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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 631412, 19 pages
Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems
1Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkovskaya Street 3, 01601 Kyiv, Ukraine
2Department of Mathematics, University of Žilina, Univerzitná 8215/1, 01026 Žilina, Slovakia
3Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic
4Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 10, 616 00 Brno, Czech Republic
5Department of Complex System Modeling, Faculty of Cybernetics, Taras, Shevchenko National University of Kyiv, Vladimirskaya Street 64, 01033 Kyiv, Ukraine
Received 30 January 2011; Accepted 31 March 2011
Academic Editor: Elena Braverman
Copyright © 2011 A. Boichuk 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.
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.
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 .
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 , . 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 . 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 ) 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., , [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  to investigation of the stability of linear perturbed systems with a single delay.
1.2. Fredholm Boundary-Value Problem
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 , 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 . 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 . 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  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) . 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  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).
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 .
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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