On Connection between Second-Order Delay Differential Equations and Integrodifferential Equations with Delay
© Leonid Berezansky et al. 2010
Received: 5 October 2009
Accepted: 11 November 2009
Published: 17 November 2009
1. Introduction and Preliminaries
The second order delay differential equation
attracts the attention of many mathematicians because of their significance in applications.
In particular, Minorsky  in 1962 considered the problem of stabilizing the rolling of a ship by an "activated tanks method" in which ballast water is pumped from one position to another. To solve this problem, he constructed several delay differential equations with damping described by (1.1).
Despite the obvious importance in applications, there are only few papers on delay differential equations with damping.
One of the methods used to study (1.1) is transforming the second-order delay differential equation to a first-order differential or integrodifferential equations with delay. A transformation of the type
has a nonoscillatory solution, then (1.1) has a nonoscillatory solution, too.
Now we will briefly describe the scheme of another transformation, different from the one used in  (in this explanation we omit exact assumptions related to the functions used, which are formulated later).
Consider an auxiliary equation
with the initial condition
Then (1.1) is transformed into the integrodifferential equation with delay
Since (1.11) is a result of transforming (1.1), qualitative properties of (1.11) such as the existence and uniqueness of solutions, oscillation and nonoscillation, stability and asymptotic behavior can imply similar qualitative properties of (1.1).
The advantage of the suggested method in comparison with the method used in  is that a second order delay equation is reduced to one first-order integrodifferential delay equation while in  a second-order equation is reduced to a system of a nonlinear inequality and a linear delay equation.
This paper aims to investigate the problems of the existence, uniqueness and solution representation of (1.11). Problems related to oscillation/nonoscillation, stability and applications to second-order equations will be studied in our forthcoming papers. Throughout this paper, will denote the matrix or vector norm used.
2. Main Results
Together with (1.11) we consider an initial condition
We will assume that the following conditions hold:
A function is called a solution of the problem (1.11), (2.1) if it is a locally absolutely continuous function on , satisfies equation (1.11) on almost everywhere, and initial conditions (2.1) for .
Let conditions (a1)–(a3) hold. Then there exists a unique solution of problem (1.11), (2.1).
is a unique solution of (1.11), (2.1).
Equality (2.17) implies
Leonid Berezansky was partially supported by grant 25/5 "Systematic support of international academic staff at Faculty of Electrical Engineering and Communication, Brno University of Technology" (Ministry of Education, Youth and Sports of the Czech Republic) and by grant 201/07/0145 of the Czech Grant Agency (Prague). Josef Diblík was supported by grant 201/08/0469 of the Czech Grant Agency (Prague), and by the Council of Czech Government grant MSM 00216 30503 and MSM 00216 30519. Zdeněk Šmarda was supported by the Council of Czech Government grant MSM 00216 30503 and MSM 00216 30529.
- Minorsky N: Nonlinear Oscillations. D. Van Nostrand, Princeton, NJ, USA; 1962:xviii+714.MATHGoogle Scholar
- Berezansky L, Braverman E: Nonoscillation of a second order linear delay differential equation with a middle term. Functional Differential Equations 1999,6(3-4):233-247.MathSciNetMATHGoogle Scholar
- Hale JK, Verduyn Lunel SM: Introduction to Functional-Differential Equations, Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993:x+447.View ArticleGoogle Scholar
- Azbelev N, Maksimov V, Rakhmatullina L: Introduction to the Theory of Linear Functional-Differential Equations, Advanced Series in Mathematical Science and Engineering. Volume 3. World Federation Publishers, Atlanta, Ga, USA; 1995:ii+172.Google Scholar
- Karaca IY: Positive solutions for boundary value problems of second-order functional dynamic equations on time scales. Advances in Difference Equations 2009, 2009:-21.Google Scholar
- Ma R, Xu Y, Gao Ch: A global description of the positive solutions of sublinear second-order discrete boundary value problems. Advances in Difference Equations 2009, 2009:-15.Google Scholar
- Azbelev NV, Maksimov VP, Rakhmatullina LF: Introduction to the Theory of Functional Differential Equations: Methods and Applications, Contemporary Mathematics and Its Applications. Volume 3. Hindawi, Cairo, Egypt; 2007:x+314.View ArticleGoogle Scholar
- Azbelev NV, Simonov PM: Stability of Differential Equations with Aftereffect, Stability and Control: Theory, Methods and Applications. Volume 20. Taylor & Francis, London, UK; 2003:xviii+222.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.