Title

OSCILLATORY AND ASYMPTOTIC PROPERTIES OF
NONLINEAR FIRST ORDER DIFFERENTIAL EQUATIONS
WITH PIECEWISE CONSTANT ARGUMENT
OF GENERALIZED TYPE

Authors

Todor Kostadinov, Valentina Proycheva
Faculty of Mathematics
Technical University of Sofia, Filial Plovdiv
25 Tsanko Dystabanov, 4000, Plovdiv, BULGARIA

Abstract

In the paper, we consider differential equations of neutral type with piecewise constant argument of generalized type, i.e., the argument is a general step function. Sufficient conditions for oscillation of all solutions of this type equations are obtained. The asymptotic behavior of the nonoscillating solutions is studied also. Appropriate examples are given to illustrate the recieved results.

History

Received: April 19, 2016
Revised: October 11, 2016
Published: November 5, 2016

AMS Classification, Key Words

AMS Subject Classification:
Key Words and Phrases: Neutral type, non oscillatory and oscillatory solution, piecewise constant argument

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

Bibliography

1

A. R. Aftebizadeh and J. Wiener, Oscillatory and periodic solutions of an equation alternately of retarded and advanced type, Apll. Anal. , 23 (1986), 219-231.

2

A. R. Aftabizadeh, J. Wiener and J. Xu, Oscillatory and periodic solutions of delay differencial equations with piecewise constant argument, Proc. Amer. Math. Soc. , 99 (1987), 673-679.

3

M. U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Analysis: Theory, Methods and Applications, 66, No. 2 (2007), 367-383.

4

M. U. Akhmet, Nonlinear Hybrid Continuous/Discrete-Time Models Atlantis Press, Amsterdam, The Netherlands, 2011.

5

A. I. Alonso, J. Hong and R. Obaya, Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences, Applied Mathematics Letters, 13, No. 2 (2000), 131-137.

6

KUo-Shou Chiu, Periodic solutions for nonlinear integro-differential systems with piecewise constant argument, The Scientic World Journal (2014), 1-4.

7

K. S. Chiu and M. Pinto, Oscillatory and periodic solutions in alternately advanced and delayed differential equations, Carpathian Journal of Mathematics, 29, No. 2 (2013), 149-158.

8

K. S. Chiu and M. Pinto, Periodic solutions of differential equations with a general piecewise constant argument and applications, Electronic Journal of Qualitative Theory of Differential Equations, 46 (2010), 1-19.

9

K. L. Cooke and J. Wiener, Retarded differential equations with piecewise constant delay, J. Math. Amal. Appl. , 99 (1984), 265-297.

10

K. I. Cooke and J. Wiener, An equation alternately of retarded an advanced type, Proceedings of the American Mathematical Society, 99 (1987), 726-732.

11

I. Gyori and G. Ladas, Linearized oscillations for equations with piecewise constant arguments, Differential Integral Equations, 2 (1988), 123-131.

12

I. Gyori, G. Ladas and L. Pakula, Conditions for oscillation of difference equations with applications to equations with piecewise constant arguments, SIAM J. Math. Anal. , 22, No. 3 (1991), 269-773.

13

I. Gyori and G. Ladas, Oscillations Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991.

14

J. K. Hale, S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, NewYork, 1993.

15

Y. Huang, Oscillations and asymptotic stability of solutions of first order neutral differential equations with piecewise constant argument, J. Math. Anal. Apll. , 149 (1990), 70-85.

16

V. Kolmanovskii, A. Myshkis, Introduction to the Theory and Application of Functional Differential Equations, Kluivert Academic Publishers, Dordrecht, 1999.

17

A. D. Myshkis, On certain problems in the theory of differential equations with deviating arguments, Uspekhi Matematicheskikh Nauk, 32 (1977), 173-202.

18

E. C. Partheniadis, Stability and oscillation of neutral delay differential equations with piecewise constant argument, Differential Integral Equations, 1 (1988), 459-472.

19

M. Pinto, Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments, Mathematical and Computer Modelling, 49, No-s: 9-10, (2009) 1750-1758.

20

M. Pinto, Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems, Journal of Difference Equations and Applications, 17, No. 2, (2011) 235-254.

21

S. M. Shah and J. Wiener, Advanced differential equations with piecewise constant argument deviations, Internat. J. Math. Math. Sci. , 6 (1983), 671-703.

22

G. Wang Existence theorem of periodic solutions for a delay nonlinear differential equation with piecewise constant argument, Journal of Mathematical Analysis and Applications, 298, No. 1 (2004), 298-307.

23

J. Wiener and A. R. Aftabizadeh, Differential equations alternately of retarded and advanced type, J. Math. Anal. Appl. , 129 (1988), 243-255.

24

J. Wiener, Differential equations with piecewise constant delays, In: Trends in the Theory and Practice of Nonlinear Differential Equations (Ed. V. Lakshmikantham), Marcel Dekker, New York, (1983) 547-580.

25

J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientic, Singapore (1993).

How to Cite?

DOI: 10.12732/ijpam.v110i3.13 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 110
Issue: 3
Pages: 547 - 562

Google Scholar; DOI (International DOI Foundation); WorldCAT.

This work is licensed under the Creative Commons Attribution International License (CC BY).

---