- Open Access
About reducing integro-differential equations with infinite limits of integration to systems of ordinary differential equations
© Goltser and Domoshnitsky; licensee Springer 2013
- Received: 21 January 2013
- Accepted: 30 May 2013
- Published: 26 June 2013
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.
- integro-differential equations
- fundamental matrix
- Cauchy matrix
- hyperbolic systems
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 . Note the idea of the chain trick used in various applications (see, for example, [9, 10]) and its developed form in the paper . 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 . Starting with this reduction, approaches to the study of stability and bifurcation of integro-differential equations were proposed in the papers [13–16]. 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 . 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 . A parametric resonance in linear almost periodic systems was studied in , and the bifurcation of steady resonance modes for integro-differential systems was investigated in . Stabilization by control in a form of integrals of solutions was studied in . The stability of partial functional differential equations on the basis of this reduction was studied in . Constructive approach to a phase transition model was presented in . 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.
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 , hematopoiesis , stability and persistence in plankton models  which will be considered below.
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 and can be also considered. If , we get a system with distributed delay, and if , 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 , , can have more than one solution or not have solutions at all (see, for example, , 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  or hyperbolicity . It is known that such systems have (under corresponding conditions) unique bounded on the axis solution. Corresponding bibliography can be found in . 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 .
where , P is an matrix and g is an n-vector function with continuous bounded elements.
We use the following definition introduced in .
Theorem 2.1 
where , , .
We present corresponding constructions, developed in  for the proof of this theorem, which will be used below in our paper.
Define the Cauchy matrices and such that , , where is a unit -matrix.
Let us prove the following assertion about the representation of bounded solutions to system (2.1).
for which the homogeneous system is of the form (2.13), (2.14).
The obtained solution is unique. If we assume the existence of two bounded solutions and , then is a bounded on the axis solution of (2.2). From hyperbolicity, it follows that it is a zero solution. □
where , is a fundamental matrix of system (2.2).
Remark 2.1 If the matrix 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 for system (2.2) with bounded variable coefficients is known (see, , p.69, Proposition 2). Similar topics were also studied in .
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
where was defined above.
We have proven the following assertion.
matrices in the kernels (3.2) be continuously differentiable and invertible for , ,
- (b)systems (3.6) be of dimension , where
be hyperbolic for every j in the sense of Definition 2.1 (for ).
Remark 3.1 If (3.2) is a finite sum, then system (3.11) is finite dimensional.
can be found in various applications. It can be reduced by the change of variable to system (3.1) with the kernel .
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 [33–37]; see also the papers [24, 38–41].
in Section 5.
3.2 Reduction to the system of ordinary differential equations of high orders
where all coefficients () and f are essentially bounded on .
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
which can be written as a system of the order . Note that for , the last equation in system (4.4) is of the form .
with the initial conditions defined by (4.10).
Concerning the kernel, it is assumed that is a bounded nonnegative function such that .
by the formulas (3.3), (3.4) and (3.5), let us require that () satisfy inequalities (2.6) under the assumption that in the condition of hyperbolicity.
As a result, we obtain an analog of Theorem 3.1 for equation (5.1) with the kernel (5.2).
matrices in the kernels (5.2) be continuously differentiable and invertible for , ,
- (b)systems (3.6) be of dimension , where
be hyperbolic for every j in the sense of Definition 2.1 (for ).
where , , and denote , , , .
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 [13–18, 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 [33–37].
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 .
where , 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.
Authors thank the reviewers for their reports, which have essentially improved the paper.
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