A new continuous dependence result for impulsive retarded functional differential equations

Márcia Federson; Jaqueline Godoy Mesquita

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 1, page 1-12
  • ISSN: 0011-4642

Abstract

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We consider a large class of impulsive retarded functional differential equations (IRFDEs) and prove a result concerning uniqueness of solutions of impulsive FDEs. Also, we present a new result on continuous dependence of solutions on parameters for this class of equations. More precisely, we consider a sequence of initial value problems for impulsive RFDEs in the above setting, with convergent right-hand sides, convergent impulse operators and uniformly convergent initial data. We assume that the limiting equation is an impulsive RFDE whose initial condition is the uniform limit of the sequence of the initial data and whose solution exists and is unique. Then, for sufficient large indexes, the elements of the sequence of impulsive retarded initial value problem admit a unique solution and such a sequence of solutions converges to the solution of the limiting Cauchy problem.

How to cite

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Federson, Márcia, and Mesquita, Jaqueline Godoy. "A new continuous dependence result for impulsive retarded functional differential equations." Czechoslovak Mathematical Journal 66.1 (2016): 1-12. <http://eudml.org/doc/276750>.

@article{Federson2016,
abstract = {We consider a large class of impulsive retarded functional differential equations (IRFDEs) and prove a result concerning uniqueness of solutions of impulsive FDEs. Also, we present a new result on continuous dependence of solutions on parameters for this class of equations. More precisely, we consider a sequence of initial value problems for impulsive RFDEs in the above setting, with convergent right-hand sides, convergent impulse operators and uniformly convergent initial data. We assume that the limiting equation is an impulsive RFDE whose initial condition is the uniform limit of the sequence of the initial data and whose solution exists and is unique. Then, for sufficient large indexes, the elements of the sequence of impulsive retarded initial value problem admit a unique solution and such a sequence of solutions converges to the solution of the limiting Cauchy problem.},
author = {Federson, Márcia, Mesquita, Jaqueline Godoy},
journal = {Czechoslovak Mathematical Journal},
keywords = {retarded functional differential equation; impulse local existence; impulse local existence uniqueness; continuous dependence on parameters},
language = {eng},
number = {1},
pages = {1-12},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A new continuous dependence result for impulsive retarded functional differential equations},
url = {http://eudml.org/doc/276750},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Federson, Márcia
AU - Mesquita, Jaqueline Godoy
TI - A new continuous dependence result for impulsive retarded functional differential equations
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 1
EP - 12
AB - We consider a large class of impulsive retarded functional differential equations (IRFDEs) and prove a result concerning uniqueness of solutions of impulsive FDEs. Also, we present a new result on continuous dependence of solutions on parameters for this class of equations. More precisely, we consider a sequence of initial value problems for impulsive RFDEs in the above setting, with convergent right-hand sides, convergent impulse operators and uniformly convergent initial data. We assume that the limiting equation is an impulsive RFDE whose initial condition is the uniform limit of the sequence of the initial data and whose solution exists and is unique. Then, for sufficient large indexes, the elements of the sequence of impulsive retarded initial value problem admit a unique solution and such a sequence of solutions converges to the solution of the limiting Cauchy problem.
LA - eng
KW - retarded functional differential equation; impulse local existence; impulse local existence uniqueness; continuous dependence on parameters
UR - http://eudml.org/doc/276750
ER -

References

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  2. Federson, M., Mesquita, J. G., 10.1016/j.jmaa.2011.04.034, J. Math. Anal. Appl. 382 (2011), 77-85. (2011) Zbl1226.34075MR2805496DOI10.1016/j.jmaa.2011.04.034
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  7. Kurzweil, J., Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J. 7 (82) (1957), 418-449. (1957) Zbl0090.30002MR0111875
  8. Lakshmikantham, V., Baĭnov, D. D., Simeonov, P. S., Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics Vol. 6 World Scientific, Singapore (1989). (1989) MR1082551
  9. Liu, X., Ballinger, G., 10.1016/S0893-9659(04)90094-8, Appl. Math. Lett. 17 (2004), 483-490. (2004) Zbl1085.34558MR2045757DOI10.1016/S0893-9659(04)90094-8
  10. Schwabik, Š., Generalized Ordinary Differential Equations, Series in Real Analysis 5 World Scientific, Singapore (1992). (1992) Zbl0781.34003MR1200241

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