Advances in Difference Equations

, 2013:187

About reducing integro-differential equations with infinite limits of integration to systems of ordinary differential equations

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DOI: 10.1186/1687-1847-2013-187

Cite this article as:
Goltser, Y. & Domoshnitsky, A. Adv Differ Equ (2013) 2013: 187. doi:10.1186/1687-1847-2013-187
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  1. Progress in Functional Differential and Difference Equations

Abstract

The purpose of this paper is to propose a method for studying integro-differential equations with infinite limits of integration. The main idea of this method is to reduce integro-differential equations to auxiliary systems of ordinary differential equations.

Results: a scheme of the reduction of integro-differential equations with infinite limits of integration to these auxiliary systems is described and a formula for representation of bounded solutions, based on fundamental matrices of these systems, is obtained.

Conclusion: methods proposed in this paper could be a basis for the Floquet theory and studies of stability, bifurcations, parametric resonance and various boundary value problems. As examples, models of tumor-immune system interaction, hematopoiesis and plankton-nutrient interaction are considered.

MSC:45J05, 45J15, 34A12, 34K05, 34K30, 47G20.

Keywords

integro-differential equations fundamental matrix Cauchy matrix hyperbolic systems 

1 Introduction

Integro-differential equations appeared very naturally in various applications (see, for example, [1, 2, 3, 4, 5]), which explains the interest in the theory of these equations (see, for example, [6, 7]). Various examples, in which the simple enough integro-differential equation
x ( t ) = X ( t , x ( t ) , 0 t F ( t , s , x ( s ) ) d s ) , Open image in new window
(1.1)

by elementary operations can be reduced to a system of ordinary differential equations, are known. In this connection, let us refer, for example, to the monograph [8]. Note the idea of the chain trick used in various applications (see, for example, [9, 10]) and its developed form in the paper [11]. Independently, the idea of a reduction to systems of ordinary differential equations in the study of stability, which was, actually, the chain trick, was presented in [12]. Starting with this reduction, approaches to the study of stability and bifurcation of integro-differential equations were proposed in the papers [13, 14, 15, 16]. The approach developed in these papers allowed researchers to define a notion of periodic integro-differential systems and to build the Floquet theory for integro-differential equations on this basis in [17]. The first known results on estimates of distance between two adjacent zeros of oscillating solutions to a linearization of equation (1.1) and results connecting oscillation behavior and the exponential stability were obtained on this basis [17]. A parametric resonance in linear almost periodic systems was studied in [18], and the bifurcation of steady resonance modes for integro-differential systems was investigated in [19]. Stabilization by control in a form of integrals of solutions was studied in [20]. The stability of partial functional differential equations on the basis of this reduction was studied in [21]. Constructive approach to a phase transition model was presented in [22]. A reduction to infinite dimensional systems was considered in [21, 23, 24]. In all these papers the limits of integration in integral terms were 0 and t, and this was very essential.

The main goal of this paper is to present a method reducing integro differential equations with infinite limits of integration
x ( t ) = f ( t , x ( t ) , + K ( t , s ) g ( s , x ( s ) ) d s ) , t ( , + ) , Open image in new window
(1.2)

to systems of ordinary differential equations. In a future we are planning to develop the ideas of noted above papers for equation (1.2). As well as we know, there are no results of this type. Important motivation in the study of integro-differential equation (1.2) can be found also in various applications of such equations in, for example, models of tumor-immune system interaction [9], hematopoiesis [10], stability and persistence in plankton models [25] which will be considered below.

Denote
u ( t ) = + K ( t , s ) g ( s , x ( s ) ) d s , Open image in new window
(1.3)
v ( t ) = t K ( t , s ) g ( s , x ( s ) ) d s , Open image in new window
(1.4)
w ( t ) = t + K ( t , s ) g ( s , x ( s ) ) d s . Open image in new window
(1.5)
Using these notations, we can write
x ( t ) = f ( t , x ( t ) , u ( t ) ) , t ( , + ) , Open image in new window
(1.6)
or
x ( t ) = f ( t , x ( t ) , v ( t ) + w ( t ) ) , t ( , + ) . Open image in new window
It is possible to represent the vector x R n Open image in new window in the form x = col { y , z } Open image in new window, where y R k Open image in new window, z R n k Open image in new window. In many applications, system (1.4) can be represented in the form
y ( t ) = Y ( t , x ( t ) , v ( t ) ) , z ( t ) = Z ( t , x ( t ) , w ( t ) ) , t ( , + ) . Open image in new window
(1.7)

The first equation in (1.7) depends on its integral part v on delay only (see (1.4)) and the second one is dependent on advance only. Note that the cases k = n Open image in new window and k = 0 Open image in new window can be also considered. If k = n Open image in new window, we get a system with distributed delay, and if k = 0 Open image in new window, the one with distributed advance. Note that a combination of distributed and concentrated deviations is also possible. Considering such systems, we do not discuss questions of existence of solutions and assume that solutions to these systems exist. Note that even for the Volterra equation, one-point problem (1.1) with the condition x ( t 0 ) = x 0 Open image in new window, t 0 > 0 Open image in new window, can have more than one solution or not have solutions at all (see, for example, [26], Chapter 1, Section 9, pp. 70-74).

