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Neutral Differential Equations

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Nonoscillation Theory of Functional Differential Equations with Applications

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Abstract

Chapter 6 deals with nonoscillation and oscillation properties of scalar linear neutral differential equations. There are two kinds of neutral equations, one of them can be integrated leading to a term with a concentrated delay and an integral term; the second type which is considered in this chapter has a derivative involved both without a delay and with one or several delays. The study of these equations is based on the functional properties of the linear operator of inner superposition (composition operator).

The main result of the chapter is the equivalence of the nonoscillation of the equation and the existence of a positive solution for a specially constructed nonlinear operator inequality. This result is applied here to obtain explicit nonoscillation conditions and prove comparison theorems. The second auxiliary result is the equivalence of oscillation properties of the neutral equation and a specially constructed equation with an infinite number of delays, such equations were considered in Chap. 4. This method allows to deduce sufficient oscillation conditions for neutral equations. This chapter also presents nonoscillation conditions and comparison results for neutral equations with positive and negative coefficients and results on existence of slowly oscillating solutions.

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References

  1. Azbelev, N.V., Berezansky, L.M., Simonov, P.M., Chistykov, A.V.: Stability of linear systems with time-lag. Differ. Equ. 23, 493–500 (1987)

    MATH  Google Scholar 

  2. Azbelev, N.V., Berezansky, L.M., Simonov, P.M., Chistykov, A.V.: Stability of linear systems with time-lag. Differ. Equ. 27, 383–388, 1165–1172 (1991)

    MATH  Google Scholar 

  3. Azbelev, N.V., Berezansky, L.M., Simonov, P.M., Chistykov, A.V.: Stability of linear systems with time-lag. Differ. Equ. 29, 153–160 (1993)

    Google Scholar 

  4. Azbelev, N.V., Maksimov, V.P., Rakhmatullina, L.F.: Introduction to the Theory of Linear Functional-Differential Equations. Advanced Series in Mathematical Science and Engineering, vol. 3. World Federation Publishers Company, Atlanta (1995)

    MATH  Google Scholar 

  5. Berezansky, L., Braverman, E.: On oscillation of a differential equation with infinite number of delays. Z. Anal. Anwend. 21, 803–816 (2002)

    MathSciNet  MATH  Google Scholar 

  6. Berezansky, L., Braverman, E.: Non-oscillation properties of linear neutral differential equations. Funct. Differ. Equ. 9, 275–288 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Berezansky, L., Braverman, E.: Oscillation criteria for linear neutral differential equations. J. Math. Anal. Appl. 286, 601–617 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berezansky, L., Larionov, A.: Positivity of the Cauchy matrix of a linear functional-differential equation. Differ. Equ. 24, 1221–1230 (1988)

    MathSciNet  Google Scholar 

  9. Candan, T., Dahiya, R.S.: Positive solutions of first-order neutral differential equations. Appl. Math. Lett. 22, 1266–1270 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, M.P., Yu, J.S., Huang, L.H.: Oscillation of first order neutral differential equations. J. Math. Anal. Appl. 185, 288–301 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Domshlak, Y.: Properties of delay differential equations with oscillating coefficients. Funct. Differ. Equ. (Isr. Semin.) 2, 59–68 (1994)

    MathSciNet  MATH  Google Scholar 

  12. Domshlak, Y., Kvinikadze, G., Stavroulakis, I.P.: Sturmian comparison method: the version for first order neutral differential equations. Math. Inequal. Appl. 5, 247–256 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Drakhlin, M.E.: Inner superposition operator in spaces of integrable functions. Izv. Vysš. Učebn. Zaved., Mat. 5, 18–24 (1986), 88 (in Russian)

    MathSciNet  Google Scholar 

  14. Drakhlin, M.E., Plyshevskaya, T.K.: On the theory of functional-differential equations. Differ. Uravn. 14, 1347–1361 (1978) (in Russian)

    MathSciNet  MATH  Google Scholar 

  15. Erbe, L.H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995)

    Google Scholar 

  16. Gopalsamy, K.: Stability and Oscillation in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht, Boston, London (1992)

    Google Scholar 

  17. Gusarenko, S.A., Domoshnitskii, A.I.: Asymptotic and oscillation properties of the first order linear scalar functional differential equations. Differ. Equ. 25, 1480–1491 (1989)

    MathSciNet  MATH  Google Scholar 

  18. Győri, I.: Oscillation and comparison results in neutral differential equations and their applications to the delay logistic equation. Comput. Math. Appl. 18, 893–906 (1989)

    Article  MathSciNet  Google Scholar 

  19. Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, New York (1991)

