Title:
|
On the behavior of the solutions to autonomous linear difference equations with continuous variable (English) |
Author:
|
Philos, Ch. G |
Author:
|
Purnaras, I. K. |
Language:
|
English |
Journal:
|
Archivum Mathematicum |
ISSN:
|
0044-8753 (print) |
ISSN:
|
1212-5059 (online) |
Volume:
|
43 |
Issue:
|
2 |
Year:
|
2007 |
Pages:
|
133-155 |
Summary lang:
|
English |
. |
Category:
|
math |
. |
Summary:
|
Autonomous linear neutral delay and, especially, (non-neutral) delay difference equations with continuous variable are considered, and some new results on the behavior of the solutions are established. The results are obtained by the use of appropriate positive roots of the corresponding characteristic equation. (English) |
Keyword:
|
difference equation with continuous variable |
Keyword:
|
delay difference equation |
Keyword:
|
neutral delay difference equation |
Keyword:
|
behavior of solutions |
Keyword:
|
characteristic equation |
MSC:
|
39A10 |
MSC:
|
39A11 |
idZBL:
|
Zbl 1164.39006 |
idMR:
|
MR2336965 |
. |
Date available:
|
2008-06-06T22:50:53Z |
Last updated:
|
2012-05-10 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108058 |
. |
Reference:
|
[1] Agarwal R. P.: Difference Equations and Inequalities.Marcel Dekker, New York, 1992. Zbl 0925.39001, MR 1155840 |
Reference:
|
[2] Arino O., Pituk M.: More on linear differential systems with small delays.J. Differential Equations 170 (2001), 381–407. Zbl 0989.34053, MR 1815189 |
Reference:
|
[3] de Bruijn N. G.: On some linear functional equations.Publ. Math. Debrecen 1 (1950), 129–134. Zbl 0036.19501, MR 0036427 |
Reference:
|
[4] Dix J. G., Philos, Ch. G., Purnaras I. K.: An asymptotic property of solutions to linear nonautonomous delay differential equations.Electron. J. Differential Equations 2005 (2005), No. 10, 1–9. Zbl 1076.34072, MR 2119064 |
Reference:
|
[5] Dix J. G., Philos, Ch. G., Purnaras I. K.: Asymptotic properties of solutions to linear non-autonomous neutral differential equations.J. Math. Anal. Appl. 318 (2006), 296–304. Zbl 1104.34054, MR 2210889 |
Reference:
|
[6] Domshlak Y.: Oscillatory properties of linear difference equations with continuous time.Differential Equations Dynam. Systems 1 (1993), 311–324. Zbl 0868.39006, MR 1259171 |
Reference:
|
[7] Driver R. D.: Some harmless delays.Delay and Functional Differential Equations and Their Applications, Academic Press, New York, 1972, 103–119. Zbl 0284.34074, MR 0385277 |
Reference:
|
[8] Driver R. D.: Linear differential systems with small delays.J. Differential Equations 21 (1976), 148–166. Zbl 0319.34067, MR 0404803 |
Reference:
|
[9] Driver R. D., Ladas G., Vlahos P. N.: Asymptotic behavior of a linear delay difference equation.Proc. Amer. Math. Soc. 115 (1992), 105–112. Zbl 0751.39001, MR 1111217 |
Reference:
|
[10] Driver R. D., Sasser D. W., Slater M. L.: The equation $x^{\prime }(t)=ax(t)+bx(t-\tau )$ with “small” delay.Amer. Math. Monthly 80 (1973), 990–995. Zbl 0292.34076, MR 0326104 |
Reference:
|
[11] Elaydi S. N.: An Introduction to Difference Equations.Springer, New York, 1996. Zbl 0840.39002, MR 1410259 |
Reference:
|
[12] Frasson M. V. S., Verduyn Lunel S. M.: Large time behaviour of linear functional differential equations.Integral Equations Operator Theory 47 (2003), 91–121. Zbl 1069.34115, MR 2015849 |
Reference:
|
[13] Graef J. R., Qian C.: Asymptotic behavior of forced delay equations with periodic coefficients.Commun. Appl. Anal. 2 (1998), 551–564. Zbl 0903.34061, MR 1637012 |
Reference:
|
[14] Györi I.: Invariant cones of positive initial functions for delay differential equations.Appl. Anal. 35 (1990), 21–41. Zbl 0662.34069, MR 1387399 |
Reference:
|
[15] Györi I.: Sharp conditions for existence of nontrivial invariant cones of nonnegative initial values of difference equations.Appl. Math. Comput. 36 (1990), 89–111. Zbl 0715.39001, MR 1049397 |
Reference:
|
[16] Kelley W. G., Peterson A. C.: Difference Equations: An Introduction with Applications.Academic Press, Inc., New York, 1991. Zbl 0733.39001, MR 1142573 |
Reference:
|
[17] Kordonis I.-G. E., Niyianni N. T., Philos, Ch. G.: On the behavior of the solutions of scalar first order linear autonomous neutral delay differential equations.Arch. Math. (Basel) 71 (1998), 454–464. Zbl 0920.34065, MR 1653396 |
Reference:
|
[18] Kordonis I.-G. E., Philos, Ch. G.: The behavior of solutions of linear integro-differential equations with unbounded delay.Comput. Math. Appl. 38 (1999), 45–50. MR 1698917 |
Reference:
|
[19] Kordonis I.-G. E., Philos, Ch. G.: On the behavior of the solutions for linear autonomous neutral delay difference equations.J. Differ. Equations Appl. 5 (1999), 219–233. Zbl 0973.39005, MR 1697057 |
Reference:
|
[20] Kordonis I.-G. E., Philos, Ch. G., Purnaras I. K.: Some results on the behavior of the solutions of a linear delay difference equation with periodic coefficients.Appl. Anal. 69 (1998), 83–104. Zbl 0903.39002, MR 1708189 |
Reference:
|
[21] Kordonis I.-G. E., Philos, Ch. G., Purnaras I. K.: On the behavior of solutions of linear neutral integrodifferential equations with unbounded delay.Georgian Math. J. 11 (2004), 337–348. Zbl 1065.34075, MR 2084992 |
Reference:
|
[22] Ladas G.: Recent developments in the oscillation of delay difference equations.Differential Equations (Colorado Springs, CO, 1989), Lecture Notes in Pure and Appl. Math. 127, Dekker, New York, 1991, 321–332. MR 1096768 |
Reference:
|
[23] Ladas G., Pakula L., Wang Z.: Necessary and sufficient conditions for the oscillation of difference equations.Panamer. Math. J. 2 (1992), 17–26. Zbl 0773.39002, MR 1145924 |
Reference:
|
[24] Lakshmikantham V., Trigiante D.: Theory of Difference Equations: Numerical Methods and Applications.Academic Press, Inc., New York, 1988. Zbl 0683.39001, MR 0939611 |
Reference:
|
[25] Mickens R.: Difference Equations.Van Nostrand Reinhold, New York, 1990. Zbl 0949.39500, MR 1158461 |
Reference:
|
[26] Norris M. J.: Unpublished notes on the delay differential equation $x^{\prime }(t)=bx(t-1)$ where $-1/e\le b<0$.October 1967. |
Reference:
|
[27] Philos, Ch. G.: Asymptotic behaviour, nonoscillation and stability in periodic first-order linear delay differential equations.Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1371–1387. Zbl 0918.34067, MR 1663984 |
Reference:
|
[28] Philos, Ch. G., Purnaras I. K.: Periodic first order linear neutral delay differential equations.Appl. Math. Comput. 117 (2001), 203–222. Zbl 1029.34061, MR 1806047 |
Reference:
|
[29] Philos, Ch. G., Purnaras I. K.: The behavior of solutions of linear Volterra difference equations with infinite delay.Comput. Math. Appl. 47 (2004), 1555–1563. Zbl 1069.39012, MR 2079865 |
Reference:
|
[30] Philos, Ch. G., Purnaras I. K.: Asymptotic properties, nonoscillation, and stability for scalar first order linear autonomous neutral delay differential equations.Electron. J. Differential Equations 2004 (2004), No. 03, 1–17. Zbl 1054.34109, MR 2036187 |
Reference:
|