For system (1.2) our method essentially uses the properties of linear nonhomogeneous systems of ODEs, possessing exponential dichotomy [27] or hyperbolicity [28]. It is known that such systems have (under corresponding conditions) unique bounded on the axis solution. Corresponding bibliography can be found in [28]. The case of autonomous systems was considered in [29, 30]. Below, in the next paragraph, we formulate, in convenient for us form, a result about the existence and structure of the solution for general non-autonomous linear systems of ODEs. This result is based on the theorem about reduction of hyperbolic systems to a block diagonal form [28].

2 Methods: about bounded solutions of linear nonhomogeneous systems

Consider
x ( t ) = P ( t ) x ( t ) + g ( t ) , t ( , + ) Open image in new window
(2.1)
and the corresponding homogeneous system
w ( t ) = P ( t ) w ( t ) , t ( , + ) , Open image in new window
(2.2)

where x , w R n Open image in new window, P is an n × n Open image in new window matrix and g is an n-vector function with continuous bounded elements.

We use the following definition introduced in [28].

Definition 2.1 We say that system (2.2) is hyperbolic if there exist constants a > 0 Open image in new window and λ > 0 Open image in new window and hyperplanes M + Open image in new window and M dim M + = k Open image in new window, dim M = n k Open image in new window such that if for t = t 0 Open image in new window, w ( t 0 ) = w 0 M + Open image in new window, then the solution w ( t , t 0 , w 0 ) Open image in new window satisfies the inequality
| w ( t , t 0 , w 0 ) | a | w 0 | e λ ( t t 0 ) , t t 0 , Open image in new window
(2.3)
and if w 0 M Open image in new window, the inequality
| w ( t , t 0 , w 0 ) | a | w 0 | e λ ( t t 0 ) , t t 0 . Open image in new window
(2.4)

Theorem 2.1 [28]

Let system (2.2) be hyperbolic. Then there exists an n × n Open image in new windowmatrix U ( t ) Open image in new windowwith bounded elements such that its inverse matrix U 1 ( t ) Open image in new windowalso possesses bounded elements and the transform w = U ( t ) η Open image in new windowreduces system (2.2) to the form
ξ ( t ) = Q + ( t ) ξ ( t ) , ζ ( t ) = Q ( t ) ζ ( t ) , Open image in new window
(2.5)

where η = col { ξ , ζ } Open image in new window, ξ R k Open image in new window, ζ R n k Open image in new window.

If we denote Φ + ( t , s ) = ϕ + ( t ) ϕ + 1 ( s ) Open image in new window, where ϕ + ( t ) Open image in new windowis a fundamental matrix of the first system in (2.5), Φ ( t , s ) = ϕ ( t ) ϕ 1 ( s ) Open image in new window, where ϕ ( t ) Open image in new windowis a fundamental matrix of the second system in (2.5), such that Φ + ( s , s ) = E + Open image in new window, Φ ( s , s ) = E Open image in new window, dim E + = k Open image in new window, dim E = n k Open image in new window, then
Φ + ( t , s ) a e λ ( t s ) , t s , Open image in new window
(2.6)
Φ ( t , s ) a e λ ( t s ) , t s . Open image in new window
(2.7)

We present corresponding constructions, developed in [28] for the proof of this theorem, which will be used below in our paper.

