# Nonoscillation Intervals for *n*-th-Order Equations

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## Abstract

In Chap. 17, *n*-th-order functional differential equations are considered. In order to create a concept of nonoscillation for functional differential equations which will be an analogue of nonoscillation theory for ordinary differential equations, we define a nonoscillation interval for the *n*-th-order homogeneous functional differential equation as interval, where each nontrivial solution can have at most *n*−1 zeros, counting every zero with its multiplicity. The study of many classical questions of the qualitative theory of *n*-th-order equations such as existence and uniqueness of solutions of the interpolation boundary value problems, positivity or corresponding regular behavior of their Green’s functions, maximum principles and stability, was considered in the general framework based on the notion of nonoscillation intervals of corresponding linear functional differential equations. It is demonstrated that if the Wronskian of the fundamental system is not equal to zero at the initial point of the interval, then the nonoscillation interval exists for functional differential equations; this is true for a particular case of the *n*-th-order functional differential equations with Volterra operators (delay equations). For second-order functional differential equations, nonvanishing Wronskian is equivalent to the Sturm theorem on separation of zeros (between two adjacent zeros of any nontrivial solution there is a zero of every other solution). The analogue of the classical Mammana theorem, establishing necessary and sufficient nonoscillation conditions through nonvanishing normal chains of the Wronskians, is proven for *n*-th-order functional differential equations.

For an *n*-th-order linear ordinary differential equation the classical Levin-Chichkin theorems claims that Green’s functions of all interpolation (De la Vallee-Poussin) problems behave regularly (preserve their signs in corresponding zones) on the nonoscillation interval. Under corresponding sign conditions on the relevant operators, this theorem is extended to *n*-th-order functional differential equations. Several possibilities to avoid these sign conditions are proposed, and various explicit tests for positivity and negativity of Green’s functions of two and multipoint boundary value problems are obtained.

## Keywords

Nonoscillation Functional Differential Equations Adjacent Zeros Volterra Operator Wronskian## References

