Skip to main content
SpringerLink
Log in

Differential Equations with Piecewise Constant Argument of Generalized Type

  • Chapter
  • First Online:
Neural Networks with Discontinuous/Impact Activations

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 9))

Abstract

Differential equations with piecewise constant argument (EPCA) were proposed for investigations in [63, 91] by founders of the theory, K. Cook, S. Busenberg, J. Wiener, and S. Shah. They are named as differential EPCA. In the last three decades, many interesting results have been obtained, and applications have been realized in this theory. Existence and uniqueness of solutions, oscillations and stability, integral manifolds and periodic solutions, and many other questions of the theory have been intensively discussed. Besides the mathematical analysis, various models in biology, mechanics, and electronics were developed by using these systems. The founders proposed that the method of investigation of these equations is based on a reduction to discrete systems. That is, only values of solutions at moments, which are integers or multiples of integers, were discussed. Moreover, systems must be linear with respect to the values of solutions, if the argument is not deviated. It reduces the theoretical depth of the investigations as well as the number of real-world problems, which can be modeled by using these equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aftabizadeh AR, Wiener J (1985) Oscillatory properties of first order linear functional-differential equations Appl Anal 20:165–187

    MathSciNet  MATH  Google Scholar 

  2. Aftabizadeh AR, Wiener J, Xu JM (1987) Oscillatory and periodic solutions of delay differential equations with piecewise constant argument. Proc Amer Math Soc 99:673–679

    Article  MathSciNet  MATH  Google Scholar 

  3. Akhmet MU (2004) Existence and stability of almost-periodic solutions of quasi-linear differential equations with deviating argument. Appl Math Lett 17:1177–1181

    Article  MathSciNet  MATH  Google Scholar 

  4. Akhmet MU (2006) On the integral manifolds of the differential equations with piecewise constant argument of generalized type. In: Agarval RP, Perera K (eds) Proceedings of the conference on differential and difference equations at the florida institute of technology, Melbourne, Florida, 1–5 August 2005. Hindawi Publishing Corporation, Nasr City, pp 11–20

    Google Scholar 

  5. Akhmet MU (2007) Integral manifolds of differential equations with piecewise constant argument of generalized type. Nonlin Anal 66:367–383

    Article  MathSciNet  MATH  Google Scholar 

  6. Akhmet MU (2008) Stability of differential equations with piecewise constant arguments of generalized type. Nonlinear Anal 68:794–803

    Article  MathSciNet  MATH  Google Scholar 

  7. Akhmet MU (2009) Almost periodic solutions of the linear differential equation with piecewise constant argument. Discrete Impuls Syst Ser A Math Anal 16:743–753

    MathSciNet  MATH  Google Scholar 

  8. Akhmet M (2011) Nonlinear hybrid continuous/discrete time models. Atlantis, Amsterdam-Paris

    Book  Google Scholar 

  9. Akhmet MU (2012) Exponentially dichotomous linear systems of differential equations with piecewise constant argument Discontinuity. Nonlin Complex 1:1–6

    Article  Google Scholar 

  10. Akhmet MU, Aruğaslan D (2009) Lyapunov-Razumikhin method for differential equations with piecewise constant argument. Discrete Contin Dyn Syst 25(2):457–466

    Article  MathSciNet  MATH  Google Scholar 

  11. Akhmet MU, Büyükadalı C (2008) Periodic solutions of the system with piecewise constant argument in the critical case. Comput. Math. Appl. 56:2034–2042

    Article  MathSciNet  MATH  Google Scholar 

  12. Akhmet MU, Büyükadalı C (2010) Differential equations with state-dependent piecewise constant argument. Nonlin Anal 72:4200–4210

    Article  MATH  Google Scholar 

  13. Akhmet MU, Yılmaz E (2009) Hopfield-type neural networks systems equations with piecewise constant argument. Int J Qual Theory Differ Equat Appl 3(1–2):8–14

    MATH  Google Scholar 

  14. Akhmet MU, Aruğaslan D, Liu X (2008) Permanence of non-autonomous ratio-dependent predator-prey systems with piecewise constant argument of generalized type. Dyn Contin Discrete Impuls Syst Ser A 15:37–52

    MATH  Google Scholar 

  15. Akhmet MU, Aruğaslan D, Yılmaz E (2010) Stability in cellular neural networks with a piecewise constant argument. J Comput Appl Math 233:2365–2373