Let
Φ ( t ) = ( w 1 ( t ) , , w n ( t ) ) Open image in new window
(2.8)
be a fundamental matrix of system (2.2), where w i ( t ) Open image in new window ( i = 1 , , n Open image in new window) are linearly independent solutions of system (2.2), M + = span ( w 1 , , w k ( t ) ) Open image in new window, M = span ( w k + 1 , , w n ( t ) ) Open image in new window. Setting v 1 ( t ) = w 1 ( t ) Open image in new window, u 1 ( t ) = w 1 ( t ) w 1 ( t ) Open image in new window, we define, for m = 2 , 3 , , k Open image in new window, the vectors
v m = w m i = 1 m 1 ( w m , u i ) u i , u m = v m v m . Open image in new window
(2.9)
For m = k + 1 Open image in new window, we set v k + 1 ( t ) = w k + 1 ( t ) Open image in new window, u k + 1 ( t ) = w k + 1 ( t ) w k + 1 ( t ) Open image in new window, and for m = k + 2 , , n Open image in new window, we define corresponding vectors according to scheme (2.9). The matrix
U ( t ) = ( u 1 ( t ) , , u n ( t ) ) Open image in new window
(2.10)
is bounded with its inverse matrix U 1 ( t ) Open image in new window and d U d t Open image in new window. The vectors u j ( t ) Open image in new window are pairwise orthogonal and u j ( t ) = 1 Open image in new window, j = 1 , , n Open image in new window. Let us set
U ( t ) = Φ ( t ) S ( t ) . Open image in new window
(2.11)
It is clear from the construction of the matrix U ( t ) Open image in new window that S ( t ) Open image in new window is a block diagonal
S ( t ) = diag ( S + ( t ) , S ( t ) ) , dim S + = k , dim S = n k . Open image in new window
(2.12)
Setting in (2.2) w = U ( t ) η Open image in new window, we get
d η d t = Q ( t ) η , Open image in new window
(2.13)
where Q ( t ) = U 1 ( P U d u d t ] = S 1 d S d t Open image in new window. It follows from (2.12) and (2.13) that Q ( t ) = diag ( Q + ( t ) , Q ( t ) ) Open image in new window, where Q + ( t ) = S + d S + d t Open image in new window, Q ( t ) = S d S d t Open image in new window. Thus system (2.13) has the form
d ξ d t = Q + ( t ) ξ , d ζ d t = Q ( t ) ζ , Open image in new window
(2.14)

where η = col ( ξ , ζ ) Open image in new window.

Define the Cauchy matrices Φ + ( t , s ) Open image in new window and Φ ( t , s ) Open image in new window such that Φ + ( t , t ) = E k Open image in new window, Φ ( t , t ) = E n k Open image in new window, where E j Open image in new window is a unit ( j × j ) Open image in new window-matrix.

Let us prove the following assertion about the representation of bounded solutions to system (2.1).

Theorem 2.2Let all elements of P ( t ) Open image in new windowand g ( t ) Open image in new windowin system (2.1) be continuous and bounded for t ( , + ) Open image in new window, and let system (2.2) be hyperbolic. Then system (2.1) has a unique bounded solution and this solution can be represented in the form
x ( t ) = U ( t ) z ( t ) , z ( t ) = + G ( t , s ) h ( s ) d s , Open image in new window
(2.15)
where
G ( t , s ) = { diag { Φ + ( t , s ) , 0 n k } , t > s , diag { 0 k , Φ ( t , s ) } , t < s , Open image in new window
(2.16)
G ( s + 0 , s ) G ( s 0 , s ) = E n , h ( t ) = U 1 ( t ) g ( t ) = { h + ( t ) , h ( t ) } . Open image in new window
(2.17)
Proof Let us substitute
y ( t ) = U ( t ) z ( t ) Open image in new window
(2.18)
into system (2.1), then we get the system
d z d t = Q ( t ) z + h ( t ) , Open image in new window
(2.19)

for which the homogeneous system is of the form (2.13), (2.14).

Consider the matrix (2.16). It follows from the properties of the matrices Φ + ( t ) Open image in new window, Φ ( t ) Open image in new window that equality (2.17) is fulfilled. It follows from hyperbolicity of system (2.14) that
Φ + ( t , s ) a e λ ( t s ) , t > s , Φ ( t , s ) a e λ ( t s ) , t < s . Open image in new window
(2.20)
It follows from (2.16) and (2.20) that the integral in (2.15) converges for bounded functions h ( t ) Open image in new window every t. Computing the derivative of Green’s matrix, we get
d G ( t , s ) d t = Q ( t ) G ( t , s ) . Open image in new window
(2.21)
Let us verify now that formula (2.15) defines the solution of equation (2.19). Representing z ( t ) Open image in new window in the form
z ( t ) = t G ( t , s ) h ( s ) d s + t + G ( t , s ) h ( s ) d s , Open image in new window
differentiating it and taking into account (2.21) and (2.17), we get
d z d t = t Q ( t ) G ( t , s ) h ( s ) d s + t + Q ( t ) G ( t , s ) h ( s ) d s + [ G ( s + 0 , s ) G ( s 0 , s ) ] h ( t ) = Q ( t ) z ( t ) + h ( t ) . Open image in new window

The obtained solution is unique. If we assume the existence of two bounded solutions z 1 Open image in new window and z 2 Open image in new window, then z 1 z 2 Open image in new window is a bounded on the axis solution of (2.2). From hyperbolicity, it follows that it is a zero solution. □

Corollary 2.1If homogeneous system (2.2) is hyperbolic and k = n Open image in new window ( k = 0 Open image in new window), then nonhomogeneous system (2.1) has a unique bounded for t ( , + ) Open image in new windowsolution, and this solution can be represented in the following form:
x ( t ) = U ( t ) z ( t ) , z ( t ) = t G ( t , s ) h ( s ) d s ( z ( t ) = t + G ( t , s ) h ( s ) d s ) , Open image in new window
(2.22)

where G ( t , s ) = Φ ( t ) Φ 1 ( s ) Open image in new window, Φ ( t ) Open image in new windowis a fundamental matrix of system (2.2).

