#### Abstract

For a system of linear functional differential equations, we consider a three-point problem with nonseparated boundary conditions determined by singular matrices. We show that, to investigate such a problem, it is often useful to reduce it to a parametric family of two-point boundary value problems for a suitably perturbed differential system. The auxiliary parametrised two-point problems are then studied by a method based upon a special kind of successive approximations constructed explicitly, whereas the values of the parameters that correspond to solutions of the original problem are found from certain numerical determining equations. We prove the uniform convergence of the approximations and establish some properties of the limit and determining functions.

#### 1. Introduction

The aim of this paper is to show how a suitable parametrisation can help when dealing with nonseparated three-point boundary conditions determined by singular matrices. We construct a suitable numerical-analytic scheme allowing one to approach a three-point boundary value problem through a certain iteration procedure. To explain the term, we recall that, formally, the methods used in the theory of boundary value problems can be characterised as analytic, functional-analytic, numerical, or numerical-analytic ones.

While the analytic methods are generally used for the investigation of qualitative properties of solutions such as the existence, multiplicity, branching, stability, or dichotomy and generally use techniques of calculus (see, e.g., [1–11] and the references in [12]), the functional-analytic ones are based mainly on results of functional analysis and topological degree theory and essentially use various techniques related to operator equations in abstract spaces [13–26]. The numerical methods, under the assumption on the existence of solutions, provide practical numerical algorithms for their approximation [27, 28]. The numerical construction of approximate solutions is usually based on an idea of the shooting method and may face certain difficulties because, as a rule, this technique requires some global regularity conditions, which, however, are quite often satisfied only locally.

Methods of the so-called numerical-analytic type, in a sense, combine, advantages of the mentioned approaches and are usually based upon certain iteration processes constructed explicitly. Such an approach belongs to the few of them that offer constructive possibilities both for the investigation of the existence of a solution and its approximate construction. In the theory of nonlinear oscillations, numerical-analytic methods of this kind had apparently been first developed in [20, 29–31] for the investigation of periodic boundary value problems. Appropriate versions were later developed for handling more general types of nonlinear boundary value problems for ordinary and functional-differential equations. We refer, for example, to the books [12, 32–34], the handbook [35], the papers [36–50], and the survey [51–57] for related references.

For a boundary value problem, the numerical-analytic approach usually replaces the problem by the Cauchy problem for a suitably perturbed system containing some artificially introduced vector parameter , which most often has the meaning of an initial value of the solution and the numerical value of which is to be determined later. The solution of Cauchy problem for the perturbed system is sought for in an analytic form by successive approximations. The functional “perturbation term,” by which the modified equation differs from the original one, depends explicitly on the parameter and generates a system of algebraic or transcendental “determining equations” from which the numerical values of should be found. The solvability of the determining system, in turn, may by checked by studying some of its approximations that are constructed explicitly.

For example, in the case of the two-point boundary value problem where , the corresponding Cauchy problem for the modified parametrised system of integrodifferential equations has the form [12] where is the unit matrix of dimension and the parameter has the meaning of initial value of the solution at the point . The expressionin (1.3) plays the role of a ”perturbation term” and its choice is, of course, not unique. The solution of problem (1.3) is sought for in an analytic form by the method of successive approximations similar to the Picard iterations. According to the formulas where for all and , one constructs the iterations , which depend upon the parameter and, for arbitrary its values, satisfy the given boundary conditions (1.2): . Under suitable assumptions, one proves that sequence (1.5) converges to a limit function for any value of .

The procedure of passing from the original differential system (1.1) to its ”perturbed” counterpart and the investigation of the latter by using successive approximations (1.5) leads one to the system of determining equationswhich gives those numerical values of the parameter that correspond to solutions of the given boundary value problem (1.1), (1.10). The form of system (1.6) is, of course, determined by the choice of the perturbation term (1.4); in some other related works, auxiliary equations are constructed in a different way (see, e.g., [42]). It is clear that the complexity of the given equations and boundary conditions has an essential influence both on the possibility of an efficient construction of approximate solutions and the subsequent solvability analysis.

The aim of this paper is to extend the techniques used in [46] for the system of linear functional differential equations of the form

subjected to the inhomogeneous three-point Cauchy-Nicoletti boundary conditions with is given and , to the case where the system of linear functional differential equations under consideration has the general form and the three-point boundary conditions are non-separated and have the form where , , and are singular matrices, . Here, is a bounded linear operator and is a given function.

