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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 326052, 22 pages
http://dx.doi.org/10.1155/2011/326052
Research Article

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

1Institute of Mathematics, Academy of Sciences of the Czech Republic, 22 Žižkova St., 61662 Brno, Czech Republic
2Department of Analysis, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary

Received 20 January 2011; Accepted 27 April 2011

Academic Editor: Yuri V. Rogovchenko

Copyright © 2011 A. Rontó and M. Rontó. 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.

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., [111] 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 [1326]. 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, 2931] 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, 3234], the handbook [35], the papers [3650], and the survey [5157] 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 𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡[𝑎,𝑏],(1.1)𝐴𝑥(𝑎)+𝐷𝑥(𝑏)=𝑑,(1.2) where 𝑥[𝑎,𝑏]𝑛,<𝑎<𝑏<+,𝑑𝑛,det𝐷0, the corresponding Cauchy problem for the modified parametrised system of integrodifferential equations has the form [12]𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))+1𝑏𝑎𝐷1𝑑𝐷1𝐴+𝟙𝑛𝑧1𝑏𝑎𝑏𝑎𝑓(𝑠,𝑥(𝑠))𝑑𝑠,𝑡[𝑎,𝑏],𝑥(𝑎)=𝑧,(1.3) where 𝟙𝑛 is the unit matrix of dimension 𝑛 and the parameter 𝑧𝑛 has the meaning of initial value of the solution at the point 𝑎. The expression1𝑏𝑎𝐷1𝑑𝐷1𝐴+𝟙𝑛𝑧1𝑏𝑎𝑏𝑎𝑓(𝑠,𝑥(𝑠))𝑑𝑠(1.4)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 𝑥𝑚+1(𝑡,𝑧)=𝑧+𝑡𝑎𝑓𝑠,𝑥𝑚(𝑠,𝑧)𝑑𝑠1𝑏𝑎𝑏𝑎𝑓𝜏,𝑥𝑚(𝜏,𝑧)𝑑𝜏𝑑𝑠+𝑡𝑎𝑏𝑎𝐷1𝑑𝐷1𝐴+𝟙𝑛𝑧,𝑚=0,1,2,,(1.5)where 𝑥0(𝑡,𝑧)=𝑧 for all 𝑡[𝑎,𝑏] and 𝑧𝑛, one constructs the iterations 𝑥𝑚(,𝑧),𝑚=1,2,, which depend upon the parameter 𝑧 and, for arbitrary its values, satisfy the given boundary conditions (1.2): 𝐴𝑥𝑚(𝑎,𝑧)+𝐷𝑥𝑚(𝑏,𝑧)=𝑑,𝑧𝑛,𝑚=1,2,. Under suitable assumptions, one proves that sequence (1.5) converges to a limit function 𝑥(,𝑧) for any value of 𝑧.

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 equations𝐷1𝑑𝐷1𝐴+𝟙𝑛𝑧𝑏𝑎𝑓𝑠,𝑥(𝑠,𝑧)𝑑𝑠=0,(1.6)which gives those numerical values 𝑧=𝑧 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 𝑛 linear functional differential equations of the form𝑥(𝑡)=𝑃0(𝑡)𝑥(𝑡)+𝑃1(𝑡)𝑥(𝛽(𝑡))+𝑓(𝑡),𝑡[0,𝑇],(1.7)