Remark 2.1 If the matrix P ( t ) Open image in new window in (2.1) is a constant one, analogous results are obtained in [29, 30]. The existence of a unique bounded solution under the assumption of the exponential dichotomy on ( , + ) Open image in new window for system (2.2) with bounded variable coefficients is known (see, [27], p.69, Proposition 2). Similar topics were also studied in [31].

3 Results: about reduction of integro-differential equations to systems of ordinary differential equations

3.1 Reduction to the system of first-order ordinary differential equations

Consider the system
x ( t ) = X ( t , x , t K ( t , s ) g ( s , x ( s ) ) d s ) , Open image in new window
(3.1)
where the kernel K ( t , s ) Open image in new window is of the form
K ( t , s ) = i , j = 1 C j Φ j ( t ) R j ( s ) = i , j = 1 C j K j ( t , s ) . Open image in new window
(3.2)
Series in (3.2) can be, for example, corresponding orthogonal expansions, series of exponents. One of the interesting cases is a finite sum in (3.2). We assume that all the matrices Φ j ( t ) Open image in new window are differentiable and invertible. We can write
K j ( t , s ) = Φ j ( t ) Φ j 1 ( s ) K j ( s , s ) . Open image in new window
(3.3)
Define the so-called multiplicative derivative [32]
P j ( t ) = d Φ j ( t ) d t Φ j 1 ( t ) , Open image in new window
(3.4)
the matrix
G j ( t , s ) = Φ j ( t ) Φ j 1 ( s ) , Open image in new window
(3.5)
is the Cauchy matrix of the system
w j ( t ) = P j ( t ) w ( t ) . Open image in new window
(3.6)
Let us set
z j ( t , t 0 ) = t G j ( t , s ) K j ( s , s ) g ( s , x ( s ) ) d s = t G j ( t , s ) h j ( s , x ( s ) ) d s , Open image in new window
(3.7)

where h j ( s , x ( s ) ) = K j ( s , s ) g ( s , x ( s ) ) Open image in new window.

If the matrix G j ( t , s ) Open image in new window, defined by (3.5), satisfies the inequality
| G j ( t , s ) | a e λ ( t s ) , t s Open image in new window
(this is the analog of (2.6)) for k = n Open image in new window, then z j ( t , t 0 ) Open image in new window in formula (3.7), according to Corollary 2.1, can be considered as a solution of the one-point problem
z j ( t ) = P j ( t ) z ( t ) + h j ( t , x ( t ) ) , z j ( t 0 ) = z j 0 , Open image in new window
(3.8)
where
z j 0 = t 0 G j ( t 0 , s ) h j ( s , x ( s ) ) d s , Open image in new window
if we consider x ( t ) Open image in new window as a known function bounded on ( , t 0 ] Open image in new window. Adding to equation (3.8) the so-called initial function (continuous and bounded on ( , t 0 ] Open image in new window)
x ( t ) = φ ( t ) , Open image in new window
(3.9)
we can consider representation (3.7) as a substitution, which leads us to the one-point problem
z j ( t ) = P j ( t ) z ( t ) + h j ( t , x ( t ) ) , z j ( t 0 ) = z j 0 , Open image in new window
(3.10)

where z j 0 Open image in new window was defined above.

We have proven the following assertion.

Theorem 3.1Let
  1. (a)

    matrices Φ j ( t ) Open image in new windowin the kernels (3.2) be continuously differentiable and invertible for t ( , + ) Open image in new window, j = 1 , 2 , Open image in new window ,

     
  2. (b)
    systems (3.6) be of dimension n j Open image in new window, where
    P j ( t ) = d Φ j ( t ) d t Φ j 1 ( t ) , Open image in new window
     

be hyperbolic for everyjin the sense of Definition  2.1 (for k = n Open image in new window).

Then the bounded solution x ( t ) R n Open image in new windowof system (3.1) with the kernel of the form (3.2) and the initial function (3.9) and the first component x ( t ) R n Open image in new windowof the solution to the countable system
x ( t ) = X ( t , x , z 1 , z 2 , ) , z j ( t ) = P j ( t ) z ( t ) + h j ( t , x ) , t [ t 0 , + ) , z j ( t 0 ) = z j 0 , x ( t 0 ) = φ ( t 0 ) , Open image in new window
(3.11)
where x R n Open image in new window, z j R n Open image in new window, h j ( t , x ) = K j ( t , t ) g ( t , x ) Open image in new window,
z j 0 = t 0 G j ( t 0 , s ) h j ( s , x ( s ) ) d s , j = 1 , 2 , 3 , Open image in new window

coincide.

