Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems
A. Boichuk,1,2J. DiblΓk,3,4D. Khusainov,5and M. RΕ―ΕΎiΔkovΓ‘2
Academic Editor: Elena Braverman
Received30 Jan 2011
Accepted31 Mar 2011
Published16 Jun 2011
Abstract
Conditions are derived of the existence of solutions of nonlinear boundary-value problems for systems of ordinary differential equations with constant coefficients and single delay (in the linear part) and with a finite number of measurable delays of argument in nonlinearity: , assuming that these solutions satisfy the initial and boundary conditions . The use of a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical form of sufficient conditions for the existence of solutions in a given space and, moreover, to the construction of an iterative process for finding the solutions of such problems in a general case when the number of boundary conditions (defined by a linear vector functional ) does not coincide with the number of unknowns in the differential system with a single delay.
1. Introduction
First, we derive some auxiliary results concerning the theory of differential equations with delay. Consider a system of linear differential equations with concentrated delay
assuming that
where is an real matrix and is an -dimensional real column-vector with components in the space (where ) of functions summable on ; the delay is a function measurable on ; is a given function. Using the denotations
where is an -dimensional zero column-vector and assuming , it is possible to rewrite (1.1), (1.2) as
where is an -dimensional column-vector defined by the formula
We will investigate (1.5) assuming that the operator maps a Banach space of absolutely continuous functions into a Banach space of functions summable on ; the operator maps the space into the space . Transformations (1.3), (1.4) make it possible to add the initial function , to nonhomogeneity generating an additive and homogeneous operation not depending on and without the classical assumption regarding the continuous connection of solution with the initial function at the point .
A solution of differential system (1.5) is defined as an -dimensional column vector-function , absolutely continuous on , with a derivative satisfying (1.5) almost everywhere on .
Such approach makes it possible to apply well-developed methods of linear functional analysis to (1.5) with a linear and bounded operator . It is well-known (see: [1, 2]) that a nonhomogeneous operator equation (1.5) with delayed argument is solvable in the space for an arbitrary right-hand side and has an -dimensional family of solutions in the form
where the kernel is an Cauchy matrix defined in the square being, for every fixed , a solution of the matrix Cauchy problem
where if , is null matrix and is identity matrix. A fundamental matrix for the homogeneous equation (1.5) has the form , [2]. Throughout the paper, we denote by an null matrix if , by an null matrix, by an identity matrix if , and by an -dimensional zero column-vector if .
A serious disadvantage of this approach, when investigating the above-formulated problem, is the necessity to find the Cauchy matrix [3, 4]. It exists but, as a rule, can only be found numerically. Therefore, it is important to find systems of differential equations with delay such that this problem can be solved directly. Below we consider the case of a system with so-called single delay [5]. In this case, the problem of how to construct the Cauchy matrix is successfully solved analytically due to a delayed matrix exponential defined below.
1.1. A Delayed Matrix Exponential
Consider a Cauchy problem for a linear nonhomogeneous differential system with constant coefficients and with a single delay
with an constant matrix , , , and an unknown vector-solution . Together with a nonhomogeneous problem (1.9), (1.10), we consider a related homogeneous problem
Denote by a matrix function called a delayed matrix exponential (see [5]) and defined as
This definition can be reduced to the following expression:
where is the greatest integer function. The delayed matrix exponential equals the unit matrix on and represents a fundamental matrix of a homogeneous system with single delay. Thus, the delayed matrix exponential solves the Cauchy problem for a homogeneous system (1.11), satisfying the unit initial conditions
and the following statement holds (see, e.g., [5], [6, Remark 1], [7, Theorem 2.1]).
Lemma 1.1. A solution of a Cauchy problem for a nonhomogeneous system with single delay (1.9), satisfying a constant initial condition
has the form
The delayed matrix exponential was applied, for example, in [6, 7] to investigation of boundary value problems of diffferential systems with a single delay and in [8] to investigation of the stability of linear perturbed systems with a single delay.
1.2. Fredholm Boundary-Value Problem
Without loss of generality, let and, with a view of the above, the problem (1.9), (1.10) can be transformed to an equation of the type (1.1) (see (1.5))
where, in accordance with (1.3),(1.4)
A general solution of problem (1.17) for a nonhomogeneous system with single delay and zero initial data has the form (1.7)
where, as can easily be verified (in view of the above-defined delayed matrix exponential) by substituting into (1.17),
is a normal fundamental matrix of the homogeneous system related to (1.17) (or (1.9)) with initial data , and the Cauchy matrix has the form
Obviously
and, therefore, the initial problem (1.17) for systems of ordinary differential equations with constant coefficients and single delay has an -parametric family of linearly independent solutions (1.16).
Now, we will deal with a general boundary-value problem for system (1.17). Using the results [2, 9], it is easy to derive statements for a general boundary-value problem if the number of boundary conditions does not coincide with the number of unknowns in a differential system with single delay.