    Google Scholar 

  20. Hajnalka, P., Karsai, J.: Positive solutions of neutral delay differential equation. Novi Sad J. Math. 32, 95–108 (2002)

    MathSciNet  MATH  Google Scholar 

  21. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993)

    MATH  Google Scholar 

  22. He, W.S., Li, W.-T.: Nonoscillation and oscillation for neutral differential equations with positive and negative coefficients. Int. J. Appl. Math. 6, 183–198 (2001)

    MathSciNet  MATH  Google Scholar 

  23. Jaros, J.: An oscillation test for a class of linear neutral differential equations. J. Math. Anal. Appl. 159, 406–411 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kolmanovskii, V., Myshkis, A.D.: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic, Dordrecht (1999)

    MATH  Google Scholar 

  25. Kurbatov, V.G.: Functional Differential Operators and Equations. Kluwer Academic, Dordrecht (1999)

    MATH  Google Scholar 

  26. Ladas, G., Schults, S.W.: On oscillations of neutral equations with mixed arguments. Hiroshima Math. J. 19, 409–429 (1989)

    MathSciNet  MATH  Google Scholar 

  27. Lalli, B.S., Zhang, B.G.: Oscillation of first order neutral differential equations. Appl. Anal. 39, 265–274 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lalli, B.S., Zhang, B.G.: Oscillation and nonoscillation of some neutral differential equations of odd order. Int. J. Math. Math. Sci. 15, 509–515 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, X.Z., Yu, J.S., Zhang, B.G.: Oscillation and nonoscillation for a class of neutral differential equations. Panam. Math. J. 3, 23–32 (1993)

    MathSciNet  MATH  Google Scholar 

  30. Lu, W.D.: Nonoscillation and oscillation for first order nonlinear neutral equations. Funkc. Ekvacioj 37, 383–394 (1994)

    MATH  Google Scholar 

  31. Lu, W.D.: Nonoscillation and oscillation of first order neutral equations with variable coefficients. J. Math. Anal. Appl. 181, 803–815 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  32. Luo, Z.G., Shen, J.H.: Oscillation and nonoscillation of neutral differential equations with positive and negative coefficients. Czechoslov. Math. J. 54(129), 79–93 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Philos, Ch.G., Purnaras, I.K.: Asymptotic properties, nonoscillation, and stability for scalar first order linear autonomous neutral delay differential equations. Electron. J. Differ. Equ. 2004(3) (2004), 17 pp.

    Google Scholar 

  34. Philos, C.G., Purnaras, I.K., Sficas, Y.G.: Oscillations in higher-order neutral differential equations. Can. J. Math. 45, 132–158 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shen, J.H.: Oscillation and existence of positive solutions for neutral differential equations. Ann. Differ. Equ. 12, 191–201 (1996)

    MATH  Google Scholar 

  36. Tanaka, S.: Oscillation of solutions of first-order neutral differential equations. Hiroshima Math. J. 32, 79–85 (2002)

    MathSciNet  MATH  Google Scholar 

  37. Yu, J.S., Chen, M.P., Zhang, H.: Oscillation and nonoscillation in neutral equations with integrable coefficients. Comput. Math. Appl. 35, 65–71 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yu, J.S., Wang, Y.C.: Nonoscillation of a neutral delay differential equation. Rad. Mat. 8, 127–133 (1992/1996)

    Google Scholar 

  39. Zhang, B.G., Gopalsamy, K.: Oscillation and nonoscillation of a class of neutral equations. In: World Congress of Nonlinear Analysts, I–IV 1992, Tampa, FL, 1992, pp. 1515–1522. de Gruyter, Berlin (1996)

    Chapter  Google Scholar 

  40. Zhang, B.G., Gopalsamy, K.: Oscillation and comparison of a class of neutral equations. Panam. Math. J. 4, 63–75 (1994)

    MathSciNet  MATH  Google Scholar 

  41. Zhang, B.G., Yu, J.S.: Oscillation and nonoscillation for neutral differential equations. J. Math. Anal. Appl. 172, 11–23 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University—Kingsville, Kingsville, USA

    Ravi P. Agarwal

  2. Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Leonid Berezansky

  3. Department of Mathematics, University of Calgary, Calgary, Canada

    Elena Braverman

  4. Department of Computer Sciences and Mathematics, Ariel University Center of Samaria, Ariel, Israel

    Alexander Domoshnitsky

Authors
  1. Ravi P. Agarwal
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  2. Leonid Berezansky
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  3. Elena Braverman
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  4. Alexander Domoshnitsky
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Agarwal, R.P., Berezansky, L., Braverman, E., Domoshnitsky, A. (2012). Neutral Differential Equations. In: Nonoscillation Theory of Functional Differential Equations with Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3455-9_6

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