scholarly journals Study of Differential Equations with Exponential Nonlinearities via the Lower and Upper Solutions’ Method

2020 ◽  
Vol 1 (2) ◽  
Author(s):  
Michal Feckan ◽  
Kateryna Marynets
Keyword(s):  
Differential Equations ◽  
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 Convergence of Antiperiodic Boundary Value Problems for First-Order Integro-Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Yameng Wang ◽  
Juan Zhang ◽  
Yufeng Sun
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Lower And Upper Solutions ◽  
Quasilinearization Method ◽  
Rapid Convergence ◽  
Uniqueness Of Solutions ◽  
First Order ◽  
Convergence Of Approximate Solutions

In this paper, we investigate the convergence of approximate solutions for a class of first-order integro-differential equations with antiperiodic boundary value conditions. By introducing the definitions of the coupled lower and upper solutions which are different from the former ones and establishing some new comparison principles, the results of the existence and uniqueness of solutions of the problem are given. Finally, we obtain the uniform and rapid convergence of the iterative sequences of approximate solutions via the coupled lower and upper solutions and quasilinearization method. In addition, an example is given to illustrate the feasibility of the method.

Well-Ordered and Non-Well-Ordered Lower and Upper Solutions for Periodic Planar Systems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alessandro Fonda ◽  
Giuliano Klun ◽  
Andrea Sfecci
Keyword(s):  
Differential Equations ◽  
Detailed Analysis ◽  
Topological Degree ◽  
Phase Plane ◽  
Periodic Problem ◽  
Lower And Upper Solutions ◽  
Systems Of Differential Equations ◽  
Planar Systems

Abstract The aim of this paper is to extend the theory of lower and upper solutions to the periodic problem associated with planar systems of differential equations. We generalize previously given definitions and we are able to treat both the well-ordered case and the non-well-ordered case. The proofs involve topological degree arguments, together with a detailed analysis of the solutions in the phase plane.

Examples of the nonexistence of a solution in the presence of upper and lower solutions

2003 ◽  
Vol 44 (4) ◽  
pp. 591-594 ◽  
Author(s):  
Patrick Habets ◽  
Rodrigo L. Pouso
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Second Order ◽  
Lower And Upper Solutions ◽  
Existence Of A Solution ◽  
Nagumo Condition

AbstractStandard results for boundary value problems involving second-order ordinary differential equations ensure that the existence of a well-ordered pair of lower and upper solutions together with a Nagumo condition imply existence of a solution. In this note we introduce some examples which show that existence is not guaranteed if no Nagumo condition is satisfied.

Iterative techniques with computer realization for initial value problems for Riemann–Liouville fractional differential equations

2020 ◽  
Vol 26 (1) ◽  
pp. 21-47 ◽  
Author(s):  
Ravi Agarwal ◽  
A. Golev ◽  
S. Hristova ◽  
D. O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Finite Interval ◽  
Lower And Upper Solutions ◽  
Successive Approximations ◽  
Initial Value ◽  
Practical Usefulness ◽  
Computer Environment ◽  
Fractional Differential ◽  
The Given

AbstractThe main aim of this paper is to suggest some algorithms and to use them in an appropriate computer environment to solve approximately the initial value problem for scalar nonlinear Riemann–Liouville fractional differential equations on a finite interval. The iterative schemes are based on appropriately defined lower and upper solutions to the given problem. A number of different cases depending on the type of lower and upper solutions are studied and various schemes for constructing successive approximations are provided. The suggested schemes are applied to some problems and their practical usefulness is illustrated.

scholarly journals Unique solution to periodic boundary value problems

1991 ◽  
Vol 4 (2) ◽  
pp. 129-136
Author(s):  
Yong Sun
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Unique Solution ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Lower And Upper Solutions ◽  
Periodic Boundary Value Problems ◽  
Right Hand

Existence of unique solution to periodic boundary value problems of differential equations with continuous or discontinuous right-hand side is considered by utilizing the method of lower and upper solutions and the monotone properties of the operator. This is subject to discussion in the present paper.

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