On Connection between Second Order Delay Differential Equations and Integrodifferential Equations with Delay

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doi:10.1155/2010/143298

Research Article

On Connection between Second-Order Delay

Differential Equations and Integrodifferential

Equations with Delay

Leonid Berezansky,

1

_{Josef Dibl´ık,}

2, 3

_{and Zden ˘ek ˇSmarda}

3

1_{Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel} 2_{Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, 66237,}

Brno University of Technology, Brno, Czech Republic

3_{Department of Mathematics, Faculty of Electrical Engineering and Communication, 61600,}

Brno University of Technology, Brno, Czech Republic

Correspondence should be addressed to Leonid Berezansky,[email protected]

Received 5 October 2009; Accepted 11 November 2009 Academic Editor: A ˘gacik Zafer

Copyrightq2010 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.

The existence and uniqueness of solutions and a representation of solution formulas are studied for the following initial value problem: ˙xt t_t

0Kt, sxhsdsft,t≥t0,x∈R

n_,_x_t _ϕ_t_, t < t0. Such problems are obtained by transforming second-order delay diﬀerential equations

¨

xt atx˙gt btxht 0 to first-order diﬀerential equations.

1. Introduction and Preliminaries

The second order delay diﬀerential equation

¨

xt atx˙gtbtxht 0 1.1

attracts the attention of many mathematicians because of their significance in applications. In particular, Minorsky1in 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 by1.1.

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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

xt t

t0

exp

t

s

uτdτ

zsds, 1.2

whereutis a nonnegative function is used in2. The following result is a restriction of2, Theorem 1to1.1.

Proposition 1.1. Ifa, b:t0,∞ → 0,∞are Lebesgue measurable and locally essentially bounded, g, h : t0,∞ → R are Lebesgue measurable functions, gt ≤ t, ht ≤ t if t ∈ t0,∞, limt→ ∞gt limt→ ∞ht ∞, there exists a locally absolutely continuous functionu:t0,∞ → 0,∞such that the inequality

˙

ut u2t atugte−

t

gtusdsbte−httusds≤₀_, 1.3

is valid for all suﬃciently larget, and the equation

˙

zt utzt atzgt0 1.4

has a nonoscillatory solution, then1.1has a nonoscillatory solution, too.

Proposition 1.1means that second order delay equation1.1is reduced to nonlinear

inequality1.3and first order delay diﬀerential equation1.4.

Now we will briefly describe the scheme of another transformation, diﬀerent from the one used in2 in this explanation we omit exact assumptions related to the functions used, which are formulated later.

Consider an auxiliary equation

˙

zt atzgtpt, t≥t0, 1.5

with the initial condition

zt ψt, t < t0, zt0 z0. 1.6

It is known, see3,4, that the unique solution of1.5,1.6has a form

zt Zt, t0zt0

t

t0

Zt, spsds− t

t0

Zt, sψgsds, 1.7

whereZt, sis the fundamental matrix of1.5andψgs 0 ifgs< t0. If we denotezt xt˙ , then1.1can be rewritten in the form

˙

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Applying1.7and equalityzt xt˙ to1.8, we have the following equation

˙

xt Zt, t0xt˙ 0−

t

t0

Zt, sbsxhsds− t

t0

Zt, sψgsds. 1.9

Define

Kt, s:Zt, sbs,

ft:Zt, t0xt˙ 0−

t

t0

Zt, sψgsds.

1.10

Then1.1is transformed into the integrodiﬀerential equation with delay

˙ xt

t

t0

Kt, sxhsdsft, t≥t0. 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 of1.1.

The advantage of the suggested method in comparison with the method used in2

is that a second order delay equation is reduced to one first-order integrodiﬀerential delay equation while in2a 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 5,6as 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 with1.11we consider an initial condition

xt ϕt, t < t0, xt0 x0. 2.1

We will assume that the following conditions hold:

a1For allc > t0, the elementskij:t0, c×t0, c → R, i, j 1,2, . . . , nof then×nmatrix functionKare measurable in the squaret0, c×t0, c, the elementsfi:t0, c → R, i1,2, . . . , nof the vector functionfare measurable in the intervalt0, c,

sup

t∈t0,c _c

t0

|Kt, s|ds <∞, _c

t0

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a2h:t0,∞ → Ris a measurable scalar function satisfyinght≤t, limt→ ∞ht ∞. a3The initial functionϕ:−∞, t0 → Rnis a Borel bounded function.