    Article  MathSciNet  MATH  Google Scholar 

  16. Akhmet MU, Aruğaslan D, Yılmaz E (2010) Stability analysis of recurrent neural networks with piecewise constant argument of generalized type. Neural Network 23:305–311

    Article  Google Scholar 

  17. Akhmet MU, Aruğaslan D, Yılmaz E (2011) Method of Lyapunov functions for differential equations with piecewise constant delay. J Comput Appl Math 235:4554–4560

    Article  MathSciNet  MATH  Google Scholar 

  18. Akhmetov MU, Perestyuk NA, Samoilenko AM (1983) Almost-periodic solutions of differential equations with impulse action. (Russian) Akad Nauk Ukrain SSR Inst Mat 49:26 (preprint)

    Google Scholar 

  19. Alonso A, Hong J (2001) Ergodic type solutions of differential equations with piecewise constant arguments. Int J Math Math Sci 28:609–619

    Article  MathSciNet  MATH  Google Scholar 

  20. Alonso A, Hong J, Obaya R (2000) Almost-periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences. Appl Math Lett 13:131–137

    Article  MathSciNet  MATH  Google Scholar 

  21. Aruğaslan D (2009) Dynamical systems with discontinuities and applications in biology and mechanics. METU, Ankara

    Google Scholar 

  22. Bao G, Wen S, Zeng Z (2012) Robust stability analysis of interval fuzzy Cohen-Grossberg neural networks with piecewise constant argument of generalized type. Neural Network 33:32–41

    Article  MATH  Google Scholar 

  23. Busenberg S, Cooke KL (1982) Models of vertically transmitted disease with sequential continuous dynamics. In: Lakshmikantham V (ed) Nonlinear phenomena in mathematical sciences. Academic, New York, pp 179–189

    Google Scholar 

  24. Büyükadalı C (2009) Nonlinear oscillations of differential equations with piecewise constant argument. METU Ankara

    Google Scholar 

  25. Cabada A, Ferreiro JB, Nieto JJ (2004) Green’s function and comparison principles for first order periodic differential equations with piecewise constant arguments. J Math Anal Appl 291:690–697

    Article  MathSciNet  MATH  Google Scholar 

  26. Carvalho LAV, Cooke KL (1988) A nonlinear equation with piecewise continuous argument. Differ Integral Equat 1(3):359–367

    MathSciNet  MATH  Google Scholar 

  27. Cooke KL, Györi I (1994) Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments. Comput Math Appl 28:81–92

    Article  MATH  Google Scholar 

  28. Cooke KL, Wiener J (1991) A survey of differential equation with piecewise continuous argument. Lecture notes in mathematics, vol 1475. Springer, Berlin, pp 1–15

    Google Scholar 

  29. Cooke KL, Turi J, Turner G (1991) Stabilization of hybrid systems in the presence of feedback delays. Preprint Series 906, Inst Math and Its Appls, University of Minnesota, Minneapolis, MN

    Google Scholar 

  30. Cooke KL, Wiener J (1987) An equation alternately of retarded and advanced type. Proc Amer Math Soc 99:726–732

    Article  MathSciNet  MATH  Google Scholar 

  31. Cooke KL, Wiener J (1984) Retarded differential equations with piecewise constant delays. J Math Anal Appl 99:265–297

    Article  MathSciNet  MATH  Google Scholar 

  32. Cooke KL, Wiener J (1986) Stability for linear equations with piecewise continuous delay. Comput Math Appl Ser A 12:695–701

    Article  MathSciNet  MATH  Google Scholar 

  33. Coppel WA (1978) Dichotomies in stability theory. Lecture notes in mathematics, Springer, New York

    MATH  Google Scholar 

  34. Corduneanu C (2004) Some remarks on functional equations with advanced-delayed operators. Computing anticipatory systems, CASYS’03-Sixth international conference, Liége, Belgium, 11-16 August 2003, American Institute of Physics, New York

    Google Scholar 

  35. Dai L (2008) Nonlinear dynamics of piecewise constant systems and implementation of piecewise constant arguments. World Scientific, Singapore

    Book  MATH  Google Scholar 

  36. Dai L, Singh MC (1994) On oscillatory motion of spring-mass systems subjected to piecewise constant forces. J Sound and Vibration 173:217–232

    Article  MATH  Google Scholar 

  37. Fink AM (1974) Almost-periodic Differential Equations. Lecture notes in mathematics, Springer, Berlin

    MATH  Google Scholar 

  38. Furumochi T (1992) Stability and oscillation for linear and nonlinear scalar equations with piecewise constant argument. Appl Anal 44:113–125