It should be noted that, due to the singularity of the matrices that determine the boundary conditions (1.10), certain technical difficulties arise which complicate the construction of successive approximations.

The following notation is used in the sequel: is the Banach space of the continuous functions with the standard uniform norm; is the usual Banach space of the vector functions with Lebesgue integrable components; is the algebra of all the square matrices of dimension with real elements; is the maximal, in modulus, eigenvalue of a matrix ; is the unit matrix of dimension ; is the zero matrix of dimension ;.

#### 2. Problem Setting and Freezing Technique

We consider the system of linear functional differential equations (1.9) subjected to the nonseparated inhomogeneous three-point boundary conditions of form (1.10). In the boundary value problem (1.1), (1.10), we suppose that is a bounded linear operator, is an integrable function, is a given vector, , , and are singular square matrices of dimension , and has the form where is nonsingular square matrix of dimension and is an arbitrary matrix of dimension . The singularity of the matrices determining the boundary conditions (1.10) causes certain technical difficulties. To avoid dealing with singular matrices in the boundary conditions and simplify the construction of a solution in an analytic form, we use a two-stage parametrisation technique. Namely, we first replace the three-point boundary conditions by a suitable parametrised family of two-point inhomogeneous conditions, after which one more parametrisation is applied in order to construct an auxiliary perturbed differential system. The presence of unknown parameters leads one to a certain system of determining equations, from which one finds those numerical values of the parameters that correspond to the solutions of the given three-point boundary value problem.

We construct the auxiliary family of two-point problems by ”freezing” the values of certain components of at the points and as follows: where and are vector parameters. This leads us to the parametrised two-point boundary condition whereand the matrix is given by the formulawith a certain rectangular matrix of dimension . It is important to point out that the matrix appearing in the two-point condition (2.3) is non-singular.

It is easy to see that the solutions of the original three-point boundary value problem (1.1), (1.10) coincide with those solutions of the two-point boundary value problem (1.1), (2.3) for which the additional condition (2.2) is satisfied.

*Remark 2.1. *The matrices and in the boundary conditions (1.10) are arbitrary and, in particular, may be singular. If the number of the linearly independent boundary conditions in (1.10) is less than , that is, the rank of the -dimensional matrix is equal to , then the boundary value problem (1.1), (1.10) may have an -parametric family of solutions.

We assume that throughout the paper the operator determining the system of equations (1.9) is represented in the form where , are bounded linear operators positive in the sense that for a.e. and any , , and such that for all . We also put , , and

#### 3. Auxiliary Estimates

In the sequel, we will need several auxiliary statements.

Lemma 3.1. *For an arbitrary essentially bounded function , the estimates
**are true for all , where
*

*Proof. *Inequality (3.1) is established similarly to [58, Lemma 3] (see also [12, Lemma 2.3]), whereas (3.2) follows directly from (3.1) if the relation
is taken into account.

Let us introduce some notation. For any , we define the -dimensional row-vector by putting Using operators (2.7) and the unit vectors (3.5), we define the matrix-valued function by setting Note that, in (3.6), means the value of the operator on the constant vector function is equal identically to , where is the vector transpose to . It is easy to see that the components of are Lebesgue integrable functions.

Lemma 3.2. *The componentwise estimate
**is true for any , where is the matrix-valued function given by formula (3.6).*

*Proof. *Let be an arbitrary function from . Then, recalling the notation for the components of , we see that

On the other hand, due to (3.5), we have and, therefore, by virtue of (3.8) and (2.6),

On the other hand, the obvious estimate
and the positivity of the operators , , imply
for a.e. and any , . This, in view of (2.7) and (3.9), leads us immediately to estimate (3.7).

#### 4. Successive Approximations

To study the solution of the auxiliary two-point parametrised boundary value problem (1.9), (2.3) let us introduce the sequence of functions , by puttingfor all , , , and , where In the sequel, we consider as a function of and treat the vectors , , and as parameters.

Lemma 4.1. *For any , , , , and , the equalities
**are true.*

The proof of Lemma 4.1 is carried out by straightforward computation. We emphasize that the matrix appearing in the two-point condition (2.3) is non-singular. Let us also put for an arbitrary . It is clear that is a positive linear operator. Using the operator , we put where is given by formula (3.6). Finally, define a constant square matrix of dimension by the formula We point out that, as before, the maximum in (4.6) is taken componentwise (one should remember that, for , a point such that may not exist).

