Systems of Functional Differential Equations on Finite Intervals

  • Ravi P. Agarwal
  • Leonid Berezansky
  • Elena Braverman
  • Alexander Domoshnitsky


In Chap. 16, boundary value problems for system of linear functional differential equations are studied, and results on positivity of Green’s matrices are obtained. A necessary and sufficient condition for nonnegativity of all the entries of the Cauchy matrix for linear systems of ordinary differential equations of the first order (obtained by Wazewski) is nonpositivity of off-diagonal coefficients. In the first part of this chapter, extension of this result to various boundary value problems for functional differential equations is obtained, and all the results include an analogue of Wazewski’s sign condition.

The main results of the chapter are based on the idea to construct a first-order functional differential equation for the r-th component of the solution vector. Even in the case of systems of ordinary differential equations, this equation is functional differential, with a sufficiently complicated operator. Next, the assertions on differential inequalities for first-order functional differential equations from Chap.  15 are used. This approach explains the importance of considering first-order equations in the operator form, which describes only properties of the operators and does not assume their concrete form. Assertions on differential inequalities for the r-th component of the solution vector or, in the other terminology, about positivity of the entries of the r-th row of Green’s matrices are obtained in a form of theorems on several equivalences, one of the equivalent properties is nonoscillation. It is demonstrated that many tests obtained from these general theorems for one of the solution’s components are unimprovable in a certain sense.


Scalar Functional Differential Equations Cauchy Matrix Wazewski Solution Vector Nonpositivity 
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  1. 2.
    Agarwal, R.P., Bohner, M., Domoshnitsky, A., Goltser, Y.: Floquet theory and stability of nonlinear integro-differential equations. Acta Math. Hung. 109, 305–330 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 5.
    Agarwal, R.P., Domoshnitsky, A.: On positivity of several components of solution vector for systems of linear functional differential equations. Glasg. Math. J. 52, 115–136 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 6.
    Agarwal, R.P., Domoshnitsky, A., Goltser, Ya.: Stability of partial functional integro-differential equations. J. Dyn. Control Syst. 12(1), 1–31 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 28.
    Azbelev, N.V., Maksimov, V.P., Rakhmatullina, L.F.: Introduction to Theory of Functional-Differential Equations. Nauka, Moscow (1991) (in Russian) zbMATHGoogle Scholar
  5. 29.
    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) zbMATHGoogle Scholar
  6. 38.
    Bellman, R.: Methods of Nonlinear Analysis. Academic Press, New York, London (1973) zbMATHGoogle Scholar
  7. 87.
    Campbell, S.A.: Delay independent stability for additive neural networks. Differ. Equ. Dyn. Syst. 9, 115–138 (2001) zbMATHGoogle Scholar
  8. 102.
    de La Vallee Poussin, Ch.J.: Sur l’equation differentielle lineaire du second ordre. J. Math. Pures Appl. 8(9), 125–144 (1929) Google Scholar
  9. 113.
    Domoshnitsky, A.: Componentwise applicability of Chaplygin’s theorem to a system of linear differential equations with delay. Differ. Equ. 26, 1254–1259 (1990) MathSciNetGoogle Scholar
  10. 115.
    Domoshnitsky, A.: New concept in the study of differential inequalities. Funct. Differ. Equ. 1, 53–59 (1993) Google Scholar
  11. 131.
    Domoshnitsky, A., Goltser, Ya.: Hopf bifurcation of integro-differential equations. In: Proceedings of the 6th Colloquium on the Qualitative Theory of Differential Equations (Szeged, 1999), No. 3, 11 pp. (electronic). Proc. Colloq. Qual. Theory Differential Equations. Electron. J. Qual. Theory Differential Equations (2000) Google Scholar
  12. 133.
    Domoshnitsky, A., Goltser, Ya.: Approach to study of bifurcations and stability of integro-differential equations. Lyapunov’s methods in stability and control. Math. Comput. Model. 36, 663–678 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 134.
    Domoshnitsky, A., Goltser, Ya.: On stability and boundary value problems for integro-differential equations. Nonlinear Anal. 63, e761–e767 (2005) CrossRefzbMATHGoogle Scholar
  14. 135.
    Domoshnitsky, A., Goltser, Ya.: Positivity of solutions to boundary value problems for infinite functional differential systems. Math. Comput. Model. 45, 1395–1404 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 136.
    Domoshnitsky, A., Koplatadze, R.: On a boundary value problem for integro-differential equations on the halfline. Nonlinear Anal. 72, 836–846 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 137.
    Domoshnitsky, A., Litsyn, E.: Positivity of the Green’s matrix of an infinite systems. Panam. Math. J. 16, 27–40 (2006) MathSciNetzbMATHGoogle Scholar
  17. 192.
    Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, New York (1991) Google Scholar
  18. 207.
    Hofbauer, J., So, J.W.-H.: Diagonal dominance and harmless off-diagonal delays. Proc. Am. Math. Soc. 128, 2675–2682 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 219.
    Kiguradze, I.: Boundary value problems for systems of ordinary differential equations. J. Sov. Math. 43, 2259–2339 (1988) CrossRefGoogle Scholar
  20. 222.
    Kiguradze, I., Pu̇z̆a, B.: On boundary value problems for systems of linear functional differential equations. Czechoslov. Math. J. 47(2), 341–373 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 223.
    Kiguradze, I., Pu̇z̆a, B.: Boundary Value Problems for Systems of Linear Functional Differential Equations. FOLIA, Masaryk University, Brno, Czech Republic (2003) zbMATHGoogle Scholar
  22. 233.
    Krasnosel’skii, M.A., Vainikko, G.M., Zabreiko, P.P., Rutitskii, Ja.B., Stezenko, V.Ja.: Approximate Methods for Solving Operator Equations. Nauka, Moscow (1969) (in Russian) Google Scholar
  23. 250.
    Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities. Academic Press, New York (1969) zbMATHGoogle Scholar
  24. 256.
    Levin, A.J.: Nonoscillation of solution of the equation x (n)+p n−1(t)x (n−1)+⋯+p 0(t)x=0. Usp. Mat. Nauk 24, 43–96 (1969) Google Scholar
  25. 266.
    Litsyn, E.: On the general theory of linear functional-differential equations. Differ. Equ. 24, 638–646 (1988) MathSciNetGoogle Scholar
  26. 267.
    Litsyn, E.: On the formula for general solution of infinite system of functional-differential equations. Funct. Differ. Equ. (Isr. Semin.) 2, 111–121 (1994) (1995) MathSciNetzbMATHGoogle Scholar
  27. 278.
    Luzin, N.N.: About method of approximate integration of Acad. S.A. Chaplygin. Usp. Mat. Nauk 6, 3–27 (1951) MathSciNetzbMATHGoogle Scholar
  28. 298.
    Persidskii, K.P.: On the stability of solutions of denumerable systems of differential equations. Izv. Akad. Nauk Kazah. SSR Ser. Math. Mech. 56(2), 3–35 (1948) (in Russian) MathSciNetGoogle Scholar
  29. 299.
    Persidskii, K.P.: Infinite systems of differential equations. Izv. Akad. Nauk Kazah. SSR Ser. Math. Mech. 4(8), 3–11 (1956) (in Russian) Google Scholar
  30. 313.
    Samoilenko, A.M., Teplinski, Yu.V.: Countable Systems of Differential Equations. VSP, Utrecht, Boston (2003) CrossRefzbMATHGoogle Scholar
  31. 325.
    Tchaplygin, S.A.: New Method of Approximate Integration of Differential Equations. GTTI, Moscow, Leningrad (1932) (in Russian) Google Scholar
  32. 332.
    Wazewski, T.: Systèmes des équations et des inégalités différentielles aux ordinaires aux deuxièmes membres monotones et leurs applications. Ann. Pol. Math. 23, 112–166 (1950) MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Ravi P. Agarwal
    • 1
  • Leonid Berezansky
    • 2
  • Elena Braverman
    • 3
  • Alexander Domoshnitsky
    • 4
  1. 1.Department of MathematicsTexas A&M University—KingsvilleKingsvilleUSA
  2. 2.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  3. 3.Department of MathematicsUniversity of CalgaryCalgaryCanada
  4. 4.Department of Computer Sciences and MathematicsAriel University Center of SamariaArielIsrael

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