- 20.Azbelev, N.V.: The zeros of the solutions of a second order linear differential equation with retarded argument. Differ. Uravn.
**7**, 1147–1157 (1971), 1339 (in Russian) MathSciNetzbMATHGoogle Scholar - 26.Azbelev, N.V., Domoshnitsky, A.: On the question of linear differential inequalities. I. Differ. Equ.
**27**, 257–263 (1991) MathSciNetzbMATHGoogle Scholar - 27.Azbelev, N.V., Domoshnitsky, A.: On the question of linear differential inequalities. II. Differ. Equ.
**27**, 641–647 (1991) MathSciNetzbMATHGoogle Scholar - 28.Azbelev, N.V., Maksimov, V.P., Rakhmatullina, L.F.: Introduction to Theory of Functional-Differential Equations. Nauka, Moscow (1991) (in Russian) zbMATHGoogle Scholar
- 37.Beesack, P.R.: On the Green’s function of an
*n*-point boundary value problem. Pac. J. Math.**12**, 801–812 (1962) MathSciNetzbMATHGoogle Scholar - 82.Berkowitz, K., Domoshnitsky, A., Maghakyan, A.: About functional differential generalization of Burger’s equation. Funct. Differ. Equ.
**17**, 53–60 (2010) MathSciNetGoogle Scholar - 93.Chichkin, E.S.: Theorem about differential inequality for multipoint boundary value problems. Izv. Vysš. Učebn. Zaved., Mat.
**2**, 170–179 (1962) Google Scholar - 100.Deift, V.A.: Conditions of nonoscillation for linear homogeneous differential equations with delayed argument. Differential Equations
**10**, 1957–1963 (1974) MathSciNetzbMATHGoogle Scholar - 101.Deift, V.A.: Condition of nonoscillation for linear homogeneous differential equations. Thesis, Alma-Ata (1977) (in Russian) Google Scholar
- 110.Domoshnitsky, A.: Extension of Sturm’s theorem to equations with time-lag. Differ. Uravn.
**19**, 1475–1482 (1983) MathSciNetGoogle Scholar - 112.Domoshnitsky, A.: Preserving the sign of the Green function of a two-point boundary value problem for an
*n*th-order functional-differential equation. Differ. Equ.**25**, 666–669 (1989) MathSciNetGoogle Scholar - 114.Domoshnitsky, A.: Factorization of a linear boundary value problem and the monotonicity of the Green operator. Differ. Equ.
**28**, 323–327 (1992) MathSciNetGoogle Scholar - 116.Domoshnitsky, A.: Sturm’s theorem for equation with delayed argument. Georgian Math. J.
**1**, 267–276 (1994) zbMATHCrossRefGoogle Scholar - 117.Domoshnitsky, A.: Unboundedness of solutions and instability of second order equations with delayed argument. Differ. Integral Equ.
**14**, 559–576 (2001) MathSciNetzbMATHGoogle Scholar - 118.Domoshnitsky, A.: Wronskian of fundamental system of delay differential equations. Funct. Differ. Equ.
**9**, 353–376 (2002) MathSciNetzbMATHGoogle Scholar - 119.Domoshnitsky, A.: About asymptotic and oscillation properties of the Dirichlet problem for delay partial differential equations. Georgian Math. J.
**10**, 495–502 (2003) MathSciNetzbMATHGoogle Scholar - 120.Domoshnitsky, A.: Maximum principle for functional equations in the space of discontinuous functions of three variables. J. Math. Anal. Appl.
**329**, 238–267 (2007) MathSciNetzbMATHCrossRefGoogle Scholar - 124.Domoshnitsky, A.: Nonoscillation interval for
*n*-th order functional differential equations. Nonlinear Anal.**71**, e2449–e2456 (2009) MathSciNetCrossRefGoogle Scholar - 126.Domoshnitsky, A., Drakhlin, M.: Nonoscillation of first order impulse differential equations with delay. J. Math. Anal. Appl.
**206**, 254–269 (1997) MathSciNetzbMATHCrossRefGoogle Scholar - 127.Domoshnitsky, A., Drakhlin, M., Litsyn, E.: On
*N*th order functional-differential equations with impulses. Mem. Differ. Equ. Math. Phys.**12**, 50–56 (1997) MathSciNetzbMATHGoogle Scholar - 128.Domoshnitsky, A., Drakhlin, M., Litsyn, E.: On boundary value problems for
*N*th order functional differential equations with impulses. Adv. Math. Sci. Appl.**8**, 987–996 (1998) MathSciNetzbMATHGoogle Scholar - 158.Gantmakher, F.R., Kreĭn, M.R.: Oscillatory Matrices and Kernels and Small Oscillations of Mechanical Systems. GosTechIzdat, Moscow, Leningrad (1950) Google Scholar
- 192.Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, New York (1991) Google Scholar
- 203.Hartman, P.: Ordinary Differential Equations. Wiley, New York, London, Sydney (1964) zbMATHGoogle Scholar
- 204.Hartman, P., Winter, A.: On non-conservative linear oscillators of low frequency. Am. J. Math.
**70**, 529–539 (1948) zbMATHCrossRefGoogle Scholar - 220.Kiguradze, I.T., Chanturia, T.A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic, Dordrecht, Boston, London (1993) zbMATHCrossRefGoogle Scholar
- 241.Labovskii, S.M.: Condition of nonvanishing of Wronskian of fundamental system of linear equation with delayed argument. Differ. Uravn.
**10**, 426–430 (1974) (in Russian) MathSciNetGoogle Scholar - 248.Ladde, G.S., Lakshmikantham, V., Zhang, B.G.: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York (1987) Google Scholar
- 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 - 282.Mammana, G.: Decomposizione delle espressioni differenziali omogenee in prodotto di fattori simbolici e applicazione relativa allo studion delle equazioni differenzi ali lineari. Math. Z.
**33**, 186–231 (1931) MathSciNetCrossRefGoogle Scholar - 289.Myshkis, A.D.: Linear Differential Equations with Retarded Argument. Nauka, Moscow (1972) (in Russian) zbMATHGoogle Scholar
- 308.Polia, G.: On the mean-value theorem corresponding to a given linear homogeneous differential equations. Trans. Am. Math. Soc.
**24**, 312–324 (1924) CrossRefGoogle Scholar - 314.Sansone, G.: Equazioni Differenziali nel Campo Reale, 2nd edn. Zanichelli, Bologna (1948) (in Italian) Google Scholar