An Overview of the Method of Lower and Upper Solutions for ODEs

2001 ◽  
pp. 3-22 ◽  
Author(s):  
C. De Coster ◽  
P. Habets
Keyword(s):  
Lower And Upper Solutions

Ordinary Differential Equations with Nonlinear Boundary Conditions

2002 ◽  
Vol 9 (2) ◽  
pp. 287-294
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Lower And Upper Solutions ◽  
Iterative Technique ◽  
Monotone Iterative

Abstract The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems.

Old and New Results for First Order Periodic ODEs without Uniqueness: a Comprehensive Study by Lower and Upper Solutions

2004 ◽  
Vol 4 (3) ◽  
Author(s):  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Lower And Upper Solutions ◽  
Elementary Approach ◽  
Qualitative Properties ◽  
First Order ◽  
The Past ◽  
The Cauchy Problem ◽  
Qualitative Properties Of Solutions ◽  
Comprehensive Study

AbstractAn elementary approach, based on a systematic use of lower and upper solutions, is employed to detect the qualitative properties of solutions of first order scalar periodic ordinary differential equations. This study is carried out in the Carathéodory setting, avoiding any uniqueness assumption, in the future or in the past, for the Cauchy problem. Various classical and recent results are recovered and generalized.

scholarly journals Stability of periodic solutions of first-order difference equations lying between lower and upper solutions

2005 ◽  
Vol 2005 (3) ◽  
pp. 865865
Author(s):  
Alberto Cabada ◽  
Victoria Otero-Espinar ◽  
Dolores Rodríguez-Vivero
Keyword(s):  
Periodic Solutions ◽  
Difference Equations ◽  
Lower And Upper Solutions ◽  
Order Difference ◽  
First Order

scholarly journals Lower and Upper Solutions for Even Order Boundary Value Problems

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 878
Author(s):  
Alberto Cabada ◽  
Lucía López-Somoza
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Periodic Boundary Conditions ◽  
Existence Of Solution ◽  
Lower And Upper Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Even Order

In this paper, we prove the existence of solutions of nonlinear boundary value problems of arbitrary even order using the lower and upper solutions method. In particular, we point out the fact that the existence of a pair of lower and upper solutions of a considered problem could imply the existence of solution of another one with different boundary conditions. We consider Neumann, Dirichlet, mixed and periodic boundary conditions.

scholarly journals Method of Lower and Upper Solutions for Elliptic Systems with Nonlinear Boundary Condition and Its Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Ruyun Ma ◽  
Ruipeng Chen ◽  
Yanqiong Lu
Keyword(s):  
Nonlinear Boundary ◽  
System Modeling ◽  
Principal Eigenvalue ◽  
Age Groups ◽  
Sufficient Conditions ◽  
Elliptic Systems ◽  
Nonlinear Boundary Conditions ◽  
Lower And Upper Solutions ◽  
Adult Age ◽  
Adult Model

We develop the method of lower and upper solutions for a class of elliptic systems with nonlinear boundary conditions. As its application, an elliptic system modeling a population divided into juvenile and adult age groups is studied, and we find sufficient conditions in terms of the principal eigenvalue of the corresponding linearized system, to guarantee the existence of coexistence states of the above juvenile-adult model.

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