Positive solutions to a singular third-order three-point boundary value problem with an indefinitely signed Green’s function

2008 ◽  
Vol 68 (7) ◽  
pp. 2104-2118 ◽  
Author(s):  
Alex P. Palamides ◽  
George Smyrlis
Keyword(s):  
Boundary Value Problem ◽  
Green’S Function ◽  
Positive Solutions ◽  
Green's Function ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order

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 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 Multiplicity of Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem with a Parameter

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Jian Liu ◽  
Hanying Feng ◽  
Xingfang Feng
Keyword(s):  
Boundary Value Problem ◽  
Green’S Function ◽  
Positive Solutions ◽  
Green's Function ◽  
Boundary Value ◽  
Index Theory ◽  
Second Order ◽  
Point Boundary ◽  
Fixed Point Index Theory ◽  
Multiplicity Of Positive Solutions

This paper is concerned with the following second-order three-point boundary value problemu″t+β2ut+λqtft,ut=0,t∈0 , 1,u0=0,u(1)=δu(η), whereβ∈(0,π/2),δ>0,η∈(0,1), andλis a positive parameter. First, Green’s function for the associated linear boundary value problem is constructed, and then some useful properties of Green’s function are obtained. Finally, existence, multiplicity, and nonexistence results for positive solutions are derived in terms of different values ofλby means of the fixed point index theory.

scholarly journals Existence of Positive Solutions for Nonlinear Third-Order Boundary Value Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Tiaoxia Dun ◽  
Pengyu Chen
Keyword(s):  
Boundary Value Problem ◽  
Green’S Function ◽  
Positive Solutions ◽  
Green's Function ◽  
Boundary Value ◽  
Index Theorem ◽  
Third Order ◽  
Existence Of Positive Solutions ◽  
Fixed Point Index Theorem ◽  
Linear Boundary Value Problem

We are concerned with the existence of positive solutions for the nonlinear third-order three-point boundary value problemu′′′(t)+λg(t)f(u(t))=0,0<t<1,u(0)=αu(η),u′(0)=u′′(1)=0, where0<η<1,0<α<1,λis a positive parameter,g:(0,1)→[0,∞),  and  f:[0,∞)→[0,∞)is continuous. We construct Green’s function for the associated linear boundary value problem and obtain some useful properties of Green’s function. Finally, by using fixed-point index theorem in cones, we establish the existence results of positive solutions for the boundary value problem an example illustrates the application of the results obtained.

scholarly journals Positive Solutions for a Class of Third-Order Three-Point Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Xiaojie Lin ◽  
Zhengmin Fu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Positive Parameter ◽  
Point Boundary ◽  
Point Index ◽  
Third Order ◽  
Index Theorems ◽  
Existence Of Positive Solutions

We investigate the problem of existence of positive solutions for the nonlinear third-order three-point boundary value problemu‴(t)+λa(t)f(u(t))=0,0<t<1,u(0)=u′(0)=0,u″(1)=∝u″(η), whereλis a positive parameter,∝∈(0,1),η∈(0,1),f:(0,∞)→(0,∞),a:(0,1)→(0,∞)are continuous. Using a specially constructed cone, the fixed point index theorems and Leray-Schauder degree, this work shows the existence and multiplicities of positive solutions for the nonlinear third-order boundary value problem. Some examples are given to demonstrate the main results.

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