[31] Philos, Ch. G., Purnaras I. K.: An asymptotic result for some delay difference equations with continuous variable.Advances in Difference Equations 2004:1 (2004), 1–10. Zbl 1079.39011, MR 2059199 |
Reference:
|
[32] Philos, Ch. G., Purnaras I. K.: An asymptotic result and a stability criterion for linear nonautonomous delay difference equations.Arch. Math. (Basel) 83 (2004), 243–255. Zbl 1060.39014, MR 2108552 |
Reference:
|
[33] Philos, Ch. G., Purnaras I. K.: On the behavior of the solutions for certain linear autonomous difference equations.J. Differ. Equations Appl. 10 (2004), 1049–1067. Zbl 1070.39003, MR 2097112 |
Reference:
|
[34] Philos, Ch. G., Purnaras I. K.: The behavior of the solutions of periodic linear neutral delay difference equations.J. Comput. Appl. Math. 175 (2005), 209–230. Zbl 1074.39012, MR 2108572 |
Reference:
|
[35] Philos, Ch. G., Purnaras I. K.: On the behavior of the solutions for certain first order linear autonomous functional differential equations.Rocky Mountain J. Math. 36 (2006), 1999–2019. MR 2305642 |
Reference:
|
[36] Philos, Ch. G., Purnaras I. K.: A result on the behavior of the solutions for scalar first order linear autonomous neutral delay differential equations.Math. Proc. Cambridge Philos. Soc. 140 (2006), 349–358. Zbl 1100.34063, MR 2212286 |
Reference:
|
[37] Philos, Ch. G., Purnaras I. K.: Asymptotic behaviour and stability to linear non-autonomous neutral delay difference equations.J. Differ. Equations Appl. 11 (2005), 503–513. Zbl 1077.39008, MR 2152554 |
Reference:
|
[38] Philos, Ch. G., Purnaras I. K.: More on the behavior of solutions to linear integrodifferential equations with unbounded delay.Funkcial. Ekvac. 48 (2005), 393–414. Zbl 1113.45011, MR 2197536 |
Reference:
|
[39] Philos, Ch. G., Purnaras I. K.: On linear Volterra difference equations with infinite delay.Advances in Difference Equations 2006 (2006), Art. 78470, pp. 1–28. Zbl 1133.39002, MR 2209681 |
Reference:
|
[40] Philos, Ch. G., Purnaras I. K.: On the behavior of the solutions to periodic linear delay differential and difference equations.J. Math. Anal. Appl. 322 (2006), 847–863. Zbl 1110.34042, MR 2250621 |
Reference:
|
[41] Philos, Ch. G., Purnaras I. K.: On periodic linear neutral delay differential and difference equations.Electron. J. Differential Equations 2006 (2006), No. 110, 1–25. Zbl 1118.34077, MR 2255225 |
Reference:
|
[42] Pituk M.: The limits of the solutions of a nonautonomous linear delay difference equation.Comput. Math. Appl. 42 (2001), 543–550. Zbl 0996.39011, MR 1838013 |
Reference:
|
[43] Pituk M.: Asymptotic behavior of a nonhomogeneous linear recurrence system.J. Math. Anal. Appl. 267 (2002), 626–642. Zbl 1002.39011, MR 1888027 |
Reference:
|
[44] Pituk M.: Cesàro summability in a linear autonomous difference equation.Proc. Amer. Math. Soc. 133 (2005), 3333–3339. Zbl 1085.39004, MR 2161157 |
Reference:
|
[45] Sharkovsky A. N., Maistrenko, Yu. L., Romanenko E. Yu.: Difference Equations and Their Applications.Kluwer Academic Publishers, Dordrecht, 1993. MR 1244579 |
Reference:
|
[46] Shen J. H.: Comparison and oscillation results for difference equations with continuous variable.Indian J. Pure Appl. Math. 31 (2000), 1633–1642. Zbl 0978.39010, MR 1816632 |
Reference:
|
[47] Yan J., Zhang F.: Oscillation for system of delay difference equations.J. Math. Anal. Appl. 230 (1999), 223–231. Zbl 0920.39003, MR 1669617 |
Reference:
|
[48] Zhang Y., Yan J., Zhao A.: Oscillation criteria for a difference equation.Indian J. Pure Appl. Math. 28 (1997), 1241–1249. Zbl 0904.39009, MR 1605506 |
. |