subjected to the inhomogeneous three-point Cauchy-Nicoletti boundary conditions 𝑥1(0)=𝑥10,,𝑥𝑝(0)=𝑥𝑝0,𝑥𝑝+1(𝜉)=𝑑𝑝+1,,𝑥𝑝+𝑞(𝜉)=𝑑𝑝+𝑞,𝑥𝑝+𝑞+1(𝑇)=𝑑𝑝+𝑞+1,,𝑥𝑛(𝑇)=𝑑𝑛,(1.8)with 𝜉(0,𝑇) is given and 𝑥=col(𝑥1,,𝑥𝑛), to the case where the system of linear functional differential equations under consideration has the general form 𝑥(𝑡)=(𝑙𝑥)(𝑡)+𝑓(𝑡),𝑡[𝑎,𝑏],(1.9)and the three-point boundary conditions are non-separated and have the form 𝐴𝑥(𝑎)+𝐵𝑥(𝜉)+𝐶𝑥(𝑏)=𝑑,(1.10)where 𝐴, 𝐵, and 𝐶 are singular matrices, 𝑑=col(𝑑1,,𝑑𝑛). Here, 𝑙=(𝑙𝑘)𝑛𝑘=1𝐶([𝑎,𝑏],𝑛)𝐿1([𝑎,𝑏],𝑛) is a bounded linear operator and 𝑓𝐿1([𝑎,𝑏],𝑛) 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:𝐶([𝑎,𝑏],𝑛) is the Banach space of the continuous functions [𝑎,𝑏]𝑛 with the standard uniform norm;𝐿1([𝑎,𝑏],𝑛) is the usual Banach space of the vector functions [𝑎,𝑏]𝑛 with Lebesgue integrable components;(𝑛) is the algebra of all the square matrices of dimension 𝑛 with real elements;𝑟(𝑄) is the maximal, in modulus, eigenvalue of a matrix 𝑄(𝑛);𝟙𝑘 is the unit matrix of dimension 𝑘;𝟘𝑖,𝑗 is the zero matrix of dimension 𝑖×𝑗;𝟘𝑖=𝟘𝑖,𝑖.

2. 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 <𝑎<𝑏<,𝑙=(𝑙𝑘)𝑛𝑘=1𝐶([𝑎,𝑏],𝑛)𝐿1([𝑎,𝑏],𝑛) is a bounded linear operator, 𝑓[a,𝑏]𝑛 is an integrable function, 𝑑𝑛 is a given vector, 𝐴, 𝐵, and 𝐶 are singular square matrices of dimension 𝑛, and 𝐶 has the form 𝐶=𝑉𝑊𝟘𝑛𝑞,𝑞𝟘𝑛𝑞,(2.1)where 𝑉 is nonsingular square matrix of dimension 𝑞<𝑛 and 𝑊 is an arbitrary matrix of dimension 𝑞×(𝑛𝑞). 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: col𝑥1(𝜉),,𝑥𝑛(𝜉)=𝜆,col𝑥𝑞+1(𝑏),,𝑥𝑛(𝑏)=𝜂,(2.2)where 𝜆=col(𝜆1,,𝜆𝑛)𝑛 and 𝜂=𝑐𝑜𝑙(𝜂1,,𝜂𝑛𝑞)𝑛𝑞 are vector parameters. This leads us to the parametrised two-point boundary condition 𝐴𝑥(𝑎)+𝐷𝑥(𝑏)=𝑑𝐵𝜆𝑁𝑞𝜂,(2.3)where𝑁𝑞=𝟘𝑞,𝑛𝑞𝟙𝑛𝑞(2.4)and the matrix 𝐷 is given by the formula𝐷=𝑉𝑊𝟘𝑛𝑞,𝑞𝟙𝑛𝑞(2.5)with a certain rectangular matrix 𝑊 of dimension 𝑞×(𝑛𝑞). 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 (𝑛𝑟)-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 𝑙=𝑙0𝑙1,(2.6)where 𝑙𝑗=(𝑙𝑗𝑘)𝑛𝑘=1𝐶([𝑎,𝑏],𝑛)𝐿1([𝑎,𝑏],𝑛),𝑗=0,1, are bounded linear operators positive in the sense that (𝑙𝑗𝑘𝑢)(𝑡)0 for a.e. 𝑡[𝑎,𝑏] and any 𝑘=1,2,,𝑛, 𝑗=0,1, and 𝑢𝐶([𝑎,𝑏],𝑛) such that min𝑡[𝑎,𝑏]𝑢𝑘(𝑡)0 for all 𝑘=1,2,,𝑛. We also put ̂𝑙𝑘=𝑙0𝑘+𝑙1𝑘, 𝑘=1,2,,𝑛, and ̂𝑙=𝑙0+𝑙1.(2.7)

3. Auxiliary Estimates

In the sequel, we will need several auxiliary statements.

