]. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.

scholarly journals Existence of Positive Solutions for an Elastic Beam Equation with Nonlinear Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Ruikuan Liu ◽  
Ruyun Ma
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Nonlinear Boundary ◽  
Elastic Beam ◽  
Index Theory ◽  
Nonlinear Boundary Conditions ◽  
Beam Equation ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

We study the existence and nonexistence of positive solutions for the following fourth-order two-point boundary value problem subject to nonlinear boundary conditionsu′′′′(t)=λf(t,u(t)),  t∈(0,1),u(0)=0,  u′(0)=μh(u(0)),  u′′(1)=0,  u′′′(1)=μg(u(1)), whereλ>0, μ≥0are parameters, andf:0, 1×0,+∞→0, +∞, h:0, +∞→0, +∞, andg:0, +∞→-∞,0are continuous. By using the fixed-point index theory, we prove that the problem has at least one positive solution forλ,  μsufficiently small and has no positive solution forλlarge enough.

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