    Article  MathSciNet  MATH  Google Scholar 

  39. Gopalsamy K (1992) Stability and oscillation in delay differential equations of population dynamics. Kluwer Academic Publishers, Dordrecht

    Book  Google Scholar 

  40. Gopalsamy K (2004) Stability of artificial neural networks with impulses. Appl Math Comput 154:783–813

    Article  MathSciNet  MATH  Google Scholar 

  41. Gopalsamy K, Liu P (1998) Persistence and global stability in a population model. J Math Anal Appl 224:59–80

    Article  MathSciNet  MATH  Google Scholar 

  42. Grove EA, Ladas G, Zhang S (1997) On a nonlinear equation with piecewise constant argument. Comm Appl Nonlinear Anal 4:67–79

    MathSciNet  MATH  Google Scholar 

  43. Györi I (1991) On approximation of the solutions of delay differential equations by using piecewise constant argument. Int J Math Math Sci 14:111–126

    Article  MATH  Google Scholar 

  44. Györi I, Ladas G (1991) Oscillation theory of delay differential equations with applications. Oxford University Press, New York

    MATH  Google Scholar 

  45. Györi I, Hartung F (2002) Numerical approximation of neutral differential equations on infinite interval. J Difference Equat Appl 8:983–999

    Article  MATH  Google Scholar 

  46. Györi I, Hartung F (2008) On numerical approximation using differential equations with piecewise-constant arguments. Period Math Hungar 56: 55–69

    Article  MathSciNet  MATH  Google Scholar 

  47. Hahn W (1967) Stability of motion. Springer, Berlin

    MATH  Google Scholar 

  48. Halanay A, Wexler D (1968) Qualitative theory of impulsive systems. Edit Acad RPR, Bucuresti, (in Romanian)

    Google Scholar 

  49. Hale JK (1971) Functional differential equations. Springer, New-York

    Book  MATH  Google Scholar 

  50. Hale JK (1997) Theory of functional differential equations. Springer, New York

    Google Scholar 

  51. Hartman P (1982) Ordinary differential equations. Birkhäuser, Boston

    MATH  Google Scholar 

  52. Hong J, Obaya R, Sanz A (2001) Almost periodic type solutions of some differential equations with piecewise constant argument. Nonlinear Anal.: TMA 45:661–688

    Google Scholar 

  53. Huang YK (1989) On a system of differential equations alternately of advanced and delay type, differential equations and applications. Ohio University Press, Athens, p 455–465

    Google Scholar 

  54. Huang YK (1990) Oscillations and asymptotic stability of solutions of first order neutral differential equations with piecewise constant argument. J Math Anal Appl 149:70–85

    Article  MathSciNet  MATH  Google Scholar 

  55. Huang Z, Wang X, Gao F (2006) The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks. Phys Lett A 350:182–191

    Article  MATH  Google Scholar 

  56. Huang LH, Chen XX (2006) Oscillation of systems of neutral differential equations with piecewise constant argument. (Chinese), J Jiangxi Norm Univ Nat Sci Ed 30:470–474

    MathSciNet  MATH  Google Scholar 

  57. Huang Z, Xia Y, Wang X (2007) The existence and exponential attractivity of κ-almost periodic sequence solution of discrete time neural networks. Nonlinear Dynam 50:13–26

    Article  MathSciNet  MATH  Google Scholar 

  58. Krasovskii NN (1963) Stability of motion, applications of Lyapunov’s second method to differential systems and equations with delay. Stanford University Press, Stanford, California

    MATH  Google Scholar 

  59. Küpper T, Yuan R (2002) On quasi-periodic solutions of differential equations with piecewise constant argument. J Math Anal Appl 267:173–193

    Article  MathSciNet  MATH  Google Scholar 

  60. Ladas G (1989) Oscillations of equations with piecewise constant mixed arguments. Differential equations and applications. Ohio University Press, Athens, pp. 64–69

    Google Scholar 

  61. Ladas G, Partheniadis EC, Schinas J (1989) Oscillation and stability of second order differential equations with piecewise constant argument. Rad Mat 5:171–178

    MathSciNet  MATH  Google Scholar 

  62. Li H, Yuan R (2008) An affirmative answer to Gopalsamy and Liu’s conjecture in a population model. J Math Anal Appl 338:1152–1168