Remark 3.1 If (3.2) is a finite sum, then system (3.11) is finite dimensional.

Remark 3.2 The system of the form
x ( t ) = X ( t , x , 0 K ( ξ ) g ( ξ , x ( t ξ ) ) d ξ ) Open image in new window
(3.12)

can be found in various applications. It can be reduced by the change of variable t ξ = s Open image in new window to system (3.1) with the kernel K ( t , s ) = K ( t s ) Open image in new window.

Remark 3.3 System (3.11) can be used for studying qualitative properties and for an approximate solution of system (3.1) of integro-differential system (3.1). An important basis is the theory of countable systems [33, 34, 35, 36, 37]; see also the papers [24, 38, 39, 40, 41].

Remark 3.4 Analogous result could be obtained for the system
x ( t ) = X ( t , x , t + K ( t , s ) g ( s , x ( s ) ) d s ) , Open image in new window

in Section 5.

3.2 Reduction to the system of ordinary differential equations of high orders

Consider the nonhomogeneous linear equation of n th order
L [ y ] y ( n ) + p 1 ( t ) y ( n 1 ) + + p n ( t ) y = f ( t ) Open image in new window
(3.13)
and the corresponding homogeneous equation
L [ z ] z ( n ) + p 1 ( t ) z ( n 1 ) + + p n ( t ) z = 0 , Open image in new window
(3.14)

where all coefficients p j Open image in new window ( j = 1 , , n Open image in new window) and f are essentially bounded on ( , + ) Open image in new window.

Let
z 1 ( t ) , , z n ( t ) Open image in new window
(3.15)
be a fundamental system of solutions of equation (3.14). Using (3.15), we can construct the solution z = ψ ( t , t 0 ) Open image in new window such that
z ( t 0 ) = ψ ( t 0 , t 0 ) = 0 , , z ( n 2 ) ( t 0 ) = 0 , z ( n 1 ) ( t 0 ) = 1 . Open image in new window
(3.16)
The function ψ ( t , s ) Open image in new window is called the Cauchy function of equation (3.13) [42, 43]. Consider the function
y ( t ) = t ψ ( t , s ) f ( s ) d s , Open image in new window
(3.17)
assuming that the integral converges. Let us verify that (3.17) is a solution of (3.13). Actually,
y ( k ) ( t ) = t ψ ( k ) ( t , s ) f ( s ) d s + ψ ( k 1 ) ( t , t ) f ( t ) = t ψ ( k ) ( t , s ) f ( s ) d s Open image in new window
(3.18)
for k = 1 , , n 1 Open image in new window, and
y ( n ) ( t ) = t ψ ( k ) ( t , s ) f ( s ) d s + f ( t ) . Open image in new window
(3.19)
It follows from (3.18), (3.19) and the equality L [ ψ ( t , t 0 ) ] = 0 Open image in new window that
L [ y ] = f ( t ) . Open image in new window
(3.20)
The obtained particular solution satisfies the initial conditions
y ( t 0 ) = t 0 ψ ( t 0 , s ) f ( s ) d s , y ( k ) ( t 0 ) = t 0 ψ ( k ) ( t 0 , s ) f ( s ) d s , k = 1 , , n 1 . Open image in new window
(3.21)
Example 3.1 For the equation
y ( n ) ( t ) = f ( t ) , Open image in new window
(3.22)
we get
ψ ( t , s ) = 1 ( n 1 ) ! ( t s ) n 1 , y ( t ) = 1 ( n 1 ) ! t ( t s ) n 1 f ( s ) d s . Open image in new window
(3.23)
Example 3.2 For the equation
j = 0 n c n j λ j y ( n j ) ( t ) = f ( t ) , Open image in new window
(3.24)
we get
ψ ( t , s ) = 1 ( n 1 ) ! ( t s ) n 1 e λ ( t s ) , y ( t ) = 1 ( n 1 ) ! t ( t s ) n 1 e λ ( t s ) f ( s ) d s . Open image in new window
(3.25)
Consider the system
x ( t ) = X ( t , x ( t ) , t F ( t , s , y ( s ) ) d s ) , t [ t 0 , + ) , Open image in new window
(3.26)
where x = col ( u , y ) Open image in new window, u R n 1 Open image in new window, y R 1 Open image in new window, and assume that
F ( t , s , y ( s ) ) = j = 1 ψ j ( t , s ) g j ( s , y ( s ) ) , Open image in new window
(3.27)
where ψ j ( t , s ) Open image in new window ( j = 1 , 2 , Open image in new window) are the Cauchy functions of corresponding linear equations
L j [ z j ] z j ( n ) + p 1 j ( t ) z j ( n 1 ) + + p n j ( t ) z j = 0 , j = 1 , 2 , . Open image in new window
(3.28)
Define
v j ( t ) = t ψ j ( t , s ) g j ( s , y ( s ) ) d s , Open image in new window
(3.29)
and set
y ( t ) = φ ( t ) , t ( , t 0 ] , Open image in new window
(3.30)
where φ is considered as a known function. Assuming that the Cauchy functions imply convergence of integrals (3.29) for all j and that the function φ is bounded, we obtained that system (3.26) is reduced to the countable system
x ( t ) = X ( t , x , v 1 , v 2 , ) , L j [ v j ] = g j ( t , y ) , x ( t 0 ) = x 0 , v j ( k ) ( t 0 ) = t ψ j ( k ) ( t , s ) g j ( s , φ ( s ) ) d s , t [ t 0 , + ) , k = 0 , , n j , j = 1 , 2 , Open image in new window
(3.31)

