Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T07:09:04.908Z Has data issue: false hasContentIssue false
  • English
  • Français

ON POSITIVITY OF SEVERAL COMPONENTS OF SOLUTION VECTOR FOR SYSTEMS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

Published online by Cambridge University Press:  04 December 2009

RAVI P. AGARWAL  and
ALEXANDER DOMOSHNITSKY
RAVI P. AGARWAL
Affiliation:
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA and Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia e-mail: agarwal@fit.edu
ALEXANDER DOMOSHNITSKY
Affiliation:
Department of Mathematics and Computer Sciences, The Ariel University Center of Samaria, 44837 Ariel, Israel e-mail: adom@ariel.ac.il
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the classical theorems about lower and upper vector functions for systems of linear differential equations very heavy restrictions on the signs of coefficients are assumed. These restrictions in many cases become necessary if we wish to compare all the components of a solution vector. The formulas of the integral representation of the general solution explain that these theorems claim actually the positivity of all elements of the Green's matrix. In this paper we define a principle of partial monotonicity (comparison of only several components of the solution vector), which assumes only the positivity of elements in a corresponding row of the Green's matrix. The main theorem of the paper claims the equivalence of positivity of all elements in the nth row of the Green's matrices of the initial and two other problems, non-oscillation of the nth component of the solution vector and a corresponding assertion about differential inequality of the de La Vallee Poussin type. Necessary and sufficient conditions of the partial monotonicity are obtained. It is demonstrated that our sufficient tests of positivity of the elements in the nth row of the Cauchy matrix are exact in corresponding cases. The main idea in our approach is a construction of an equation for the nth component of the solution vector. In this sense we can say that an analog of the classical Gauss method for solving systems of functional differential equations is proposed in the paper.

Keywords

34K11
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 1 , January 2010 , pp. 115 - 136
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Agarwal, R. P. and Domoshnitsky, A., Nonoscillation of the first order differential equations with unbounded memory for stabilization by control signal, Appl. Math. Comput. 173 (2006), 177–195.Google Scholar
2.Azbelev, N. V., Maksimov, V. P. and Rakhmatullina, L. F., Introduction to the theory of functional differential equations, Advanced Series in Math. Science and Engineering 3 (World Federation Publisher Company, Atlanta, GA, 1995).Google Scholar
3.Campbell, S. A., Delay independent stability for additive neural networks, Differential Equations Dynam. Syst. 9 (3–4) (2001), 115–138.Google Scholar
4.Corduneanu, C.. Integral equations and applications (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
5.Diblik, J. and Svoboda, Z., An existence criterion of positive solutions of p-type retarded functional differential equations. J. Comput. Appl. Math. 147 (2002), 315–331.CrossRefGoogle Scholar
6.Domoshnitsky, A. and Goltser, Ya., One approach to study stability of integro–differential equations, Nonlinear Anal. 47 (2001), 3885–3896.CrossRefGoogle Scholar
7.Domoshnitsky, A. and Sheina, M. V., Nonnegativity of Cauchy matrix and stability of systems with delay, Differentsial'nye uravnenija 25 (1989), 201–208.Google Scholar
8.Drozdov, A. D. and Kolmanovskii, V. B., Stability in viscoelasticity (North Holland, Amsterdam, 1994).Google Scholar
9.Erbe, L. H., Kong, Q. and Zhang, B. G., Oscillation theory for functional differential equations (Marcel Dekker, New York, 1995).Google Scholar
10.Golden, J. M. and Graham, G. A. C., Boundary value problems in linear viscoelasticity (Springer-Verlag, Berlin, 1988).CrossRefGoogle Scholar
11.Gyori, I., Interaction between oscillation and global asymptotic stability in delay differential equations, Differential Integr. Equations 3 (1990), 181–200.CrossRefGoogle Scholar
12.Gyori, I. and Hartung, F., Fundamental solution and asymptotic stability of linear delay differential equations, Dyn. Continuous, Discrete Impulsive Syst. 13 (2) (2006), 261–288.Google Scholar
13.Gyori, I. and Ladas, G., Oscillation theory of delay differential equations (Clarendon, Oxford, 1991).CrossRefGoogle Scholar
14.Hino, Y., Murakami, S. and Naito, T., Functional differential equations with infinite delay, Lecture Notes in Mathematics, 1473 (Springer-Verlag, Berlin, 1991).CrossRefGoogle Scholar
15.Hofbauer, J. and So, J. W.-H., Diagonal dominance and harmless off-diagonal delays, Proc. Amer. Math. Soc. 128 (2000), 2675–2682.CrossRefGoogle Scholar
16.Kantorovich, L. V., Vulich, B. Z. and Pinsker, A. G., Functional analysis in semi-ordered spaces (GosTechIzdat, Moscow, 1950 (in Russian).Google Scholar
17.Kiguradze, I.. Boundary value problems for systems of ordinary differential equations. Itogi Nauki Tech., Ser. Sovrem., Probl. Mat., Novejshie Dostizh. 30 (1987), 3–103 (in Russian). English transl.: J. Sov. Math. 43(2) (1988), 2259–2339.Google Scholar
18.Kiguradze, I. and Puza, B., On boundary value problems for systems of linear functional differential equations, Czechoslovak Math. J. 47 (2) (1997), 341–373.CrossRefGoogle Scholar
19.Kiguradze, I. and Puza, B., Boundary value problems for systems of linear functional differential equations (FOLIA, Brno, Czech Republic, 2002).Google Scholar
20.Krasnosel'skii, M. A., Vainikko, G. M., Zabreiko, P. P., Rutitskii, Ja. B. and Stezenko, V. Ja., Approximate methods for solving operator equations (Nauka, Moscow, 1969) (in Russian).Google Scholar
21.Lakshmikantham, V. and Leela, S., Differential and integral inequalities (Academic Press, New York, 1969).Google Scholar
22.Lakshmikantham, V., Wen, L. and Zhang, B., Theory of differential equations with unbounded delay (Kluwer Academic Publishers, Dordrecht, 1994).CrossRefGoogle Scholar
23.Levin, A. Ju., Non-oscillation of solution of the equation x (n) + p n−1(t)x (n−1) + ··· + p 0(t)x = 0. Uspekhi Math. Nauk 24 (1969), 43–96.Google Scholar
24.Luzin, N. N., About method of approximate integration of acad. S. A. Chaplygin. Uspekhi Math. Nauk 6(6) (1951), 3–27.Google Scholar
25.Tchaplygin, S. A., New method of approximate integration of differential equations (GTTI, Moscow, 1932).Google Scholar
26.de La Vallee Poussin, Ch. J., Sur l'equation differentielle lineaire du second ordre, J. Math. Pura et Appl. 8 (9) (1929), 125–144.Google Scholar
27.Wazewski, T., Systemes des equations et des inegalites differentielled aux deuxieme membres et leurs applications, Ann. Polon. Math. 23 (1950), 112–166.Google Scholar