    Article  MathSciNet  MATH  Google Scholar 

  63. Li X, Wang Z (2004) Global attractivity for a logistic equation with piecewise constant arguments. Differences and differ Equat, Fields Inst Commun 42:215–222

    Google Scholar 

  64. Li H, Muroya Y, Yuan R (2009) A sufficient condition for global asymptotic stability of a class of logistic equations with piecewise constant delay. Nonlinear Anal: Real World Appl 10:244–253

    Article  MathSciNet  MATH  Google Scholar 

  65. Lin LC, Wang GQ (1991) Oscillatory and asymptotic behavior of first order nonlinear differential equations with retarded argument [t]. Chinese Sci Bull 36:889–891

    MathSciNet  MATH  Google Scholar 

  66. Liu P, Gopalsamy K (1999) Global stability and chaos in a population model with piecewise constant arguments. Appl Math Comput 101:63–88

    Article  MathSciNet  MATH  Google Scholar 

  67. Liu Y (2001) Global stability in a differential equation with piecewisely constant arguments. J Math Res Exposition 21:31–36

    MathSciNet  MATH  Google Scholar 

  68. Malkin IG (1944) Stability in the case of constantly acting disturbances. Prikl Mat 8:241–245

    MathSciNet  MATH  Google Scholar 

  69. Matsunaga H, Hara T, Sakata S (2001) Global attractivity for a logistic equation with piecewise constant argument. Nonlinear Differ Equat Appl 8:45–52

    Article  MathSciNet  MATH  Google Scholar 

  70. Minh NV, Dat TT (2007) On the almost automorphy of bounded solutions of differential equations with piecewise constant argument. J Math Anal Appl 326:165–178

    Article  MathSciNet  MATH  Google Scholar 

  71. Muroya Y (2002) Persistence, contractivity and global stability in logistic equations with piecewise constant delays. J Math Anal Appl 270:602–635

    Article  MathSciNet  MATH  Google Scholar 

  72. Myshkis AD (1977) On certain problems in the theory of differential equations with deviating argument. Russian Math. Surv 32:181–213

    Article  MATH  Google Scholar 

  73. Nieto JJ, Rodriguez-Lopez R (2005) Green’s function for second order periodic boundary value problems with piecewise constant argument. J Math Anal Appl 304:33–57

    Article  MathSciNet  MATH  Google Scholar 

  74. Papaschinopoulos G (1994) Exponential dichotomy, topological equivalence and structural stability for differential equations with piecewise constant argument. Analysis 14:239–247

    MathSciNet  MATH  Google Scholar 

  75. Papaschinopoulos G (1994) On asymptotic behavior of the solutions of a class of perturbed differential equations with piecewise constant argument and variable coefficients. J Math Anal Appl 185:490–500

    Article  MathSciNet  MATH  Google Scholar 

  76. Papaschinopoulos G, Schinas J (1992) Existence stability and oscillation of the solutions of first order neutral delay differential equations with piecewise constant argument. Appl Anal 44:99–111

    Article  MathSciNet  MATH  Google Scholar 

  77. Papaschinopoulos G, Stefanidou G, Efraimidis P (2007) Existence, uniqueness and asymptotic behavior of the solutions of a fuzzy differential equation with piecewise constant argument. Inform Sci 177:3855–3870

    Article  MathSciNet  MATH  Google Scholar 

  78. Partheniadis EC (1988) Stability and oscillation of neutral delay differential equations with piecewise constant argument. Differ Integral Equat 1:459–472

    MathSciNet  MATH  Google Scholar 

  79. Pinto M (2009) Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments. Math Comput Modeling 49:1750–1758

    Article  MathSciNet  MATH  Google Scholar 

  80. Pinto M (2011) Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems. J Differ Equat Appl 17:235–254

    Article  MathSciNet  MATH  Google Scholar 

  81. Razumikhin BS (1956) Stability of delay systems. Prikl Mat Mekh (Russian) 20:500–512

    Google Scholar 

  82. Rouche N, Habets P, Laloy M (1977) Stability theory by Liapunov’s direct method. Springer, New York

    Book  MATH  Google Scholar 

  83. Samoilenko AM, Perestyuk NA (1995) Impulsive differential equations. World Scientific, Singapore

    MATH  Google Scholar 

  84. Seifert G (1991) On an interval map associated with a delay logistic equation with discontinuous delays, Delay differential equations and dynamical systems (Claremont, CA, 1990), Lecture notes in Mathematics vol 1475. Springer, Berlin, pp. 243–249

