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 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 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.

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