Theorem 4.2. *If the spectral radius of the matrix satisfies the inequality
**
then, for arbitrary fixed , , and : *(1)*the sequence of functions (4.1) converges uniformly in for any fixed to a limit function
*(2)*the limit function possesses the properties
*(3)*function (4.8) is a unique absolutely continuous solution of the integro-functional equation
*(4)*the error estimate
**holds, where is given by the equality
*

In (3.6), (4.11) and similar relations, the signs , ≤, ≥, as well as the operators ””, ””, ””, and so forth, applied to vectors or matrices are understood componentwise.

*Proof. *The validity of assertion 1 is an immediate consequence of the formula (4.1). To obtain the other required properties, we will show, that under the conditions assumed, sequence (4.1) is a Cauchy sequence in the Banach space equipped with the standard uniform norm. Let us put
for all , , , , and . Using Lemma 3.2 and taking equality (3.4) into account, we find that (4.1) yields
for arbitrary fixed , , and , where is the function given by (3.3) and is defined by formula (4.12).

According to formulae (4.1), for all , arbitrary fixed , , and and we have

Equalities (4.13) and (4.15) imply that for all , arbitrary fixed ,, and ,

Applying inequality (3.7) of Lemma 3.2 and recalling formulae (4.5) and (4.6), we get

Using (4.17) recursively and taking (4.14) into account, we obtain
for all , , , , and . Using (4.18) and (4.13), we easily obtain that, for an arbitrary ,

Therefore, by virtue of assumption (4.7), it follows that
for all , , , , , and . We see from (4.20) that (4.1) is a Cauchy sequence in the Banach space and, therefore, converges uniformly in for all :
that is, assertion 2 holds. Since all functions of the sequence (4.1) satisfy the boundary conditions (2.3), by passing to the limit in (2.3) as we show that the function satisfies these conditions.

Passing to the limit as in (4.1), we show that the limit function is a solution of the integro-functional equation (4.10). Passing to the limit as in (4.20) we obtain the estimate
for a.e. and arbitrary fixed , , , and . This completes the proof of Theorem 4.2.

We have the following simple statement.

Proposition 4.3. *If, under the assumptions of Theorem 4.2, one can specify some values of , , and , such that the limit function possesses the property
**
then, for these ,it is also a solution of the boundary value problem (1.9), (2.3).*

*Proof. *The proof is a straightforward application of the above theorem.

#### 5. Some Properties of the Limit Function

Let us first establish the relation of the limit function to the auxiliary two-point boundary value problem (1.9), (2.3). Along with system (1.9), we also consider the system with a constant forcing term in the right-hand side and the initial condition where is a control parameter.

We will show that for arbitrary fixed , , and , the parameter can be chosen so that the solution of the initial value problem (5.1), (5.2) is, at the same time, a solution of the two-point parametrised boundary value problem (5.1), (2.3).

Proposition 5.1. *Let , , and be arbitrary given vectors. Assume that condition (4.7) is satisfied. Then a solution of the initial value problem (5.1), (5.2) satisfies the boundary conditions (2.3) if and only if coincides with and
**where
**and is the limit function of sequence (4.1).*

*Proof. *The assertion of Proposition 5.1 is obtained by analogy to the proof of [50, Theorem 4.2]. Indeed, let , , and be arbitrary.

If is given by (5.3), then, due to Theorem 4.2, the function has properties (4.9) and satisfies equation (4.10), whence, by differentiation, equation (5.1) with the above-mentioned value of is obtained. Thus, is a solution of (5.1), (5.2) with of form (5.3) and, moreover, this function satisfies the two-point boundary conditions (2.3).

Let us fix an arbitrary and assume that the initial value problem (5.1), (5.2) has a solution satisfies the two-point boundary conditions (2.3). Then
for all . By assumption, satisfies the two-point conditions (2.3) and, therefore, (5.5) yields
whence we find that can be represented in the form

On the other hand, we already know that the function , satisfies the two-point conditions (2.3) and is a solution of the initial value problem (5.1), (5.2) with , where the value is defined by formula (5.4). Consequently,

Putting
and taking (5.5), (5.8) into account, we obtain

Recalling the definition (5.4) of and using formula (5.7), we obtain
and, therefore, equality (5.10) can be rewritten as

Applying Lemma 3.2 and recalling notation (4.6), we get
for an arbitrary . By virtue of condition (4.7), inequality (5.13) implies that
as . According to (5.9), this means that coincides with , and, therefore, by (5.11), , which brings us to the desired conclusion.

