scholarly journals Monotone-iterative technique of Lakshmikantham for the initial value problem for a differential equation with a step function

2002 ◽  
Vol 30 (1) ◽  
pp. 57-62
Author(s):  
S. G. Hristova ◽  
M. A. Rangelova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Special Kind ◽  
Step Function ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Initial Value ◽  
Monotone Iterative ◽  
The Given

The initial value problem for a special kind of differential equations with a step function is studied. The monotone-iterative technique of Lakshmikantham for approximate finding of the solutions of the given problem is well grounded.

scholarly journals Initial Value Problem For Nonlinear Fractional Differential Equations With ψ-Caputo Derivative Via Monotone Iterative Technique

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 57 ◽  
Author(s):  
Choukri Derbazi ◽  
Zidane Baitiche ◽  
Mouffak Benchohra ◽  
Alberto Cabada
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Caputo Derivative ◽  
Monotone Iterative Technique ◽  
Extremal Solutions ◽  
Iterative Technique ◽  
Initial Value ◽  
Monotone Iterative ◽  
Fractional Differential

In this article, we discuss the existence and uniqueness of extremal solutions for nonlinear initial value problems of fractional differential equations involving the ψ -Caputo derivative. Moreover, some uniqueness results are obtained. Our results rely on the standard tools of functional analysis. More precisely we apply the monotone iterative technique combined with the method of upper and lower solutions to establish sufficient conditions for existence as well as the uniqueness of extremal solutions to the initial value problem. An illustrative example is presented to point out the applicability of our main results.

scholarly journals Application of Lakshmikantham's monotone-iterative technique to the solution of the initial value problem for impulsive integro-differential equations

1993 ◽  
Vol 6 (1) ◽  
pp. 25-34 ◽  
Author(s):  
D. D. Bainov ◽  
S. G. Hristova
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Initial Value ◽  
Volterra Type ◽  
Monotone Iterative

In the present paper, a technique of V. Lakshmikantham is applied to approximate finding of extremal quasisolutions of an initial value problem for a system of impulsive integro-differential equations of Volterra type.

Monotone iterative technique for impulsive Riemann-Liouville fractional differential equations

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3381-3395 ◽  
Author(s):  
Renu Chaudhary ◽  
Dwijendra Pandey
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Measure Of Noncompactness ◽  
Semigroup Theory ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Monotone Iterative ◽  
Fractional Differential

In this article, Monotone iterative technique coupled with the method of lower and upper solutions is employed to discuss the existence and uniqueness of mild solution to an impulsive Riemann-Liouville fractional differential equation. The results are obtained using the concept of measure of noncompactness, semigroup theory and generalized Gronwall inequality for fractional differential equations. At last, an example is given to illustrate the applications of the main results.

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.

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.

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