Existence and Stability Results for Impulsive Integro-DifferentialEquations

Zeng Lin, Wei Wei, JinRong Wang

DOI Number
-
First page
119
Last page
130

Abstract


In this paper, we study a new class of impulsiveintegro-differential equations for which the impulses are notinstantaneous. By using fixed point approach and techniques ofanalysis, we present the existence and uniqueness theorem and derivean interesting stability result in the sense of generalizedUlam-Hyers-Rassias.

Keywords


Existence, Stability, Impulsive, Integro-differential equations.

Full Text:

PDF

References


bibitem{Samo} A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, vol.14 of World Scientific Series on Nonlinear

Science. Series A: Monographs and Treatises, World Scientific,

Singapore, (1995).

bibitem{Bain} D. D. Bainov, V. Lakshmikantham, P. S. Simeonov, Theory of impulsive differential equations, vol. 6 of Series in

Modern Applied Mathematics, World Scientific, Singapore, (1989).

bibitem{Benchohra} M. Benchohra, J. Henderson, S. Ntouyas, Impulsive differential

equations and inclusions, vol. 2 of Contemporary Mathematics and Its

Applications, Hindawi, New York, NY, USA, 2006.

bibitem{Hernandez} E. Hern'{a}ndez, D. O'Regan, On a new class of abstract impulsive

differential equations, Proc. Amer. Math. Soc., 141(2013),

-1649.

bibitem{Pierri} M. Pierri, D. O'Regan, V. Rolnik, Existence of solutions for

semi-linear abstract differential equations with not instantaneous

impulses, Appl. Math. Comput., 219(2013), 6743-6749.

bibitem{Ulam} S. M. Ulam, A collection of mathematical problems, Interscience

Publishers, New York, 1968.

bibitem{Hyers41} D. H. Hyers, On the stability of the linear functional equation,

Proc. Nat. Acad. Sci., 27(1941), 222-224.

bibitem{Hyers} D. H. Hyers, G.

Isac, Th. M. Rassias, Stability of functional equations in several

variables, Birkh"{a}user, 1998.

bibitem{Rassias} Th. M. Rassias, On the stability of linear mappings in Banach

spaces, Proc. Amer. Math. Soc., 72(1978), 297-300.

bibitem{Jung2} S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in

mathematical analysis, Hadronic Press, Palm Harbor, 2001.

bibitem{Jung2-add} S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in

nonlinear analysis, Springer, New York, 2011.

bibitem{Cadariu1} L. Cu{a}dariu, Stabilitatea

Ulam-Hyers-Bourgin pentru ecuatii functionale, Ed. Univ. Vest

Timic{s}oara, Timic{s}ara, 2007.

bibitem{Ibrahim} R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional

differential equations. Int. J. Math., 23(2012), 1250056.

bibitem{Jung} S.-M. Jung, T. S. Kim, K. S. Lee, A fixed point approach to the stability of quadratic functional equation, Bull. Korean Math. Soc.,

(2006), 531-541.

bibitem{Jung-add1} S.-M. Jung, Th. M. Rassias, Generalized Hyers-Ulam stability of

Riccati differential equation, Math. Inequal. Appl., 11(2008),

-782.

bibitem{Jung-add3} J. Brzdc{e}k, S.-M. Jung, A note on stability of an operator linear

equation of the second order, Abstr. Appl. Anal., 2011(2011),

Article ID 602713, 15 pages.

bibitem{Andras-NATMA} Sz. Andr'{a}s, J. J. Kolumb'{a}n, On the Ulam-Hyers stability of

first order differential systems with nonlocal initial conditions,

Nonlinear Anal.:TMA, 82(2013), 1-11.

bibitem{Andras-AMC} Sz. Andr'{a}s, A. R. M'{e}sz'{a}ros, Ulam-Hyers stability of

dynamic equations on time scales via Picard operators, Appl. Math.

Comput., 219(2013), 4853-4864.

bibitem{Burger} M. Burger, N. Ozawa, A. Thom, On Ulam stability, Isr. J. Math.,

(2013), 109-129.

bibitem{Cimpean} D. S. Cimpean, D. Popa, Hyers-Ulam stability of Euler's equation,

Appl. Math. Lett., 24(2011), 1539-1543.

bibitem{Hegyi} B. Hegyi, S.-M. Jung, On the stability of Laplace's equation, Appl.

Math. Lett., 26(2013), 549-552.

bibitem{Rezaei} H. Rezaei, S. M. Jung, Th. M. Rassias, Laplace transform and

Hyers-Ulam stability of linear differential equations, J. Math.

Anal. Appl., 403(2013), 244-251.

bibitem{Lungu} N. Lungu, D. Popa, Hyers-Ulam stability of a first order partial differential equation,

J. Math. Anal. Appl., 385(2012), 86-91.

bibitem{Popa} D. Popa, I. Rac{s}a, On the Hyers-Ulam stability of the linear differential equation,

J. Math. Anal. Appl., 381(2011), 530-537.

bibitem{Rus2009} I. A. Rus, Ulam stability of ordinary

differential equations, Studia Univ. ``Babec{s} Bolyai"

Mathematica, 54(2009), 125-133.

bibitem{Rus2010} I. A. Rus, Ulam stabilities of ordinary differential equations in a

Banach space, Carpathian J. Math., 26(2010), 103-107.

bibitem{XuTZ} T. Z. Xu, On the stability of multi-Jensen mappings in

$beta$-normed spaces, Appl. Math. Lett., 25(2012), 1866-1870.

bibitem{Wang-JMAA} J. Wang, M. Fev{c}kan, Y. Zhou, Ulam's type stability of impulsive ordinary differential

equations, J. Math. Anal. Appl., 395(2012), 258-264.

bibitem{Diaz} J. B. Diaz, B. Margolis, A fixed point theorem of the alternative,

for contractions on a generalized complete metric space, Bull. Amer.

Math. Soc., 74(1968), 305-309.


Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)