Lemma 3.1. For an arbitrary essentially bounded function 𝑢[𝑎,𝑏], the estimates ||||𝑡𝑎𝑢(𝜏)1𝑏𝑎𝑏𝑎𝑢(𝑠)𝑑𝑠𝑑𝜏||||𝛼(𝑡)esssup𝑠[𝑎,𝑏]𝑢(𝑠)essinf𝑠[𝑎,𝑏]𝑢(𝑠)(3.1)𝑏𝑎4esssup𝑠[𝑎,𝑏]𝑢(𝑠)essinf𝑠[𝑎,𝑏]𝑢(𝑠)(3.2)are true for all 𝑡[𝑎,𝑏], where 𝛼(𝑡)=(𝑡𝑎)1𝑡𝑎𝑏𝑎,𝑡[𝑎,𝑏].(3.3)

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 max𝑡[𝑎,𝑏]𝛼(𝑡)=14(𝑏𝑎)(3.4) is taken into account.

Let us introduce some notation. For any 𝑘=1,2,,𝑛, we define the 𝑛-dimensional row-vector 𝑒𝑘 by putting 𝑒𝑘=(0,0,,0,𝑘1,1,0,,0).(3.5) Using operators (2.7) and the unit vectors (3.5), we define the matrix-valued function 𝐾𝑙[𝑎,𝑏](𝑛) by setting 𝐾𝑙=̂𝑙𝑒1,̂𝑙𝑒2,,̂𝑙𝑒𝑛.(3.6) Note that, in (3.6), ̂𝑙𝑒𝑖 means the value of the operator ̂𝑙 on the constant vector function is equal identically to 𝑒𝑖, where 𝑒𝑖 is the vector transpose to 𝑒𝑖. It is easy to see that the components of 𝐾𝑙 are Lebesgue integrable functions.

Lemma 3.2. The componentwise estimate ||(𝑙𝑥)(𝑡)||𝐾𝑙(𝑡)max𝑠[𝑎,𝑏]||𝑥(𝑠)||,𝑡[𝑎,𝑏],(3.7)is true for any 𝑥𝐶([𝑎,𝑏],𝑛), where 𝐾𝑙[𝑎,𝑏](𝑛) is the matrix-valued function given by formula (3.6).

Proof. Let 𝑥=(𝑥𝑘)𝑛𝑘=1 be an arbitrary function from 𝐶([𝑎,𝑏],𝑛). Then, recalling the notation for the components of 𝑙, we see that 𝑙𝑥=𝑛𝑖=1𝑒𝑖𝑙𝑖𝑥.(3.8)
On the other hand, due to (3.5), we have 𝑥=𝑛𝑘=1𝑒𝑘𝑥𝑘 and, therefore, by virtue of (3.8) and (2.6), 𝑙𝑥=𝑛𝑖=1𝑒𝑖𝑙𝑖𝑥=𝑛𝑖=1𝑒𝑖𝑙𝑖𝑛𝑘=1𝑒𝑘𝑥𝑘=𝑛𝑖=1𝑒𝑖𝑛𝑘=1𝑙0𝑖𝑒𝑘𝑥𝑘𝑙1𝑖𝑒𝑘𝑥𝑘.(3.9)
On the other hand, the obvious estimate 𝜎𝑥𝑘(𝑡)max𝑠[𝑎,𝑏]||𝑥𝑘(𝑠)||,𝑡[𝑎,𝑏],𝑘=1,2,,𝑛,𝜎{1,1},(3.10)and the positivity of the operators 𝑙𝑗, 𝑗=0,1, imply 𝑙𝑗𝑖𝜎𝑥𝑘(𝑡)=𝜎𝑙𝑗𝑖𝑥𝑘(𝑡)𝑙j𝑖max𝑠[𝑎,𝑏]||𝑥𝑘(𝑠)||(3.11)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).

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 𝑥𝑚[𝑎,𝑏]×3𝑛𝑞𝑛,𝑚0, by putting𝑥𝑚+1(𝑡,𝑧,𝜆,𝜂)=𝜑𝑧,𝜆,𝜂(𝑡)+𝑡𝑎𝑙𝑥𝑚(,𝑧,𝜆,𝜂)(𝑠)+𝑓(𝑠)𝑑𝑠𝑡𝑎𝑏𝑎𝑏𝑎𝑙𝑥𝑚(,𝑧,𝜆,𝜂)(𝑠)+𝑓(𝑠)𝑑𝑠,𝑚=0,1,2,,𝑥0(𝑡,𝑧,𝜆,𝜂)=𝜑𝑧,𝜆,𝜂(𝑡)(4.1)for all 𝑡[𝑎,𝑏], 𝑧𝑛, 𝜆𝑛, and 𝜂𝑛𝑞, where 𝜑𝑧,𝜆,𝜂(𝑡)=𝑧+𝑡𝑎𝑏𝑎𝐷1𝑑𝐵𝜆+𝑁𝑞𝜂𝐷1𝐴+𝟙𝑛𝑧.(4.2) In the sequel, we consider 𝑥𝑚 as a function of 𝑡 and treat the vectors 𝑧, 𝜆, and 𝜂 as parameters.

Lemma 4.1. For any 𝑚0, 𝑡[𝑎,𝑏], 𝑧𝑛, 𝜆𝑛, and 𝜂𝑛𝑞, the equalities 𝑥𝑚(𝑎,𝑧,𝜆,𝜂)=z,𝐴𝑥𝑚(𝑎,𝑧,𝜆,𝜂)+𝐷𝑥𝑚(𝑏,𝑧,𝜆,𝜂)=𝑑𝐵𝜆+𝑁𝑞𝜂,(4.3)are true.

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 (𝑦)(𝑡)=1𝑡𝑎𝑏𝑎𝑡𝑎𝑦(𝑠)𝑑𝑠+𝑡𝑎𝑏𝑎𝑏𝑡𝑦(𝑠)𝑑𝑠,𝑡[𝑎,𝑏],(4.4)for an arbitrary 𝑦𝐿1([𝑎,𝑏],𝑛). It is clear that 𝐿1([𝑎,𝑏],𝑛)𝐶([𝑎,𝑏],𝑛) is a positive linear operator. Using the operator , we put 𝒬𝑙=𝐾𝑙𝑒1,𝐾𝑙𝑒2,,𝐾𝑙𝑒𝑛,(4.5)where 𝐾𝑙 is given by formula (3.6). Finally, define a constant square matrix 𝑄𝑙 of dimension 𝑛 by the formula 𝑄𝑙=max𝑡[𝑎,𝑏]𝒬𝑙(𝑡).(4.6) 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 𝑄𝑙=𝒬𝑙(𝑡) may not exist).

Theorem 4.2. If the spectral radius of the matrix 𝑄𝑙 satisfies the inequality 𝑟𝑄𝑙<1,(4.7) then, for arbitrary fixed 𝑧𝑛, 𝜆𝑛, and 𝜂𝑛𝑞: (1)the sequence of functions (4.1) converges uniformly in 𝑡[𝑎,𝑏] for any fixed (𝑧,𝜆,𝜂)3𝑛𝑞 to a limit function 𝑥(𝑡,𝑧,𝜆,𝜂)=lim𝑚+𝑥𝑚(𝑡,𝑧,𝜆,𝜂);(4.8)(2)the limit function 𝑥(,𝑧,𝜆,𝜂) possesses the properties 𝑥(𝑎,𝑧,𝜆,𝜂)=𝑧,𝐴𝑥(𝑎,𝑧,𝜆,𝜂)+𝐷𝑥(𝑏,𝑧,𝜆,𝜂)=𝑑𝐵𝜆+𝑁𝑞𝜂;(4.9)(3)function (4.8) is a unique absolutely continuous solution of the integro-functional equation 𝑥(𝑡)=𝑧+𝑡𝑎𝑏𝑎𝐷1𝑑𝐵𝜆+𝑁𝑞𝜂𝐷1𝐴+𝟙𝑛𝑧+𝑡𝑎((𝑙𝑥)(𝑠)+𝑓(𝑠))𝑑𝑠𝑡𝑎𝑏𝑎𝑏𝑎((𝑙𝑥)(𝑠)+𝑓(𝑠))𝑑𝑠,𝑡[𝑎,𝑏];(4.10)(4)the error estimate max𝑡[𝑎,𝑏]||𝑥(𝑡,𝑧,𝜆,𝜂)𝑥𝑚(𝑡,𝑧,𝜆,𝜂)||𝑏𝑎4𝑄𝑚𝑙𝟙𝑛𝑄𝑙1𝜔(𝑧,𝜆,𝜂)(4.11)holds, where 𝜔3𝑛𝑞𝑛 is given by the equality 𝜔(𝑧,𝜆,𝜂)=esssup𝑠[𝑎,𝑏]𝑙𝜑𝑧,𝜆,𝜂(𝑠)+𝑓(𝑠)essinf𝑠[𝑎,𝑏]𝑙𝜑𝑧,𝜆,𝜂(𝑠)+𝑓(𝑠).(4.12)