We consider a boundary-value problem
assuming that
or, using (1.18), its equivalent form
where is an -dimensional constant vector-column is an -dimensional linear vector-functional defined on the space of an -dimensional vector-functions
absolutely continuous on . Such problems for functional-differential equations are of Fredholm's type (see, e.g., [1, 2]). In order to formulate the following result, we need several auxiliary abbreviations. We set
We define an -dimensional matrix (orthogonal projection)
projecting space to of the matrix .
Moreover, we define an -dimensional matrix (orthogonal projection)
projecting space to of the transposed matrix , where is an identity matrix and is an -dimensional matrix pseudoinverse to the -dimensional matrix . Denote and . Since
we have .
We will denote by an -dimensional matrix constructed from linearly independent rows of the matrix . Denote . Since
we have . By we will denote an -dimensional matrix constructed from linearly independent columns of the matrix . Finally, we define
and a generalized Green operator
where
is a generalized Green matrix corresponding to the boundary-value problem (1.25) (the Cauchy matrix has the form (1.21)).
In [6, Theorem 4], the following result (formulating the necessary and sufficient conditions of solvability and giving representations of the solutions , of the boundary-value problem (1.25) in an explicit analytical form) is proved.
Theorem 1.2. If , then: (i)the homogeneous problem
corresponding to problem (1.25) has exactly linearly independent solutions
(ii)nonhomogeneous problem (1.25) is solvable in the space if and only if and satisfy linearly independent conditions
(iii)in that case the nonhomogeneous problem (1.25) has an -dimensional family of linearly independent solutions represented in an analytical form
2. Perturbed Weakly Nonlinear Boundary Value Problems
As an example of applying Theorem 1.2, we consider a problem of the branching of solutions , of systems of nonlinear ordinary differential equations with a small parameter and with a finite number of measurable delays , of argument of the form
satisfying the initial and boundary conditions
and such that its solution , satisfying
for a sufficiently small , for , turns into one of the generating solutions (1.38); that is, for a . We assume that the vector-operator satisfies
where is sufficiently small. Using denotations (1.3), (1.4), and (1.6), it is easy to show that the perturbed nonlinear boundary value problem (2.1), (2.2) can be rewritten in the form
In (2.5), is an constant matrix, is a single delay defined by , ,
is an -dimensional column vector, where , and is an -dimensional column vector given by
The operator maps the space into the space
that is, . Using denotation (1.3) for the operator , , we have the following representation:
where
is the characteristic function of the set
Assume that the generating boundary value problem
being a particular case of (2.5) for , has solutions for nonhomogeneities and that satisfy conditions (1.37). In such a case, by Theorem 1.2, the problem (2.12) possesses an -dimensional family of solutions of the form (1.38).
Problem 1. Below, we consider the following problem: derive the necessary and sufficient conditions indicating when solutions of (2.5) turn into solutions (1.38) of the boundary value problem (2.12) for .
Using the theory of generalized inverse operators [2], it is possible to find conditions for the solutions of the boundary value problem (2.5) to be branching from the solutions of (2.5) with . Below, we formulate statements, solving the above problem. As compared with an earlier result [10, page 150], the present result is derived in an explicit analytical form. The progress was possible by using the delayed matrix exponential since, in such a case, all the necessary calculations can be performed to the full.
Theorem 2.1 (necessary condition). Consider the system (2.1); that is,
where , , with the initial and boundary conditions (2.2); that is,
and assume that, for nonhomogeneities
and for , the generating boundary value problem
corresponding to the problem (1.25), has exactly an -dimensional family of linearly independent solutions of the form (1.38). Moreover, assume that the boundary value problem (2.13), (2.14) has a solution which, for , turns into one of solutions in (1.38) with a vector-constant . Then, the vector satisfies the equation
where
Proof. We consider the nonlinearity in system (2.13), that is, the term as an inhomogeneity, and use Theorem 1.2 assuming that condition (1.37) is satisfied. This gives
In this integral, letting , we arrive at the required condition (2.17).
Corollary 2.2. For periodic boundary-value problems, the vector-constant has a physical meaning-it is the amplitude of the oscillations generated. For this reason, (2.17) is called an equation generating the amplitude [11]. By analogy with the investigation of periodic problems, it is natural to say (2.17) is an equation for generating the constants of the boundary value problem (2.13), (2.14). If (2.17) is solvable, then the vector constant specifies the generating solution corresponding to the solution of the original problem such that
Also, if (2.17) is unsolvable, the problem (2.13), (2.14) has no solution in the analyzed space. Note that, here and in what follows, all expressions are obtained in the real form and hence, we are interested in real solutions of (2.17), which can be algebraic or transcendental.