in the sense that the solution of (3.26) coincides with the component x of the solution vector of (3.31).

4 Results: examples of reduction of integro-differential equations to systems of ordinary differential equations

Example 4.1 Model of tumor-immune system [9]
x ( t ) = x ( f ( x ) Φ ( x , y ) ) , y = β ( z ) y μ ( x ) y + σ q ( x ) + θ ( t ) , z = t K ( t s ) x ( s ) d s . Open image in new window
(4.1)
This system is an example of two-dimensional system (3.26) with distributed delay of x. In [9] the following kernel
K ( t ) = E r l λ , n t = λ n ( n 1 ) ! t n 1 e λ t , Open image in new window
(4.2)
is used and the case n = 1 Open image in new window is studied in detail. It is clear from Example 3.1 (see (3.24) and (3.25)) that the substitution
z ( t ) = 1 ( n 1 ) ! t ( t s ) n 1 e λ ( t s ) x ( s ) d s Open image in new window
(4.3)
reduces system (4.1) with the kernel (4.2) to the system of ordinary differential equations
x ( t ) = x ( f ( x ) Φ ( x , y ) ) , y = β ( z ) y μ ( x ) y + σ q ( x ) + θ ( t ) , j = 0 n c n j λ j z ( n j ) ( t ) = λ n x ( t ) , Open image in new window
(4.4)

which can be written as a system of the order n + 2 Open image in new window. Note that for n = 1 Open image in new window, the last equation in system (4.4) is of the form z + z = λ x Open image in new window.

Example 4.2 Model of hematopoiesis [10]. This model can be written in the form
P ( t ) = δ ( t ) P ( t ) β ( t ) P ( t ) 1 + P n ( t ) + α ( t ) 0 K ( τ ) P ( t τ ) 1 + P n ( t τ ) d τ . Open image in new window
(4.5)
The coefficients α, β and δ in (4.5) are positive ω-periodic functions, the kernel K satisfies the condition 0 K ( τ ) d τ = 1 Open image in new window. The change of variable t τ = s Open image in new window and then the substitution of the type (4.3) in the case of the kernel (4.2) reduces integro-differential equation (4.5) to the system of ordinary differential equations
P ( t ) = δ ( t ) P ( t ) β ( t ) P ( t ) 1 + P n ( t ) + α ( t ) z ( t ) , j = 0 n c n j λ j z ( n j ) ( t ) = λ n P ( t ) 1 + P n ( t ) . Open image in new window
(4.6)
Consider now the case of both distributed and concentrated delays in the system
y ( t ) = Y ( t , x ( t ) , t K ( t , s ) q ( s , y ( s ) ) d s ) , z ( t ) = Z ( t , x ( t ) , z ( t τ ) ) . Open image in new window
(4.7)
Let us describe the process of reduction, which is similar to the process described in the Section 3.1. The vector x is of the form x = col ( y , z ) Open image in new window. Denoting
v ( t ) = t K ( t , s ) q ( s , y ( s ) ) d s , Open image in new window
(4.8)
we make the substitution (compare with (3.7))
v ( t ) = t G j ( t , s ) h j ( s , y ( s ) ) d s . Open image in new window
(4.9)
Introduce the initial functions
y ( t ) = φ ( t ) , t ( , t 0 ] , z ( t ) = ψ ( t ) , t ( τ , 0 ] . Open image in new window
(4.10)
Under the assumption of convergence of the integrals, substitution (4.8) reduces (4.7) to the system
y ( t ) = Y ( t , x ( t ) , v 1 ( t ) , v 2 ( t ) , ) , z ( t ) = Z ( t , x ( t ) , z ( t τ ) ) , v j ( t ) = P j ( t ) v j ( t ) + h j ( t , y ( t ) ) , j = 1 , 2 , Open image in new window
(4.11)

with the initial conditions defined by (4.10).

