Leonid Berezansky,¹ Josef Diblík,^2,3 and Zdenĕk Šmarda³

Copyright © 2010 Leonid Berezansky et al. 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.

Abstract

The existence and uniqueness of solutions and a representation of solution formulas are studied for the following initial value problem: x˙(t)+∫t0tK(t,s)x(h(s))ds=f(t),  t≥t0,  x∈ℝn, x(t)=φ(t), t<t0. Such problems are obtained by transforming second-order delay differential equations x¨(t)+a(t)x˙(g(t))+b(t)x(h(t))=0 to first-order differential equations.

1. Introduction and Preliminaries

The second order delay differential equation

(1.1) attracts the attention of many mathematicians because of their significance in applications.

In particular, Minorsky [1] 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

(1.2) where is a nonnegative function is used in [2]. The following result is a restriction of [2, Theorem ] to (1.1).

Proposition 1.1. If are Lebesgue measurable and locally essentially bounded, are Lebesgue measurable functions, if , there exists a locally absolutely continuous function such that the inequality (1.3) is valid for all sufficiently large , and the equation (1.4) has a nonoscillatory solution, then (1.1) has a nonoscillatory solution, too.

Proposition 1.1 means that second order delay equation (1.1) is reduced to nonlinear inequality (1.3) and first order delay differential equation (1.4).

Now we will briefly describe the scheme of another transformation, different from the one used in [2] (in this explanation we omit exact assumptions related to the functions used, which are formulated later).

Consider an auxiliary equation

(1.5) with the initial condition

(1.6) It is known, see [3, 4], that the unique solution of (1.5), (1.6) has a form

(1.7) where is the fundamental matrix of (1.5) and if .

If we denote , then (1.1) can be rewritten in the form

(1.8) Applying (1.7) and equality to (1.8), we have the following equation

(1.9) Define (1.10) Then (1.1) is transformed into the integrodifferential equation with delay

(1.11) 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 [2] is that a second order delay equation is reduced to one first-order integrodifferential delay equation while in [2] a second-order equation is reduced to a system of a nonlinear inequality and a linear delay equation.

Similar in a sense problems for delay difference equations were studied in [5, 6] as well.

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

(2.1) We will assume that the following conditions hold:

(a1) For all , the elements of the matrix function are measurable in the square , the elements of the vector function are measurable in the interval ,

(2.2)

(a2) is a measurable scalar function satisfying . (a3) The initial function is a Borel bounded function.

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 .

Theorem 2.1. Let conditions (a1)–(a3) hold. Then there exists a unique solution of problem (1.11), (2.1).

Proof. It is sufficient to prove that there exists a unique solution of (1.11), (2.1) on the interval for any .

Denote

(2.3) Then and (1.11), (2.1) takes the form (2.4) where (2.5) Denote the characteristic function of the interval . We will assume that if . Since (2.6) we have (2.7) Hence problem (2.4) can be transformed into (2.8) We have (2.9) Denote (2.10) Finally, problem (2.4) has the form (2.11) where . Consider the linear integral operator (2.12) in the space of all Lebesgue integrable functions with the norm .

We have

(2.13) Hence the integral operator is a compact Volterra operator and its spectral radius is equal to zero [4, 7, 8]. Then the integral equation (2.11) has a unique solution . Consequently (2.14) is a unique solution of (1.11), (2.1).

Let be the zero matrix and the identity matrix.

Definition 2.2. For each , the solution of the problem (2.15) is called the fundamental matrix of (1.11). (Here is the partial derivative of with respect to its first argument.)

Theorem 2.3. Let conditions (a1)–(a3) hold. Then the unique solution of (1.11), (2.1) can be represented in the form (2.16) for where is defined by (2.5).

Proof. In the proof we will use notation defined in the proof of Theorem 2.1. The existence and uniqueness of a solution of (1.11), (2.1) is a consequence of Theorem 2.1. Thus, we will only prove the solution representation formula (2.16). Problem (1.11), (2.1) is equivalent to (2.4). We need to show that the function (2.17) where is the fundamental matrix of (1.11) is the solution of problem (2.4). For convenience, we will write instead of assuming that , if .

Equality (2.17) implies

(2.18) We consider the left-hand side of (2.4) if assuming to have the form (2.17). With the help of the last relation, we have (2.19) Since (2.20) we obtain (2.21)

Acknowledgments

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.

References

1. N. Minorsky, Nonlinear Oscillations, D. Van Nostrand, Princeton, NJ, USA, 1962.
2. L. Berezansky and E. Braverman, “Nonoscillation of a second order linear delay differential equation with a middle term,” Functional Differential Equations, vol. 6, no. 3-4, pp. 233–247, 1999.
3. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993.
4. N. Azbelev, V. Maksimov, and L. Rakhmatullina, Introduction to the Theory of Linear Functional-Differential Equations, vol. 3 of Advanced Series in Mathematical Science and Engineering, World Federation Publishers, Atlanta, Ga, USA, 1995.
5. I. Y. Karaca, “Positive solutions for boundary value problems of second-order functional dynamic equations on time scales,” Advances in Difference Equations, vol. 2009, Article ID 829735, 21 pages, 2009.
6. R. Ma, Y. Xu, and Ch. Gao, “A global description of the positive solutions of sublinear second-order discrete boundary value problems,” Advances in Difference Equations, vol. 2009, Article ID 671625, 15 pages, 2009.
7. N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Functional Differential Equations: Methods and Applications, vol. 3 of Contemporary Mathematics and Its Applications, Hindawi, Cairo, Egypt, 2007.
8. N. V. Azbelev and P. M. Simonov, Stability of Differential Equations with Aftereffect, vol. 20 of Stability and Control: Theory, Methods and Applications, Taylor & Francis, London, UK, 2003.