A function x : R → Rn _{is called a solution of the problem} _1.11_, _2.1 _{if it is a}

locally absolutely continuous function ont0,∞, satisfies equation1.11 ont ≥ t0 almost everywhere, and initial conditions2.1fort≤t0.

Theorem 2.1. Let conditions (a1)–(a3) hold. Then there exists a unique solution of problem1.11,

2.1.

Proof. It is suﬃcient to prove that there exists a unique solution of1.11,2.1on the interval t0, cfor anyc > t0.

Denote

xht ⎧ ⎨ ⎩

xht ifht≥t0,

0 ifht< t0,

ϕht ⎧ ⎨ ⎩

ϕht ifht< t0,

0 ifht≥t0.

2.3

Thenxht xht ϕht, t≥t0and1.11,2.1takes the form

˙ xt

t

t0

Kt, sxhsdsgt, t≥t0, xt0 x0, 2.4

where

gt ft−ψt

ψt t

t0

Kt, qϕhqdq.

2.5

Denoteχα,βtthe characteristic function of the intervalα, β. We will assume thatχα,βt≡

0 ifα≥β. Since

xt xt0

t

t0

˙

xsds, 2.6

we have

xht

xt0

max{ht,t0}

t0

˙ xsds

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Hence problem2.4can be transformed into

Finally, problem2.4has the form

yt t

t0

Bt, sysdsrt, t≥t0, 2.11

whereyt xt, r˙ t Atxt0 gt. Consider the linear integral operator

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We have

sup

t0≤t≤c _c

t0

|Bt, s|ds≤ sup

t0≤t≤c _c

t0 _t

s

|Kt, τ|dτds

≤c−t0· sup

t0≤t≤c _c

t0

|Kt, τ|dτ <∞.

2.13

Hence the integral operator T is a compact Volterra operator and its spectral radius is equal to zero4,7,8. Then the integral equation2.11has a unique solutionyt, t ≥ t0. Consequently

xt ⎧ ⎪ ⎨ ⎪ ⎩

xt0

t

t0

ysds, t≥t0, ϕt, t < t0

2.14

is a unique solution of1.11,2.1.

LetΘbe then×nzero matrix andIthe identityn×nmatrix.

Definition 2.2. For eachs≥t0, the solutionXX·, sof the problem

˙ Xt, s

_t

s

Kt, τXhτ, sdτ Θ, t≥s,

Xt, s Θ, t < s, Xs, s I,

2.15

is called the fundamental matrix of1.11.Here ˙Xt, sis the partial derivative ofXt, swith 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

xt Xt, t0x0

t

t0

Xt, sfsds− t

t0

Xt, sψsds 2.16

fort≥t0whereψis defined by2.5.

Proof. In the proof we will use notation defined in the proof ofTheorem 2.1. The existence and uniqueness of a solution of1.11,2.1is a consequence ofTheorem 2.1. Thus, we will only prove the solution representation formula2.16. Problem1.11,2.1is equivalent to 2.4. We need to show that the function

xt Xt, t0x0

_t

t0

Xt, sgsds, 2.17

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Equality2.17implies the last relation, we have

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we obtain

˙ xt

_t

t0

Kt, sxhsdsgt _t

t0

˙ Xt, s

_t

s

Kt, ξXhξ, sdξ

gsdsgt.

2.21

Acknowledgments

Leonid Berezansky was partially supported by grant 25/5 “Systematic support of inter-national 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 AgencyPrague. Josef Dibl´ık was supported by grant 201/08/0469 of the Czech Grant AgencyPrague, and by the Council of Czech Government grant MSM 00216 30503 and MSM 00216 30519. Zdenˇek ˇSmarda was supported by the Council of Czech Government grant MSM 00216 30503 and MSM 00216 30529.

References

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3 J. K. Hale and S. M. Verduyn Lunel,Introduction to Functional-Diﬀerential Equations, vol. 99 ofApplied Mathematical Sciences, Springer, New York, NY, USA, 1993.

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