scholarly journals Integral inequalities of Gronwall type for piecewise continuous functions

1997 ◽  
Vol 10 (1) ◽  
pp. 89-94 ◽  
Author(s):  
Drumi D. Bainov ◽  
Snezhana G. Hristova
Keyword(s):  
Differential Equations ◽  
Integral Inequality ◽  
Initial Value Problem ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Continuous Functions ◽  
Integral Inequalities ◽  
Initial Value ◽  
Volterra Type ◽  
Piecewise Continuous

In this paper we generalize the integral inequality of Gronwall and study the continuous dependence of the solution of the initial value problem for nonlinear impulsive integro-differential equations of Volterra type on the initial conditions.

SOME PROPERTIES OF THE GENERALIZED SOLUTIONS OF AN INITIAL VALUE PROBLEM FOR FUNCTIONAL-DIFFERENTIAL INCLUSION WITH MULTIPLE-VALUED IMPULSES

2018 ◽  
pp. 805-812
Author(s):  
Olga Viktorovna Filippova ◽  
Andrey Igorevich Shindiapin
Keyword(s):  
Differential Inclusion ◽  
Initial Value Problem ◽  
Continuous Dependence ◽  
Continuous Functions ◽  
Generalized Solutions ◽  
Initial Value ◽  
Functional Differential Inclusion ◽  
Starting Conditions ◽  
Piecewise Continuous ◽  
Functional Differential

Deviation estimates in space of piecewise continuous functions of a set of the generalized decisions from beforehand given function are received. The continuous dependence of the generalized decisions on starting conditions is established.

scholarly journals BOUNDEDNESS OF IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH VARIABLE IMPULSIVE PERTURBATIONS

2008 ◽  
Vol 77 (2) ◽  
pp. 331-345 ◽  
Author(s):  
I. M. Stamova
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Boundedness Of Solutions ◽  
Initial Value ◽  
Razumikhin Technique ◽  
Piecewise Continuous ◽  
Impulsive Perturbations ◽  
Functional Differential

AbstractIn the present paper an initial value problem for impulsive functional differential equations with variable impulsive perturbations is considered. By means of piecewise continuous functions coupled with the Razumikhin technique, sufficient conditions for boundedness of solutions of such equations are found.

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.

Convergent and asymptotic expansions of solutions of second-order differential equations with a large parameter

2014 ◽  
Vol 12 (05) ◽  
pp. 523-536 ◽  
Author(s):  
Chelo Ferreira ◽  
José L. López ◽  
Ester Pérez Sinusía
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Initial Conditions ◽  
Special Functions ◽  
Auxiliary Problem ◽  
Second Order ◽  
Large Parameter ◽  
Initial Value

We consider the second-order linear differential equation [Formula: see text] where x ∈ [0, X], X > 0, α ∈ (-∞, 2), Λ is a large complex parameter and g is a continuous function. The asymptotic method designed by Olver [Asymptotics and Special Functions (Academic Press, New York, 1974)] gives the Poincaré-type asymptotic expansion of two independent solutions of the equation in inverse powers of Λ. We add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green's function of an auxiliary problem, we transform the initial value problem into a Volterra integral equation of the second kind. Then using a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence has also the property of being an asymptotic expansion for large Λ (not of Poincaré-type) of the solution of the problem. Moreover, we show that the idea may be applied also to nonlinear differential equations with a large parameter.

A Class of Integral Inequality and Application

2013 ◽  
Vol 785-786 ◽  
pp. 1395-1398 ◽  
Author(s):  
Wu Sheng Wang
Keyword(s):  
Differential Equations ◽  
Upper Bound ◽  
Integral Inequality ◽  
Unknown Function ◽  
Initial Conditions ◽  
Integral Inequalities ◽  
Inequality Technique ◽  
Bound Estimation

We discuss a class of generalized retarded nonlinear integral inequalities, which not only include nonlinear compound function of unknown function but also include retarded items, and give upper bound estimation of the unknown function by integral inequality technique. This estimation can be used as tool in the study of differential equations with the initial conditions.

scholarly journals The System of Differential Equations Associated With Involutions

2021 ◽  
Author(s):  
Farrukh Nuriddin ugli Dekhkonov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Initial Value Problem ◽  
Initial Conditions ◽  
Higher Order ◽  
System Of Differential Equations ◽  
Initial Value ◽  
Order Ordinary Differential Equation

In this paper, we consider with a class of system of differential equations whose argument transforms are involution. In this an initial value problem for a differential equation with involution is reduced to an initial value problem for a higher order ordinary differential equation. Than either two initial conditions are necessary for a solution, the equation is then reduced to a boundary value problem for a higher order ODE.

scholarly journals Practical stability of the solutions of impulsive systems of differential-difference equations via the method of comparison and some applications to population dynamics

2002 ◽  
Vol 43 (4) ◽  
pp. 525-539 ◽  
Author(s):  
D. D. Bainov ◽  
A. B. Dishliev ◽  
I. M. Stamova
Keyword(s):  
Population Dynamics ◽  
Difference Equations ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Practical Stability ◽  
Differential Inequalities ◽  
Impulsive Systems ◽  
Initial Value ◽  
Piecewise Continuous

AbstractIn this paper we consider an initial value problem for systems of impulsive differential-difference equations is considered. Making use of the method of comparison and differential inequalities for piecewise continuous functions, sufficient conditions for practical stability of the solutions of such systems are obtained. Applications to population dynamics are also given.

scholarly journals On nonlinear mixed fractional integro-differential equations with positive constant coefficient

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5623-5638 ◽  
Author(s):  
Shivaji Tate ◽  
V.V. Kharat ◽  
H.T. Dinde
Keyword(s):  
Differential Equations ◽  
Positive Constant ◽  
Integral Inequality ◽  
Constant Coefficient ◽  
Continuous Dependence ◽  
Mixed Type ◽  
Initial Conditions

In this paper, we study the existence and other properties of the solution of nonlinear mixed fractional integro-differential equations with constant coefficient. Also with the help of integral inequality of mixed type, we prove the continuous dependence of the solutions on the initial conditions.

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