Example 4.3 The model of the plankton-nutrient interaction [25]
N ( t ) = D ( N 0 N ( t ) ) a P ( t ) U ( N ( t ) ) + γ t F ( t s ) P ( s ) d s , P ( t ) = P { a 1 U ( N ( t τ ) ) ( y + D 1 ) } . Open image in new window
(4.12)
The initial functions
P ( t ) = ψ ( t ) , t ( , t 0 ] , N ( t ) = φ ( t ) , t ( τ , 0 ] . Open image in new window
(4.13)
The description of all parameters can be found in the paper [25]. U ( N ) Open image in new window is a known function. Concerning the function U ( N ) Open image in new window, it is assumed that
U ( 0 ) = 0 , d U d N > 0 , lim N U ( N ) = 1 . Open image in new window
(4.14)
A particular case of U ( N ) Open image in new window is
U ( N ) = N k + N , where  k > 0 . Open image in new window
(4.15)

Concerning the kernel, it is assumed that F ( t ) Open image in new window is a bounded nonnegative function such that + F ( t ) d t = 1 Open image in new window.

In [25] the properties of system (4.12) are considered in various particular cases of the kernel F ( t ) Open image in new window. The most general of them is the following:
F ( t ) = α e α t , where  α > 0 . Open image in new window
(4.16)
It is clear from (4.2) for n = 1 Open image in new window that
F ( t ) = E r l α , 1 ( t ) . Open image in new window
(4.17)
The substitution (4.3) for n = 1 Open image in new window is of the form
z ( t ) = t e α ( t s ) P ( s ) d s Open image in new window
(4.18)
and it reduces system (4.12) to the system
N ( t ) = D ( N 0 N ( t ) ) a P ( t ) U ( N ( t ) ) + γ z ( t ) , P ( t ) = P { a 1 U ( N ( t τ ) ) ( y + D 1 ) } , z ( t ) + z ( t ) = α P ( t ) . Open image in new window
(4.19)

5 Results: systems with advanced argument

Using results of Section 2 and approach of Section 3 (Section 3.1), we describe reduction of the integro-differential system
x ( t ) = X ( t , x , t + K ( t , s ) g ( s , x ( s ) ) d s ) , Open image in new window
(5.1)
and the kernel
K ( t , s ) = i , j = 1 C j Φ j ( t ) R j ( s ) = i , j = 1 C j K j ( t , s ) Open image in new window
(5.2)
to a system of ordinary differential equations. Introducing the matrix G j ( t , s ) Open image in new window and the equation
w j ( t ) = P j ( t ) w ( t ) , Open image in new window

by the formulas (3.3), (3.4) and (3.5), let us require that G j ( t , s ) Open image in new window ( j = 1 , 2 , Open image in new window) satisfy inequalities (2.6) under the assumption that k = 0 Open image in new window in the condition of hyperbolicity.

Introduce the substitution
z j ( t , t 0 ) = t + G j ( t , s ) h j ( s , x ( s ) ) d s . Open image in new window
(5.3)
According to Corollary 2.1, we can consider (5.3) for k = 0 Open image in new window as the solution of the one-point problem for system (3.8), supposing x ( t ) Open image in new window is a known function
x ( t ) = ψ ( t ) , t [ t 0 , + ) . Open image in new window
(5.4)

As a result, we obtain an analog of Theorem 3.1 for equation (5.1) with the kernel (5.2).

Theorem 5.1Let
  1. (a)

    matrices Φ j ( t ) Open image in new windowin the kernels (5.2) be continuously differentiable and invertible for t ( , + ) Open image in new window, j = 1 , 2 , Open image in new window ,

     
  2. (b)
    systems (3.6) be of dimension n j Open image in new window, where
    P j ( t ) = d Φ j ( t ) d t Φ j 1 ( t ) , Open image in new window
     

be hyperbolic for everyjin the sense of Definition  2.1 (for k = n Open image in new window).

Then the bounded solution x ( t ) R n Open image in new windowof system (5.1) with the kernel of the form (5.2) and the end function (5.4) and the first component x ( t ) R n Open image in new windowof the solution to the system
x ( t ) = X ( t , x , z 1 , z 2 , ) , z j ( t ) = P j ( t ) z ( t ) + h j ( t , x ) , t [ t 0 , + ) , z j ( t 0 ) = z j 0 , x ( t 0 ) = ψ ( t 0 ) , Open image in new window
(5.5)
where x R n Open image in new window, z j R n Open image in new window, h j ( t , x ) = K j ( t , t ) g ( t , x ) Open image in new window,
z j 0 = t 0 G j ( t 0 , s ) h j ( s , x ( s ) ) d s , j = 1 , 2 , 3 , Open image in new window

coincide.