Sufficient conditions for the existence of solutions of the boundary-value problem (2.13), (2.14) can be derived using results in [10, page 155] and [2]. By changing the variables in system (2.13), (2.14)
we arrive at a problem of finding sufficient conditions for the existence of solutions of the problem
and such that
Since the vector function is continuously differentiable with respect to and continuous in in the neighborhood of the point
we can separate its linear term as a function depending on and terms of order zero with respect to
where
We now consider the vector function in (2.22) as an inhomogeneity and we apply Theorem 1.2 to this system. As the result, we obtain the following representation for the solution of (2.22):
In this expression, the unknown vector of constants is determined from a condition similar to condition (1.37) for the existence of solution of problem (2.22):
where
is a matrix, and
The unknown vector function is determined by using the generalized Green operator as follows:
Let be an matrix orthoprojector , and let be a matrix-orthoprojector . Equation (2.28) is solvable with respect to if and only if
For
the last condition is always satisfied and (2.28) is solvable with respect to up to an arbitrary vector constant from the null space of the matrix
To find a solution of (2.28) such that
it is necessary to solve the following operator system:
The operator system (2.36) belongs to the class of systems solvable by the method of simple iterations, convergent for sufficiently small (see [10, page 188]). Indeed, system (2.36) can be rewritten in the form
where is a -dimensional column vector, is a linear operator
where
and is a nonlinear operator
In view of the structure of the operator containing zero blocks on and below the main diagonal, the inverse operator
exists. System (2.37) can be transformed into
where
is a contraction operator in a sufficiently small neighborhood of the point
Thus, the solvability of the last operator system can be established by using one of the existing versions of the fixed-point principles [12] applicable to the system for sufficiently small . It is easy to prove that the sufficient condition for the existence of solutions of the boundary value problem (2.13), (2.14) means that the constant of the equation for generating constant (2.17) is a simple root of equation (2.17) [2]. By using the method of simple iterations, we can find the solution of the operator system and hence the solution of the original boundary value problem (2.13), (2.14). Now, we arrive at the following theorem.
Theorem 2.3 (sufficient condition). Assume that the boundary value problem (2.13), (2.14) satisfies the conditions listed above and the corresponding linear boundary value problem (1.25) has an -dimensional family of linearly independent solutions of the form (1.38). Then, for any simple root of the equation for generating the constants (2.17), there exist at least one solution of the boundary value problem (2.13), (2.14). The indicated solution is such that
and, for , turns into one of the generating solutions (1.38) with a constant ; that is, . This solution can be found by the method of simple iterations, which is convergent for a sufficiently small .
Corollary 2.4. If the number of unknown variables is equal to the number of boundary conditions (and hence ), the boundary value problem (2.13), (2.14) has a unique solution. In such a case, the problems considered for functional-differential equations are of Fredholm's type with a zero index. By using the procedure proposed in [2] with some simplifying assumptions, we can generalize the proposed method to the case of multiple roots of equation (2.17) to determine sufficient conditions for the existence of solutions of the boundary-value problem (2.13), (2.14).
3. Example
We will illustrate the above proved theorems on the example of a weakly perturbed linear boundary value problem. Consider the following simplest boundary value problem-a periodic problem for the delayed differential equation:
where , are matrices, , , are measurable functions. Using the symbols and (see (1.3), (1.4), (2.9)), we arrive at the following operator system:
where is an matrix , and
We will consider the simplest case with . Utilizing the delayed matrix exponential, it can be easily verified that in this case, the matrix
is a normal fundamental matrix for the homogeneous generating system
Then,
To illustrate the theorems proved above, we will find the conditions for which the boundary value problem (3.1) has a solution that, for , turns into one of solutions (1.38) of the generating problem. In contrast to the previous works [7, 9], we consider the case when the unperturbed boundary-value problem
has an -parametric family of linear-independent solutions of the form(1.38)
For this purpose, it is necessary and sufficient for the vector function
to satisfy the condition of type (1.37)
Then, according to the Theorem 2.1, the constant must satisfy (2.17), that is, the equation
which in our case is a linear algebraic system
with the matrix in the form
According to Corollary 2.4, if , the problem (3.1) for the case has a unique solution with the properties
for , , and for measurable delays that which satisfy the criterion (3.10) of the existence of a generating solution where
A solution of the boundary value problem (3.1) can be found by the convergent method of simple iterations (see Theorem 2.3).
If, for example, , where , , then
The matrix can be rewritten in the form
and the unique solvability condition of the boundary value problem (3.1) takes the form
It is easy to see that if the vector function is nonlinear in , for example as a square, then (3.11) generating the constants will be a square-algebraic system and, in this case, the boundary value problem (3.1) can have two solutions branching from the point .
Acknowledgments
The first and the fourth authors were supported by the Grant no. 1/0090/09 of the Grant Agency of Slovak Republic (VEGA) and by the project APVV-0700-07 of Slovak Research and Development Agency. The second author was supported by the Grant no. P201/11/0768 of Czech Grant Agency, by the Council of Czech Government MSM 0021630503 and by the Project FEKT/FSI-S-11-1-1159. The third author was supported by the Project no. M/34-2008 of Ukrainian Ministry of Education, Ukraine.
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