    Google Scholar 

  85. Seifert G (2000) Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence. J Differ Equat 164:451–458

    Article  MathSciNet  MATH  Google Scholar 

  86. Shah SM, Wiener J (1983) Advanced differential equations with piecewise constant argument deviations. Internat J Math Sci 6:671–703

    Article  MathSciNet  MATH  Google Scholar 

  87. So JWH, Yu JS (1995) Global stability in a logistic equation with piecewise constant arguments. Hokkaido Math J 24:269–286

    Article  MathSciNet  MATH  Google Scholar 

  88. Wang G (2004) Existence theorem of periodic solutions for a delay nonlinear differential equation with piecewise constant arguments. J Math Anal Appl 298:298–307

    Article  MathSciNet  MATH  Google Scholar 

  89. Wang L, Yuan R, Zhang C (2009) Corrigendum to: “On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument”. [R.Yuan J (2005) Math Anal Appl 303:103–118], J Math Anal Appl 349(1):299-300

    Google Scholar 

  90. Wexler D (1966) Solutions périodiques et presque-périodiques des systémes d’équations différetielles linéaires en distributions. J Differ Equat 2:12–32

    Article  MathSciNet  MATH  Google Scholar 

  91. Wiener J (1983) Differential equations with piecewise constant delays. Trends in the theory and practice of nonlinear differential equations. Marcel Dekker, New York, pp 547–552

    Google Scholar 

  92. Wiener J (1984) Pointwise initial value problems for functional-differential equations. Differential equations, North-Holland, pp 571–580

    Google Scholar 

  93. Wiener J (1993) Generalized solutions of functional differential equations. World Scientific, Singapore

    MATH  Google Scholar 

  94. Wiener J, Aftabizadeh AR (1988) Differential equations alternately of retarded and advanced types. J Math Anal Appl 129:243–255

    Article  MathSciNet  MATH  Google Scholar 

  95. Wiener J, Shah SM (1987) Continued fractions arising in a class of functional-differential equations. J Math Phys Sci 21:527–543

    MathSciNet  MATH  Google Scholar 

  96. Xia Y, Huang Z, Han M (2007) Existence of almost periodic solutions for forced perturbed systems with piecewise constant argument. J Math Anal Appl 333:798–816

    Article  MathSciNet  MATH  Google Scholar 

  97. Yuan R (1999) Almost periodic solutions of a class of singularly perturbed differential equations with piecewise constant argument. Nonlinear Anal 37:841–859

    Article  MathSciNet  MATH  Google Scholar 

  98. Yuan R (2002) The existence of almost periodic solutions of retarded differential equations with piecewise constant argument. Nonlinear Anal 48:1013–1032

    Article  MathSciNet  MATH  Google Scholar 

  99. Yuan R (2005) On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument. J Math Anal Appl 303:103–118

    Article  MathSciNet  MATH  Google Scholar 

  100. Yang X (2006) Existence and exponential stability of almost periodic solution for cellular neural networks with piecewise constant argument. Acta Math Appl Sin 29:789–800

    MathSciNet  Google Scholar 

  101. Yuan R, Hong J (1996) Almost periodic solutions of differential equations with piecewise constant argument. Analysis 16:171–180

    MathSciNet  MATH  Google Scholar 

  102. Yuan R, Hong J (1997) The existence of almost periodic solutions for a class of differential equations with piecewise constant argument. Nonlinear Anal 28(8):1439–1450

    Article  MathSciNet  MATH  Google Scholar 

  103. Yang P, Liu Y, Ge W (2006) Green’s function for second order differential equations with piecewise constant argument. Nonlinear Anal 64:1812–1830

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Middle East Technical University (METU) Üniversiteler Mah. Dumlupinar, Ankara, Turkey

    Marat Akhmet

  2. Department of Mathematics, Adnan Menderes University Merkez Kampüsü Aytepe Mevkii, Aydın, Turkey

    Enes Yılmaz

Authors
  1. Marat Akhmet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Enes Yılmaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Akhmet, M., Yılmaz, E. (2014). Differential Equations with Piecewise Constant Argument of Generalized Type. In: Neural Networks with Discontinuous/Impact Activations. Nonlinear Systems and Complexity, vol 9. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8566-7_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-8566-7_2

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-8565-0

  • Online ISBN: 978-1-4614-8566-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation