On Nonseparated Three-Point Boundary Value Problems for Linear Functional Differential Equations

and Applied Analysis 3 in 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


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 z, 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 z and generates a system of algebraic or transcendental "determining equations" from which the numerical values of z 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 det D / 0, the corresponding Cauchy problem for the modified parametrised system of integrodifferential equations has the form 12 where 1 n is the unit matrix of dimension n and the parameter z ∈ R n has the meaning of initial value of the solution at the point a.The expression 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 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 equations 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: C a, b , R n is the Banach space of the continuous functions a, b → R n with the standard uniform norm; L 1 a, b , R n is the usual Banach space of the vector functions a, b → R n with Lebesgue integrable components; L R n is the algebra of all the square matrices of dimension n with real elements; 1 k is the unit matrix of dimension k; 0 i,j is the zero matrix of dimension i × j;

Problem Setting and Freezing Technique
We consider the system of n 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 given vector, A, B, and C are singular square matrices of dimension n, and C has the form where V is nonsingular square matrix of dimension q < n and W is an arbitrary matrix of dimension q × n − q .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 x at the points ξ and b as follows: where λ col λ 1 , . . ., λ n ∈ R n and η col η 1 , . . ., η n−q ∈ R n−q are vector parameters.This leads us to the parametrised two-point boundary condition where and the matrix D is given by the formula with a certain rectangular matrix W of dimension q × n − q .It is important to point out that the matrix D 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 A and B in the boundary conditions 1.10 are arbitrary and, in particular, may be singular.If the number r of the linearly independent boundary conditions in 1.10 is less than n, that is, the rank of the n × 3n -dimensional matrix A, B, C is equal to r, then the boundary value problem 1.1 , 1.10 may have an n − r -parametric family of solutions.
We assume that throughout the paper the operator l determining the system of equations 1.9 is represented in the form where

Auxiliary Estimates
In the sequel, we will need several auxiliary statements.are true for all t ∈ a, b , where 3.9 On the other hand, the obvious estimate and the positivity of the operators l j , j 0, 1, imply for a.e.t ∈ a, b and any k, j 1, 2, . . ., n, σ ∈ {−1, 1}.This, in view of 2.7 and 3.9 , leads us immediately to estimate 3.7 .

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 x m : a, b × R 3n−q → R n , m ≥ 0, by putting In the sequel, we consider x m as a function of t and treat the vectors z, λ, and η as parameters.
Lemma 4.1.For any m ≥ 0, t ∈ a, b , z ∈ R n , λ ∈ R n , and η ∈ R n−q , the equalities The proof of Lemma 4.1 is carried out by straightforward computation.We emphasize that the matrix D appearing in the two-point condition 2.3 is non-singular.Let us also put Using the operator M, we put where K l is given by formula 3.6 .Finally, define a constant square matrix Q l of dimension n by the formula We point out that, as before, the maximum in 4.6 is taken componentwise one should remember that, for n > 1, a point t * ∈ a, b such that Q l Q l t * may not exist .
Theorem 4.2.If the spectral radius of the matrix Q l satisfies the inequality then, for arbitrary fixed z ∈ R n , λ ∈ R n , and η ∈ R n−q : 1 the sequence of functions 4.1 converges uniformly in t ∈ a, b for any fixed z, λ, η ∈ R 3n−q to a limit function x m t, z, λ, η ; 4.8 4.9 Abstract and Applied Analysis 9 3 function 4.8 is a unique absolutely continuous solution of the integro-functional equation holds, where ω : R 3n−q → R n is given by the equality ω z, λ, η : ess sup In 3.6 , 4.11 and similar relations, the signs | • |, ≤, ≥, as well as the operators "max", "ess sup", "ess inf", 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 C a, b , R n equipped with the standard uniform norm.Let us put , and m ≥ 0. Using Lemma 3.2 and taking equality 3.4 into account, we find that 4.1 yields for arbitrary fixed z, λ, and η, where α is the function given by 3.3 and ω • is defined by formula 4.12 . 10

Abstract and Applied Analysis
According to formulae 4.1 , for all t ∈ a, b , arbitrary fixed z, λ, and η and m 1, 2, . . .we have

4.16
Applying inequality 3.7 of Lemma 3.2 and recalling formulae 4.5 and 4.6 , we get

4.17
Using 4.17 recursively and taking 4.14 into account, we obtain

4.18
for all m ≥ 1, t ∈ a, b , z ∈ R n , λ ∈ R n , and η ∈ R n−q .Using 4.18 and 4.13 , we easily obtain that, for an arbitrary j ∈ N,

4.19
Therefore, by virtue of assumption 4.7 , it follows that for all m ≥ 1, j ≥ 1, t ∈ a, b , z ∈ R n , λ ∈ R n , and η ∈ R n−q .We see from 4.20 that 4.1 is a Cauchy sequence in the Banach space C a, b , R n and, therefore, converges uniformly in t ∈ a, b for all z, λ, η ∈ R 3n−q : that is, assertion 2 holds.Since all functions x m t, z, λ, η of the sequence 4.1 satisfy the boundary conditions 2.3 , by passing to the limit in 2.3 as m → ∞ we show that the function x ∞ •, z, λ, η satisfies these conditions.Passing to the limit as m → ∞ in 4.1 , we show that the limit function is a solution of the integro-functional equation 4.10 .Passing to the limit as j → ∞ in 4.20 we obtain the estimate then, for these z, λ, and η,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.

Some Properties of the Limit Function
Let us first establish the relation of the limit function x ∞ •, z, λ, η 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 x t lx t f t μ, t ∈ a, b , 5.1 and the initial condition where μ col μ 1 , . . ., μ n is a control parameter.We will show that for arbitrary fixed z ∈ R n , λ ∈ R n , and η∈ n−q , the parameter μ can be chosen so that the solution x •, z, λ, η, μ 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. where and x ∞ •, z, λ, η 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 z ∈ R n , λ ∈ R n , and η ∈ R n−q be arbitrary.
If μ is given by 5.3 , then, due to Theorem 4.2, the function x ∞ •, z, λ, η has properties 4.9 and satisfies equation 4.10 , whence, by differentiation, equation 5.1 with the abovementioned value of μ is obtained.Thus, x ∞ •, z, λ, η is a solution of 5.1 , 5.2 with μ of form 5.3 and, moreover, this function satisfies the two-point boundary conditions 2.3 .
Let us fix an arbitrary μ ∈ R n and assume that the initial value problem 5.1 , 5.2 has a solution y satisfies the two-point boundary conditions 2. whence we find that μ can be represented in the form ly s f s s ds .

5.7
On the other hand, we already know that the function x ∞ •, z, λ, η , satisfies the twopoint conditions 2.3 and is a solution of the initial value problem 5.1 , 5.2 with μ μ z,λ,η , where the value μ z,λ,η is defined by formula 5.4 .Consequently,

5.10
Recalling the definition 5.4 of μ z,λ,η and using formula 5.7 , we obtain
We show that one can choose certain values of parameters z z * , λ λ * , η η * for which the function x ∞ •, z * , λ * , η * is the solution of the original three-point boundary value problem 1.9 , 1.10 .Let us consider the function Δ : R 3n−q → R n given by formula for all z, λ, and η, where x ∞ 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 x ∞ •, z, λ, η is a solution of the threepoint boundary value problem 1.9 , 1.10 if and only if the triplet z, λ, η satisfies the system of 3n − q algebraic equations Δ z, λ, η 0, 5.17 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 z, λ, η 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 .

5.22
Consider the sequence of vectors c m , m 0, 1, . .., determined by the recurrence relation

5.29
Due to assumption 4.7 , lim m → ∞ Q m l 0. Therefore, passing to the limit in 5.29 as m → ∞ 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" Δ : R 3n−q → R n given by equality 5.15 .Proposition 5.4.Under condition 3.10 , formula 5.15 determines a well-defined function Δ : R 3n−q → R n , which, moreover, satisfies the estimate for all z j , λ j , η j , j 0, 1, where the n × n -matrices G and R l are defined by the equalities Proof.According to the definition 5.15 of Δ, we have whence, due to Lemma 3.2,

5.34
Using Proposition 5.3, we find

5.35
On the other hand, recalling 4.2 and 5.21 , we get It follows immediately from 5.16 that

5.37
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.

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 z ∈ R n , according to 4.1 , we analytically construct the sequence of functions x m •, z, λ, η depending on the parameters z, λ, η and satisfying the auxiliary two-point boundary condition 2.3 .
3 We construct the algebraic determining system 5.17 , 5.18 , and 5.19 with respect 3n − q scalar variables.
4 Using a suitable numerical method, we approximately find a root z * ∈ R n , λ * ∈ R n , η * ∈ R n−q 6.1 of the determining system 5.17 , 5.18 , and 5.19 .

6.2
This solution 6.2 can also be obtained by solving the Cauchy problem x a z * 6.3 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 x ∞ •, z, λ, η .This problem can often be overcome by considering certain approximations of form 4.1 , which, unlike the function x ∞ •, z, λ, η , are known in the analytic form.In practice, this means that we fix a suitable m ≥ 1, construct the corresponding function x m •, z, λ, η according to relation 4.1 , and define the function Δ m : R 3n−q → R n by putting for arbitrary z, λ, and η.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 mth approximate determining system Δ m z, λ, η 0, e 1 x m ξ, z, λ, η λ 1 , e 2 x m ξ, z, λ, η λ 2 , . . ., e n x m ξ, z, λ, η λ n , e q 1 x m b, z, λ, η η 1 , . . ., e n x m b, z, λ, η η n−q , 6.5 where e i , i 1, 2, . . ., n, 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 m 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 .

1 ,
are bounded linear operators posi-tive in the sense that l j k u t ≥ 0 for a.e.t ∈ a, b and any k 1, 2, . . ., n, j 0, 1, and

Lemma 3 . 1 .
For an arbitrary essentially bounded function u : a, b → R, the estimates dτ ≤ α t ess sup s∈ a,b u s − ess inf s∈ a s∈ a,b lϕ z,λ,η s f s − ess inf s∈ a,b lϕ z,λ,η s f s .4.12

3 . 5 . 1 .
Proposition Let z ∈ R n , λ ∈ R n , and η ∈ R n−q be arbitrary given vectors.Assume that condition 4.7 is satisfied.Then a solution x • of the initial value problem 5.1 , 5.2 satisfies the boundary conditions 2.3 if and only if x • coincides with x ∞ •, z, λ, η and μ μ z,λ,η , 5.3 s ds μ t − a , 5.5 for all t ∈ a, b .By assumption, y satisfies the two-point conditions 2.3 and, therefore, 5.5 yields Ay a Dy b Az D z b a ly s f s s ds μ b − a d − Bλ N q η, 5.6 , t ∈ a, b .
t ∈ a, b .By virtue of condition 4.7 , inequality 5.13 implies that max

5 . 28 which
estimate, in view of 5.23 and 5.24 , coincides with the required inequality 5.25 .Thus, 5.25 is true for any m.Using 5.23 and 5.25 , we obtain max t∈ a,b R n , one constructs the iterations x m •, z , m 1, 2, . .., which depend upon the parameter z and, for arbitrary its values, satisfy the given boundary conditions 1.2 : Ax m a, z Dx m b, z d, z ∈ R n , m 1, 2, . ... Under suitable assumptions, one proves that sequence 1.5 converges to a limit function x ∞ •, z for any value of z.
1, 2, . . ., 1.5 where x 0 t, z : z for all t ∈ a, b and z ∈ Note that, in 3.6 , le * i means the value of the operator l on the constant vector function is equal identically to e * i , where e * i is the vector transpose to e i .It is easy to see that the components of K l are Lebesgue integrable functions.
Let us introduce some notation.For any k 1, 2, . . ., n, we define the n-dimensional row-vector e k by putting e k : 0, 0, . . ., 0, is true for any x ∈ C a, b , R n , where K l : a, b → L R n is the matrix-valued function given by formula 3.6 .
If, under the assumptions of Theorem 4.2, one can specify some values of z, λ, and η, such that the limit function x ∞ •, z, λ, η possesses the property λ, η 4.22 for a.e.t ∈ a, b and arbitrary fixed z, λ, η, and m 1, 2, . ... This completes the proof of Theorem 4.2.We have the following simple statement.Proposition 4.3.