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.

scholarly journals Existence of Positive Solutions of Lotka-Volterra Competition Model with Nonlinear Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yang Zhang ◽  
Mingxin Wang ◽  
Yuwen Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Diffusion Coefficients ◽  
Nonlinear Boundary ◽  
Upper And Lower Solutions ◽  
Nonlinear Boundary Conditions ◽  
Competition Model ◽  
Boundary Problems ◽  
Existence Of Positive Solutions

A Lotka-Volterra competition model with nonlinear boundary conditions is considered. First, by using upper and lower solutions method for nonlinear boundary problems, we investigate the existence of positive solutions in weak competition case. Next, we prove that-d1Δu=u(a1-b1u-c1v),x∈Ω;-d2Δv=v(a2-b2u-c2v),x∈Ω;∂u/∂ν+f(u)=0,x∈∂Ω;∂v/∂ν+g(v)=0,x∈∂Ω, has no positive solution when one of the diffusion coefficients is sufficiently large.

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

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Jian-Ping Sun ◽  
Xian-Qiang Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Nonlinear Boundary ◽  
Quadratic Function ◽  
Elastic Beam ◽  
Nonlinear Boundary Conditions ◽  
Beam Equation ◽  
Iterative Schemes ◽  
Elastic Beam Equation ◽  
Monotone Positive Solutions

This paper is concerned with the existence of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions. By applying monotone iteration method, we not only obtain the existence of monotone positive solutions but also establish iterative schemes for approximating the solutions. It is worth mentioning that these iterative schemes start off with zero function or quadratic function, which is very useful and feasible for computational purpose. An example is also included to illustrate the main results obtained.

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 HIGHER-ORDER BOUNDARY-VALUE PROBLEMS

2005 ◽  
Vol 48 (2) ◽  
pp. 445-464 ◽  
Author(s):  
Lingju Kong ◽  
Qingkai Kong
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Boundary Conditions ◽  
Existence Of Positive Solutions ◽  
Eigenvalue Parameter ◽  
Even Order

AbstractWe consider a class of even-order boundary-value problems with nonlinear boundary conditions and an eigenvalue parameter $\lambda$ in the equations. Sufficient conditions are obtained for the existence and non-existence of positive solutions of the problems for different values of $\lambda$.

scholarly journals Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions

2019 ◽  
Vol 39 (5) ◽  
pp. 675-689
Author(s):  
D. D. Hai ◽  
X. Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Positive Parameter ◽  
One Dimensional ◽  
Existence Of Positive Solutions ◽  
The One

We prove the existence of positive solutions for the \(p\)-Laplacian problem \[\begin{cases}-(r(t)\phi (u^{\prime }))^{\prime }=\lambda g(t)f(u),& t\in (0,1),\\au(0)-H_{1}(u^{\prime }(0))=0,\\cu(1)+H_{2}(u^{\prime}(1))=0,\end{cases}\] where \(\phi (s)=|s|^{p-2}s\), \(p\gt 1\), \(H_{i}:\mathbb{R}\rightarrow\mathbb{R}\) can be nonlinear, \(i=1,2\), \(f:(0,\infty )\rightarrow \mathbb{R}\) is \(p\)-superlinear or \(p\)-sublinear at \(\infty\) and is allowed be singular \((\pm\infty)\) at \(0\), and \(\lambda\) is a positive parameter.

scholarly journals Positive Solutions for a Singular Superlinear Fourth-Order Equation with Nonlinear Boundary Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Dongliang Yan
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Order Equation ◽  
Positive Parameter ◽  
Fourth Order Equation ◽  
Existence Of Positive Solutions ◽  
Small Positive Parameter

We show the existence of positive solutions for a singular superlinear fourth-order equation with nonlinear boundary conditions. u⁗x=λhxfux, x∈0,1,u0=u′0=0,u″1=0, u⁗1+cu1u1=0, where λ > 0 is a small positive parameter, f:0,∞⟶ℝ is continuous, superlinear at ∞, and is allowed to be singular at 0, and h: [0, 1] ⟶ [0, ∞) is continuous. Our approach is based on the fixed-point result of Krasnoselskii type in a Banach space.

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