In (3.6), (4.11) and similar relations, the signs ||, ≤, ≥, as well as the operators ”max”, ”esssup”, ”essinf”, 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],𝑛) equipped with the standard uniform norm. Let us put 𝑟𝑚(𝑡,𝑧,𝜆,𝜂)=𝑥𝑚+1(𝑡,𝑧,𝜆,𝜂)𝑥𝑚(𝑡,𝑧,𝜆,𝜂)(4.13)for all 𝑧𝑛, 𝜆𝑛, 𝜂𝑛𝑞, 𝑡[𝑎,𝑏], and 𝑚0. Using Lemma 3.2 and taking equality (3.4) into account, we find that (4.1) yields ||𝑥1(𝑡,𝑧,𝜆,𝜂)𝑥0(𝑡,𝑧,𝜆,𝜂)||=||||𝑡𝑎𝑙𝜑𝑧,𝜆,𝜂(𝑠)+𝑓(𝑠)𝑑𝑠𝑡𝑎𝑏𝑎𝑏𝑎𝑙𝜑𝑧,𝜆,𝜂(𝑠)+𝑓(𝑠)𝑑𝑠||||𝛼(𝑡)𝜔(𝑧,𝜆,𝜂)𝑏𝑎4𝜔(𝑧,𝜆,𝜂),(4.14)for arbitrary fixed 𝑧, 𝜆, and 𝜂, where 𝛼 is the function given by (3.3) and 𝜔() is defined by formula (4.12).
According to formulae (4.1), for all 𝑡[𝑎,𝑏], arbitrary fixed 𝑧, 𝜆, and 𝜂 and 𝑚=1,2, we have 𝑟𝑚(𝑡,𝑧,𝜆,𝜂)=𝑡𝑎𝑙𝑥𝑚(,𝑧,𝜆,𝜂)𝑥𝑚1(,𝑧,𝜆,𝜂)(𝑠)𝑑𝑠𝑡𝑎𝑏𝑎𝑏𝑎𝑙𝑥𝑚(,𝑧,𝜆,𝜂)𝑥𝑚1(,𝑧,𝜆,𝜂)(𝑠)𝑑𝑠=1𝑡𝑎𝑏𝑎𝑡𝑎𝑙𝑥𝑚(,𝑧,𝜆,𝜂)𝑥𝑚1(,𝑧,𝜆,𝜂)(𝑠)𝑑𝑠𝑡𝑎𝑏𝑎𝑏𝑡𝑙𝑥𝑚(,𝑧,𝜆,𝜂)𝑥𝑚1(,𝑧,𝜆,𝜂)(𝑠)𝑑𝑠.(4.15)
Equalities (4.13) and (4.15) imply that for all 𝑚=1,2,, arbitrary fixed 𝑧,𝜆,𝜂 and 𝑡[𝑎,𝑏], ||𝑟𝑚(𝑡,𝑧,𝜆,𝜂)||1𝑡𝑎𝑏𝑎𝑡𝑎||𝑙𝑟𝑚1(,𝑧,𝜆,𝜂)(𝑠)||𝑑𝑠+𝑡𝑎𝑏𝑎𝑏𝑡||𝑙𝑟𝑚1(,𝑧,𝜆,𝜂)(𝑠)||𝑑𝑠.(4.16)
Applying inequality (3.7) of Lemma 3.2 and recalling formulae (4.5) and (4.6), we get ||𝑟𝑚(𝑡,𝑧,𝜆,𝜂)||1𝑡𝑎𝑏𝑎𝑡𝑎𝐾𝑙(𝑠)max𝜏[𝑎,𝑏]||𝑟𝑚1(𝜏,𝑧,𝜆,𝜂)||𝑑𝑠+𝑡𝑎𝑏𝑎𝑏𝑡𝐾𝑙(𝑠)max𝜏[𝑎,𝑏]||𝑟𝑚1(𝜏,𝑧,𝜆,𝜂)||𝑑𝑠=1𝑡𝑎𝑏𝑎𝑡𝑎𝐾𝑙(𝑠)𝑑𝑠+𝑡𝑎𝑏𝑎𝑏𝑡𝐾𝑙(𝑠)𝑑𝑠max𝜏[𝑎,𝑏]||𝑟𝑚1(𝜏,𝑧,𝜆,𝜂)||=𝒬𝑙(𝑡)max𝜏[𝑎,𝑏]||𝑟𝑚1(𝜏,𝑧,𝜆,𝜂)||𝑄𝑙max𝜏[𝑎,𝑏]||𝑟𝑚1(𝜏,𝑧,𝜆,𝜂)||.(4.17)
Using (4.17) recursively and taking (4.14) into account, we obtain ||𝑟𝑚(𝑡,𝑧,𝜆,𝜂)||𝑄𝑚𝑙max𝜏[𝑎,𝑏]||𝑟0(𝜏,𝑧,𝜆,𝜂)||𝑏𝑎4𝑄𝑚𝑙𝜔(𝑧,𝜆,𝜂),(4.18)for all 𝑚1, 𝑡[𝑎,𝑏], 𝑧𝑛, 𝜆𝑛, and 𝜂𝑛𝑞. Using (4.18) and (4.13), we easily obtain that, for an arbitrary 𝑗, ||𝑥𝑚+𝑗(𝑡,𝑧,𝜆,𝜂)𝑥𝑚(𝑡,𝑧,𝜆,𝜂)||=||𝑥𝑚+𝑗(𝑡,𝑧,𝜆,𝜂)𝑥𝑚+𝑗1(𝑡,𝑧,𝜆,𝜂)+𝑥𝑚+𝑗1(𝑡,𝑧,𝜆,𝜂)𝑥𝑚+𝑗2(𝑡,𝑧,𝜆,𝜂)++𝑥𝑚+1(𝑡,𝑧,𝜆,𝜂)𝑥m(𝑡,𝑧,𝜆,𝜂)||𝑗1𝑖=0||𝑟𝑚+𝑖(𝑡,𝑧,𝜆,𝜂)||𝑏𝑎4𝑗1𝑖=0𝑄𝑚+𝑖𝑙𝜔(𝑧,𝜆,𝜂).(4.19)
Therefore, by virtue of assumption (4.7), it follows that ||𝑥𝑚+𝑗(𝑡,𝑧,𝜆,𝜂)𝑥𝑚(𝑡,𝑧,𝜆,𝜂)||𝑏𝑎4𝑄𝑚𝑙+𝑖=0𝑄𝑖𝑙𝜔(𝑧,𝜆,𝜂)=𝑏𝑎4𝑄𝑚𝑙𝟙𝑛𝑄𝑙1𝜔(𝑧,𝜆,𝜂)(4.20) for all 𝑚1, 𝑗1, 𝑡[𝑎,𝑏], 𝑧𝑛, 𝜆𝑛, and 𝜂𝑛𝑞. We see from (4.20) that (4.1) is a Cauchy sequence in the Banach space 𝐶([𝑎,𝑏],𝑛) and, therefore, converges uniformly in 𝑡[𝑎,𝑏] for all (𝑧,𝜆,𝜂)3𝑛𝑞: lim𝑚𝑥𝑚(𝑡,𝑧,𝜆,𝜂)=𝑥(𝑡,𝑧,𝜆,𝜂),(4.21)that is, assertion 2 holds. Since all functions 𝑥𝑚(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 ||𝑥(𝑡,𝑧,𝜆,𝜂)𝑥𝑚(𝑡,𝑧,𝜆,𝜂)||𝑏𝑎4𝑄𝑚𝑙𝟙𝑛𝑄𝑙1𝜔(z,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. 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 𝐷1𝑑𝐵𝜆+𝑁𝑞𝜂𝐷1𝐴+𝟙𝑛𝑧=𝑏𝑎𝑙𝑥(,𝑧,𝜆,𝜂)(𝑠)+𝑓(𝑠)𝑑𝑠=0,(4.23) then, for these 𝑧,𝜆,and𝜂,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 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 𝑥(𝑎)=𝑧,(5.2) where 𝜇=col(𝜇1,,𝜇𝑛)is a control parameter.