6 Results: about systems with both delayed and advanced argument

Let us consider system (1.2) with distributed delay and advance. Denote x = col ( y , z ) Open image in new window in such a form that system (1.2) can be written in the form
y ( t ) = Y ( t , x ( t ) , t K 1 ( t , s ) g 1 ( s , y ( s ) ) d s ) , z ( t ) = Z ( t , x ( t ) , t + K 2 ( t , s ) g 2 ( s , z ( s ) ) d s ) . Open image in new window
(6.1)
Using the technique of Sections 3 and 5, we introduce
u j ( t ) = t G j 1 ( t , s ) h j 1 ( s , x ( s ) ) d s , Open image in new window
(6.2)
v j ( t ) = t + G j 2 ( t , s ) h j 2 ( s , x ( s ) ) d s , Open image in new window
(6.3)

where h j i ( t , x ) = K i ( t , t ) g j ( t , x ) Open image in new window, i = 1 , 2 Open image in new window, j = 1 , 2 , Open image in new window and denote P j 1 ( t ) = d Φ j d t Φ j 1 ( t ) Open image in new window, j = 1 , 2 , Open image in new window , P r 2 ( t ) = d Φ r d t Φ r 1 ( t ) Open image in new window, r = 1 , 2 , Open image in new window .

Requiring G j i ( t , s ) Open image in new window satisfies inequality (2.6), the first for k = n Open image in new window and the second for k = 0 Open image in new window, we obtain that solution x = col ( y , z ) Open image in new window of system (6.1) satisfies also the following problem:
y ( t ) = Y ( t , x ( t ) , u 1 ( t ) , u 2 ( t ) , ) , z ( t ) = Z ( t , x ( t ) , v 1 ( t ) , v 2 ( t ) , ) , u j ( t ) = P j 1 ( t ) u j ( t ) + h j 1 ( t , y ( t ) ) , j = 1 , 2 , , v r ( t ) = P r 2 ( t ) u r ( t ) + h r 1 ( t , z ( t ) ) , r = 1 , 2 , Open image in new window
(6.4)
with conditions
y ( t 0 ) = φ ( t 0 ) , u j ( t 0 ) = t 0 G j 1 ( t 0 , s ) h j 1 ( s , x ( s ) ) d s , j = 1 , 2 , , z ( t ) = ψ ( t 0 ) , v r ( t 0 ) = t 0 + G r 2 ( t 0 , s ) h r 2 ( s , x ( s ) ) d s , r = 1 , 2 , , y ( t ) = φ ( t ) , t ( , t 0 ) , z ( t ) = ψ ( t ) , t ( t 0 < + ) . Open image in new window
(6.5)

7 Conclusions

The method described above allows us to reduce systems of integro-differential systems with distributed delay and/or advance to systems of ordinary differential equations. For Volterra systems of the type (1.1), it was a basis for studying stability, bifurcation, Floquet theory, parametric resonance, stabilization and oscillation properties for integro-differential equations with ordinary [13, 14, 15, 16, 17, 18, 20] and partial [21, 22] derivatives. We could extend the main results of these works to integro-differential equation (1.2).

Generally speaking, after the reduction, we get infinity dimensional systems of ordinary differential equations. For their analysis, the theory of countable differentiable systems could be used [33, 34, 35, 36, 37].

In the study of various biological systems, the linear chain trick method was used (see, for example, [9, 10]). It is clear (see Section 4) that our approach includes the linear trick method. Note also the use of W-transform, which also allows researchers to reduce integro-differential equations to systems of ordinary differential equations [11].

The proposed method allows us also to study generalized and impulsive systems. For example, in the case of discontinuous solutions described by Heaviside functions H a ( t ) Open image in new window, we can use its connection with δ-function: H a ( t ) = t δ a ( ξ ) d ξ Open image in new window and to get to a system of integro-differential equations. Introducing the sequence, for example,
E r l λ , n t = λ n ( n 1 ) ! t n 1 e λ t , Open image in new window

where λ Open image in new window, we can consider the obtained system of integro-differential equations as an approximation of generalized equations. This allows us in corresponding cases to reduce the study of a generalized and impulsive system to the analysis of the sequence of integro-differential equations, and consequently to the analysis of the corresponding sequence of systems of ordinary differential equations.

Acknowledgements

Authors thank the reviewers for their reports, which have essentially improved the paper.

Copyright information

© Goltser and Domoshnitsky; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics and Computer SciencesAriel University of SamariaArielIsrael