A MONOTONE-ITERATIVE METHOD FOR FINDING PERIODIC SOLUTION OF AN IMPULSIVE DIFFERENTIAL EQUATIONS AND APPLICATION

2008 ◽  
Vol 01 (02) ◽  
pp. 247-256
Author(s):  
JIAWEI DOU
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Upper And Lower Solutions ◽  
Impulsive Differential Equations ◽  
Iteration Process ◽  
Impulsive System ◽  
Monotone Iterative Method ◽  
Initial Value ◽  
Monotone Iterative ◽  
Monotone Iterations

In this paper, using the method of upper and lower solutions and its associated monotone iterations, we establish a new monotone-iterative scheme for finding periodic solutions of an impulsive differential equations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for impulsive differential equations initial value problem. This method is constructive and can be used to develop a computational algorithm for numerical solution of the periodic impulsive system. Our existence result improves a result established in [1]. The result is applied to a model of mutualism of Lotka–Volterra type which involves interactions among a mutualist-competitor, a competitor and a mutualist, the existence of positive periodic solutions of the model is obtained.

scholarly journals Monotone-Iterative Method for the Initial Value Problem with Initial Time Difference for Differential Equations with “Maxima”

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
S. Hristova ◽  
A. Golev
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Upper And Lower Solutions ◽  
Successive Approximations ◽  
Initial Time ◽  
Time Interval ◽  
Monotone Iterative Method ◽  
Initial Value ◽  
Monotone Iterative ◽  
The Given

The object of investigation of the paper is a special type of functional differential equations containing the maximum value of the unknown function over a past time interval. An improved algorithm of the monotone-iterative technique is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. Additionally, all initial value problems for successive approximations have both initial time and initial functions different. It allows us to construct sequences of successive approximations as well as sequences of initial functions, which are convergent to the solution and to the initial function of the given initial value problem, respectively. The suggested algorithm is realized as a computer program, and it is applied to several examples, illustrating the advantages of the suggested scheme.

Monotone Iterative Techniques and a Periodic Boundary Value Problem for First Order Differential Equations with a Functional Argument

2000 ◽  
Vol 7 (2) ◽  
pp. 373-378
Author(s):  
Aiqin Qi ◽  
Yansheng Liu
Keyword(s):  
Differential Equations ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Initial Value ◽  
First Order ◽  
Monotone Iterative ◽  
Periodic Boundary Value Problems ◽  
Functional Argument ◽  
First Order Differential Equations

Abstract This paper is concerned with periodic boundary value problems involving first order differential equations with functional arguments. The main feature of the paper is that the existence of maximal and minimal solutions is obtained by constructing sequences of upper and lower solutions of the initial value problems and not by establishing the comparison principle.

scholarly journals A monotone iterative method for boundary value problems of parametric differential equations

2001 ◽  
Vol 14 (2) ◽  
pp. 183-187 ◽  
Author(s):  
Xinzhi Liu ◽  
Farzana A. McRae
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Extremal Solutions ◽  
Monotone Iterative Method ◽  
Monotone Iterative ◽  
Monotone Sequences

This paper studies boundary value problems for parametric differential equations. By using the method of upper and lower solutions, monotone sequences are constructed and proved to converge to the extremal solutions of the boundary value problem.

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 Monotone iterative technique for non-autonomous semilinear differential equations with nonlocal condition

2019 ◽  
Vol 52 (1) ◽  
pp. 29-39 ◽  
Author(s):  
Arshi Meraj ◽  
Dwijendra Narain Pandey
Keyword(s):  
Differential Equations ◽  
Measure Of Noncompactness ◽  
Nonlocal Condition ◽  
Mild Solutions ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Monotone Iterative Method ◽  
Evolution System ◽  
Monotone Iterative ◽  
Norm Space

AbstractThe objective of this article is to discuss the existence and uniqueness of mild solutions for a class of non-autonomous semilinear differential equations with nonlocal condition via monotone iterative method with upper and lower solutions in an ordered complete norm space X, using evolution system and measure of noncompactness.

Method of upper and lower solutions for coupled system of nonlinear fractional integro-differential equations with advanced arguments

2017 ◽  
Vol 67 (1) ◽  
pp. 89-98
Author(s):  
Neda Khodabakhshi ◽  
S. Mansour Vaezpour ◽  
J. Juan Trujillo
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Upper And Lower Solutions ◽  
Coupled System ◽  
Extremal Solutions ◽  
Monotone Iterative Method ◽  
Monotone Iterative

AbstractIn this paper, by means of upper and lower solutions, we develop monotone iterative method for the existence of extremal solutions for coupled system of nonlinear fractional integro-differential equations with advanced arguments. We illustrate this technique with the help of an example.

scholarly journals Existence and uniqueness of extremal mild solutions for non-autonomous nonlocal integro-differential equations via monotone iterative technique

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 2985-2993 ◽  
Author(s):  
Arshi Meraj ◽  
Dwijendra Pandey
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Measure Of Noncompactness ◽  
Nonlocal Condition ◽  
Mild Solutions ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Method ◽  
Evolution System ◽  
Ordered Banach Space ◽  
Monotone Iterative

In this work, we will discuss the existence and uniqueness of extremal mild solutions for non-autonomous integro-differential equations having nonlocal condition via monotone iterative method with upper and lower solutions in an ordered Banach space X, using evolution system and measure of noncompactness.

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