scholarly journals On a Third-Order Three-Point Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
A. Guezane-Lakoud ◽  
N. Hamidane ◽  
R. Khaldi
Keyword(s):  
Boundary Value Problem ◽  
Growth Condition ◽  
Nontrivial Solution ◽  
Polynomial Growth ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Nonlinear Alternative ◽  
Generalized Polynomial ◽  
Polynomial Growth Condition

We consider a third-order three-point boundary value problem. We introduce a generalized polynomial growth condition to obtain the existence of a nontrivial solution by using Leray-Schauder nonlinear alternative, then we give an example to illustrate our results.

scholarly journals Existence of solution for nonlinear fourth-order three-point boundary value problem

2018 ◽  
Vol 38 (1) ◽  
pp. 67-82
Author(s):  
Zouaoui Bekri ◽  
Slimane Benaicha
Keyword(s):  
Boundary Value Problem ◽  
Nontrivial Solution ◽  
Fourth Order ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Existence Of Solution ◽  
Point Boundary ◽  
Nonlinear Alternative

In this paper, we study the existence of nontrivial solution for the fourth-order three- point boundary value problem having the following formu(4) (t) + f (t, u(t)) = 0, 0 < t < 1,u(0) = α(η), u'(0) = u''(0) = 0, u(1) = βu(η),where η ∈ (0, 1), α, β ∈ R, f ∈ C ([0, 1] × R, R). We give sufficient conditions that allow us to obtain the existence of a nontrivial solution. And by using the Leray-Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate 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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