scholarly journals Existence of Solutions of Nonlinear Mixed Two-Point Boundary Value Problems for Third-Order Nonlinear Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Yongxin Gao ◽  
Fengqin Wang
Keyword(s):  
Differential Equation ◽  
Boundary Value Problems ◽  
Nonlinear Differential Equation ◽  
Solution Method ◽  
Existence Of Solutions ◽  
Lower Solution ◽  
Point Boundary ◽  
Third Order ◽  
Upper And Lower Solution ◽  
Lower Solution Method

The authors use the upper and lower solution method to study the existence of solutions of nonlinear mixed two-point boundary value problems for third-order nonlinear differential equation  y′′′=f(x,y,y′,y′′),  y′(b)=h(y′(a)),  p(y(a),y(b),y′(a),y′(b))=0,  g(y(a),y(b),y′(a),y′(b),y′′(a),y′′(b))=0. Some new existence results are obtained by developing the upper and lower solution method. Some applications are also presented.

scholarly journals Existence of Unbounded Solutions for a Third-Order Boundary Value Problem on Infinite Intervals

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hairong Lian ◽  
Junfang Zhao
Keyword(s):  
Differential Equation ◽  
Solution Method ◽  
Existence Of Solutions ◽  
Lower Solution ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
Unbounded Solutions ◽  
Sturm Liouville ◽  
Infinite Intervals ◽  
Lower Solution Method

We generalize the unbounded upper and lower solution method to a third-order ordinary differential equation on the half line subject to the Sturm-Liouville boundary conditions. By using such techniques and the Schäuder fixed point theorem, some criteria are presented for the existence of solutions and positive ones to the problem discussed.

scholarly journals Upper and Lower Solution Method for Fourth-Order Four-Point Boundary Value Problem on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Ilkay Yaslan Karaca
Keyword(s):  
Boundary Value Problem ◽  
Time Scales ◽  
Fourth Order ◽  
Solution Method ◽  
Boundary Value ◽  
Lower Solution ◽  
Point Boundary ◽  
Upper And Lower Solution ◽  
Lower Solution Method ◽  
Order Four

We consider a fourth-order four-point boundary value problem for dynamic equations on time scales. By the upper and lower solution method, some results on the existence of solutions of the fourth-order four-point boundary value problem on time scales are obtained. An example is also included to illustrate our results.

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