scholarly journals A generalized upper and lower solutions method for nonlinear second order ordinary differential equations

1992 ◽  
Vol 5 (2) ◽  
pp. 157-165 ◽  
Author(s):  
Juan J. Nieto ◽  
Alberto Cabada
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Nonlinear Boundary ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Second Order ◽  
Iterative Technique ◽  
Monotone Iterative ◽  
Carathéodory Function ◽  
Minimal And Maximal Solutions

The purpose of this paper is to study a nonlinear boundary value problem of second order when the nonlinearity is a Carathéodory function. It is shown that a generalized upper and lower solutions method is valid, and the monotone iterative technique for finding the minimal and maximal solutions is developed.

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.

scholarly journals MultiPoint BVPs for Second-Order Functional Differential Equations with Impulses

2009 ◽  
Vol 2009 ◽  
pp. 1-16
Author(s):  
Xuxin Yang ◽  
Zhimin He ◽  
Jianhua Shen
Keyword(s):  
Differential Equations ◽  
Upper And Lower Solutions ◽  
Functional Differential Equations ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Second Order ◽  
Monotone Iterative ◽  
Functional Differential ◽  
Extreme Solutions ◽  
Multipoint Boundary Value Problem

This paper is concerned about the existence of extreme solutions of multipoint boundary value problem for a class of second-order impulsive functional differential equations. We introduce a new concept of lower and upper solutions. Then, by using the method of upper and lower solutions introduced and monotone iterative technique, we obtain the existence results of extreme solutions.

Unique positive solution for a fractional boundary value problem

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Keyu Zhang ◽  
Jiafa Xu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Fractional Boundary Value Problem ◽  
Unique Positive Solution ◽  
Monotone Iterative ◽  
Liouville Fractional Derivative

AbstractIn this work we consider the unique positive solution for the following fractional boundary value problem $\left\{ \begin{gathered} D_{0 + }^\alpha u(t) = - f(t,u(t)),t \in [0,1], \hfill \\ u(0) = u'(0) = u'(1) = 0. \hfill \\ \end{gathered} \right. $ Here α ∈ (2, 3] is a real number, D 0+α is the standard Riemann-Liouville fractional derivative of order α. By using the method of upper and lower solutions and monotone iterative technique, we also obtain that there exists a sequence of iterations uniformly converges to the unique solution.

scholarly journals Monotone iterations for differential equations with a parameter

1997 ◽  
Vol 10 (3) ◽  
pp. 273-278 ◽  
Author(s):  
Tadeusz Jankowski ◽  
V. Lakshmikantham
Keyword(s):  
Differential Equations ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Extremal Solutions ◽  
Iterative Technique ◽  
Monotone Iterative ◽  
Monotone Iterations

Consider the problem {y′(t)=f(t,y(t),λ),t∈J=[0,b],y(0)=k0,G(y,λ)=0.. Employing the method of upper and lower solutions and the monotone iterative technique, existence of extremal solutions for the above equation are proved.

scholarly journals Existence for Certain Systems of Nonlinear Fractional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Zhaowen Zheng ◽  
Xiujuan Zhang ◽  
Jing Shao
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence Of Solutions ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Comparison Result ◽  
Nonlinear Fractional Differential Equations ◽  
Monotone Iterative ◽  
Fractional Differential

By establishing a comparison result and using the monotone iterative technique, combining with the method of upper and lower solutions, the existence of solutions for systems of nonlinear fractional differential equations is considered. An example is given to demonstrate the applicability of our results.

scholarly journals Minimal and maximal solutions to systems of differential equations with a singular matrix

2003 ◽  
Vol 45 (2) ◽  
pp. 223-231
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Monotone Iterative Technique ◽  
Extremal Solutions ◽  
Iterative Technique ◽  
Singular Matrix ◽  
Monotone Iterative ◽  
Maximal Solutions ◽  
Systems Of Differential Equations ◽  
Minimal And Maximal Solutions

AbstractThe monotone iterative technique is applied to a system of ordinary differential equations with a singular matrix. The existence of extremal solutions is proved.

scholarly journals Minimal and maximal solutions to first-order differential equations with piecewise constant generalized delay

2021 ◽  
Vol 40 (1) ◽  
pp. 175-186
Author(s):  
Kuo-Shou Chiu
Keyword(s):  
Differential Equations ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Constant Delay ◽  
Piecewise Constant ◽  
First Order ◽  
Monotone Iterative ◽  
Minimal And Maximal Solutions ◽  
Generalized Type ◽  
First Order Differential Equations

In this paper we employ the method of maximal and minimal solutions coupled with comparison principles and the monotone iterative technique to obtain results of existence and approximation of solutions for differential equations with piecewise constant delay of generalized type (DEPCAG).

scholarly journals Monotone iterative technique via initial time different coupled lower and upper solutions for fractional differential equations

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1031-1039 ◽  
Author(s):  
Ali Yakar ◽  
Hadi Kutlay
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Extremal Solutions ◽  
Lower And Upper Solutions ◽  
Initial Time ◽  
Iterative Technique ◽  
Monotone Iterative ◽  
Fractional Differential

In this paper, we investigate the extremal solutions for a class of nonlinear fractional differential equations with order q 2 (0; 1) by means of monotone iterative technique via initial time different coupled upper and lower solutions.

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