We show that one can choose certain values of parameters for which the function is the solution of the original three-point boundary value problem (1.9), (1.10). Let us consider the function given by formula with for all , , and , where is the limit function (4.8).

The following statement shows the relation of the limit function (4.8) to the solution of the original three-point boundary value problem (1.9), (1.10).

Theorem 5.2. *Assume condition (4.7). Then the function is a solution of the three-point boundary value problem (1.9), (1.10) if and only if the triplet satisfies the system of algebraic equations
*

*Proof. *It is sufficient to apply Proposition 5.1 and notice that the differential equation in (5.1) coincides with (1.9) if and only if the triplet satisfies (5.17). On the other hand, (5.18) and (5.19) bring us from the auxiliary two-point parametrised conditions to the three-point conditions (1.10).

Proposition 5.3. *Assume condition (4.7). Then, for any , , the estimate
**holds, where
*

*Proof. *Let us fix two arbitrary triplets , , and put
Consider the sequence of vectors , , determined by the recurrence relation
with

Let us show that
for all . Indeed, estimate (5.25) is obvious for . Assume that

It follows immediately from (4.1) that
whence, by virtue of (5.21), estimate (3.7) to Lemma 3.2, and assumption (5.26),
which estimate, in view of (5.23) and (5.24), coincides with the required inequality (5.25). Thus, (5.25) is true for any . Using (5.23) and (5.25), we obtain

Due to assumption (4.7), . Therefore, passing to the limit in (5.29) as and recalling notation (5.22), we obtain the estimate
which, in view of (5.24), coincides with (5.20).

Now we establish some properties of the “determining function” given by equality (5.15).

Proposition 5.4. *Under condition (3.10), formula (5.15) determines a well-defined function , which, moreover, satisfies the estimate
**for all , , where the -matrices and are defined by the equalities
*

*Proof. *According to the definition (5.15) of , we have
whence, due to Lemma 3.2,

Using Proposition 5.3, we find

On the other hand, recalling (4.2) and (5.21), we get

It follows immediately from (5.16) that

Therefore, (5.35) and (5.36) yield the estimate
which, in view of (5.32), coincides with (5.31).

Properties stated by Propositions 5.3 and 5.4 can be used when analysing conditions guaranteeing the solvability of the determining equations.

#### 6. On the Numerical-Analytic Algorithm of Solving the Problem

Theorems 4.2 and 5.2 allow one to formulate the following numerical-analytic algorithm for the construction of a solution of the three-point boundary value problem (1.9), (1.10).(1)For any vector , according to (4.1), we analytically construct the sequence of functions depending on the parameters and satisfying the auxiliary two-point boundary condition (2.3).(2)We find the limit of the sequence satisfying (2.3).(3)We construct the algebraic determining system (5.17), (5.18), and (5.19) with respect scalar variables.(4)Using a suitable numerical method, we (approximately) find a root of the determining system (5.17), (5.18), and (5.19).(5)Substituting values (6.1) into , we obtain a solution of the original three-point boundary value problem (1.9), (1.10) in the form This solution (6.2) can also be obtained by solving the Cauchy problem for (1.9).

The fundamental difficulty in the realization of this approach arises at point (2) and is related to the analytic construction of the function . This problem can often be overcome by considering certain approximations of form (4.1), which, unlike the function , are known in the analytic form. In practice, this means that we fix a suitable , construct the corresponding function according to relation (4.1), and define the function by putting for arbitrary , , and . To investigate the solvability of the three-point boundary value problem (1.9), (1.10), along with the determining system (5.17), (5.18), and (5.19), one considers the th approximate determining systemwhere , , are the vectors given by (5.15).

It is natural to expect (and, in fact, can be proved) that, under suitable conditions, the systems (5.17), (5.18), (5.19), and (6.5) are “close enough” to one another for sufficiently large. Based on this circumstance, existence theorems for the three-point boundary value problem (1.9), (1.10) can be obtained by studying the solvability of the approximate determining system (6.5) (in the case of periodic boundary conditions, see, e.g., [35]).

#### Acknowledgments

This research was carried out as part of the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project with support by the European Union, co-financed by the European Social Fund.