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.

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 Multiple solutions for a fourth order equation with nonlinear boundary conditions: theoretical and numerical aspects

2019 ◽  
pp. 335-348
Author(s):  
Cristiane Aparecida Pendeza Martinez ◽  
André Luís Machado Martinez ◽  
Glaucia Maria Bressan ◽  
Emerson Vitor Castelani ◽  
Roberto Molina de Souza
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Order Equation ◽  
Fourth Order Equation

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 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 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 Iterative Method for Solving a Beam Equation with Nonlinear Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Quang A. Dang ◽  
Nguyen Thanh Huong
Keyword(s):  
Iterative Method ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Order Equation ◽  
Beam Equation ◽  
Linear Boundary ◽  
Order Problem ◽  
Fourth Order Problem

In this paper, we propose an iterative method for solving a beam problem which is described by a nonlinear fourth-order equation with nonlinear boundary conditions. The method reduces this nonlinear fourth-order problem to a sequence of linear second-order problems with linear boundary conditions. The convergence of the method is proved, and some numerical examples demonstrate the efficiency of the method.

Sign in / Sign up

Export Citation Format

Share Document