scholarly journals Iteration for a Third-Order Three-Point BVP with Sign-Changing Green's Function

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Ya-Hong Zhao ◽  
Xing-Long Li
Keyword(s):  
Iterative Method ◽  
Boundary Value Problem ◽  
Green’S Function ◽  
Positive Solution ◽  
Green's Function ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Monotone Positive Solution

We are concerned with the following third-order three-point boundary value problem:u‴(t)=f(t,u(t)),t∈[0,1],u′(0)=u(1)=0,u″(η)+αu(0)=0, whereα∈[0,2)andη∈[2/3,1). Although corresponding Green's function is sign-changing, we still obtain the existence of monotone positive solution under some suitable conditions onfby applying iterative method. An example is also included to illustrate the main results obtained.

scholarly journals Positive Solutions of a Third-Order Three-Point BVP with Sign-Changing Green’s Function

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Li-Juan Gao ◽  
Jian-Ping Sun
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Green’S Function ◽  
Positive Solutions ◽  
Green's Function ◽  
Point Theorem ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order

We are concerned with the following third-order three-point boundary value problem:u′′′t=ft, ut,   t∈0, 1,   u′0=u1=0and u′′η-αu′1=0,whereα∈0, 1andη∈(14+α)/(24-3α),1. Although the corresponding Green’s function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions onfby using the two-fixed-point theorem due to Avery and Henderson.

scholarly journals GREEN’S FUNCTION AND EXISTENCE OF SOLUTIONS FOR A THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEM

2019 ◽  
Vol 24 (2) ◽  
pp. 171-178 ◽  
Author(s):  
Sergey Smirnov
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Point Boundary ◽  
Nonlinear Boundary Value Problems ◽  
Uniqueness Of Solutions ◽  
Third Order ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results

The solutions of third-order three-point boundary value problemx′′′+f(t,x) = 0, t∈[a,b],x(a) =x′(a) = 0, x(b) =kx(η),whereη∈(a,b),k∈R,f∈C([a,b]×R,R) andf(t,0)6= 0, are the subject of thisinvestigation. In order to establish existence and uniqueness results for the solutions,attention is focused on applications of the corresponding Green’s function. As anapplication, also one example is given to illustrate the result.Keywords:Green’s function, nonlinear boundary value problems, three-point boundaryconditions, existence and uniqueness of solutions.

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