On the behavior of the solutions to autonomous linear difference equations with continuous variable

Christos G. Philos; Ioannis K. Purnaras

Archivum Mathematicum (2007)

  • Volume: 043, Issue: 2, page 133-155
  • ISSN: 0044-8753

Abstract

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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.

How to cite

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Philos, Christos G., and Purnaras, Ioannis K.. "On the behavior of the solutions to autonomous linear difference equations with continuous variable." Archivum Mathematicum 043.2 (2007): 133-155. <http://eudml.org/doc/250170>.

@article{Philos2007,
abstract = {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.},
author = {Philos, Christos G., Purnaras, Ioannis K.},
journal = {Archivum Mathematicum},
keywords = {difference equation with continuous variable; delay difference equation; neutral delay difference equation; behavior of solutions; characteristic equation; difference equation with continuous variable; delay difference equation; neutral delay difference equation; behavior of solutions; characteristic equation; asymptotic behavior},
language = {eng},
number = {2},
pages = {133-155},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the behavior of the solutions to autonomous linear difference equations with continuous variable},
url = {http://eudml.org/doc/250170},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Philos, Christos G.
AU - Purnaras, Ioannis K.
TI - On the behavior of the solutions to autonomous linear difference equations with continuous variable
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 2
SP - 133
EP - 155
AB - 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.
LA - eng
KW - difference equation with continuous variable; delay difference equation; neutral delay difference equation; behavior of solutions; characteristic equation; difference equation with continuous variable; delay difference equation; neutral delay difference equation; behavior of solutions; characteristic equation; asymptotic behavior
UR - http://eudml.org/doc/250170
ER -

References

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  1. Agarwal R. P., Difference Equations and Inequalities, Marcel Dekker, New York, 1992. (1992) Zbl0925.39001MR1155840
  2. Arino O., Pituk M., More on linear differential systems with small delays, J. Differential Equations 170 (2001), 381–407. Zbl0989.34053MR1815189
  3. de Bruijn N. G., On some linear functional equations, Publ. Math. Debrecen 1 (1950), 129–134. (1950) Zbl0036.19501MR0036427
  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. Zbl1076.34072MR2119064
  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. Zbl1104.34054MR2210889
  6. Domshlak Y., Oscillatory properties of linear difference equations with continuous time, Differential Equations Dynam. Systems 1 (1993), 311–324. (1993) Zbl0868.39006MR1259171
  7. Driver R. D., Some harmless delays, Delay and Functional Differential Equations and Their Applications, Academic Press, New York, 1972, 103–119. (1972) Zbl0284.34074MR0385277
  8. Driver R. D., Linear differential systems with small delays, J. Differential Equations 21 (1976), 148–166. (1976) Zbl0319.34067MR0404803
  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. (1992) Zbl0751.39001MR1111217
  10. Driver R. D., Sasser D. W., Slater M. L., The equation x ' ( t ) = a x ( t ) + b x ( t - τ ) with “small” delay, Amer. Math. Monthly 80 (1973), 990–995. (1973) Zbl0292.34076MR0326104
  11. Elaydi S. N., An Introduction to Difference Equations, Springer, New York, 1996. (1996) Zbl0840.39002MR1410259
  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. Zbl1069.34115MR2015849
  13. Graef J. R., Qian C., Asymptotic behavior of forced delay equations with periodic coefficients, Commun. Appl. Anal. 2 (1998), 551–564. (1998) Zbl0903.34061MR1637012
  14. Györi I., Invariant cones of positive initial functions for delay differential equations, Appl. Anal. 35 (1990), 21–41. (1990) Zbl0662.34069MR1387399
  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. (1990) Zbl0715.39001MR1049397
  16. Kelley W. G., Peterson A. C., Difference Equations: An Introduction with Applications, Academic Press, Inc., New York, 1991. (1991) Zbl0733.39001MR1142573
  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. (1998) Zbl0920.34065MR1653396
  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. (1999) MR1698917
  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. (1999) Zbl0973.39005MR1697057
  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. (1998) Zbl0903.39002MR1708189
  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. Zbl1065.34075MR2084992
  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. (1989) MR1096768
  23. Ladas G., Pakula L., Wang Z., Necessary and sufficient conditions for the oscillation of difference equations, Panamer. Math. J. 2 (1992), 17–26. (1992) Zbl0773.39002MR1145924
  24. Lakshmikantham V., Trigiante D., Theory of Difference Equations: Numerical Methods and Applications, Academic Press, Inc., New York, 1988. (1988) Zbl0683.39001MR0939611
  25. Mickens R., Difference Equations, Van Nostrand Reinhold, New York, 1990. (1990) Zbl0949.39500MR1158461
  26. Norris M. J., Unpublished notes on the delay differential equation x ' ( t ) = b x ( t - 1 ) where - 1 / e b < 0 , October 1967. (1967) 
  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. (1998) Zbl0918.34067MR1663984
  28. Philos, Ch. G., Purnaras I. K., Periodic first order linear neutral delay differential equations, Appl. Math. Comput. 117 (2001), 203–222. Zbl1029.34061MR1806047
  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. Zbl1069.39012MR2079865
  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. Zbl1054.34109MR2036187
  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. Zbl1079.39011MR2059199
  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. Zbl1060.39014MR2108552
  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. Zbl1070.39003MR2097112
  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. Zbl1074.39012MR2108572
  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. (1999) MR2305642
  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. Zbl1100.34063MR2212286
  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. Zbl1077.39008MR2152554
  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. Zbl1113.45011MR2197536
  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. Zbl1133.39002MR2209681
  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. Zbl1110.34042MR2250621
  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. Zbl1118.34077MR2255225
  42. Pituk M., The limits of the solutions of a nonautonomous linear delay difference equation, Comput. Math. Appl. 42 (2001), 543–550. Zbl0996.39011MR1838013
  43. Pituk M., Asymptotic behavior of a nonhomogeneous linear recurrence system, J. Math. Anal. Appl. 267 (2002), 626–642. Zbl1002.39011MR1888027
  44. Pituk M., Cesàro summability in a linear autonomous difference equation, Proc. Amer. Math. Soc. 133 (2005), 3333–3339. Zbl1085.39004MR2161157
  45. Sharkovsky A. N., Maistrenko, Yu. L., Romanenko E. Yu., Difference Equations and Their Applications, Kluwer Academic Publishers, Dordrecht, 1993. (1993) MR1244579
  46. Shen J. H., Comparison and oscillation results for difference equations with continuous variable, Indian J. Pure Appl. Math. 31 (2000), 1633–1642. Zbl0978.39010MR1816632
  47. Yan J., Zhang F., Oscillation for system of delay difference equations, J. Math. Anal. Appl. 230 (1999), 223–231. (1999) Zbl0920.39003MR1669617
  48. Zhang Y., Yan J., Zhao A., Oscillation criteria for a difference equation, Indian J. Pure Appl. Math. 28 (1997), 1241–1249. (1997) Zbl0904.39009MR1605506

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