We will show that for arbitrary fixed 𝑧𝑛, 𝜆𝑛, and 𝜂𝑛𝑞, 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), (2.3).

Proposition 5.1. Let 𝑧𝑛, 𝜆𝑛, and 𝜂𝑛𝑞 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=𝜇𝑧,𝜆,𝜂,(5.3)where 𝜇𝑧,𝜆,𝜂=1𝑏𝑎𝐷1𝑑𝐵𝜆+𝑁𝑞𝜂𝐷1𝐴+𝟙𝑛𝑧1𝑏𝑎𝑏𝑎𝑙𝑥(,𝑧,𝜆,𝜂)(𝑠)+𝑓(𝑠)𝑑𝑠(5.4)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 𝑧𝑛, 𝜆𝑛, and 𝜂𝑛𝑞 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 above-mentioned 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 𝜇𝑛 and assume that the initial value problem (5.1), (5.2) has a solution 𝑦 satisfies the two-point boundary conditions (2.3). Then 𝑦(𝑡)=𝑧+𝑡𝑎[(𝑙𝑦)(𝑠)+𝑓(𝑠)]𝑑𝑠+𝜇(𝑡𝑎),(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+𝐷𝑧+𝑏𝑎((𝑙𝑦)(𝑠)+𝑓(𝑠))(𝑠)𝑑𝑠+𝜇(𝑏𝑎)=𝑑𝐵𝜆+𝑁𝑞𝜂,(5.6) whence we find that 𝜇 can be represented in the form 𝜇=1𝑏𝑎𝐷1𝑑𝐵𝜆+𝑁𝑞𝜂(𝐴+𝐷)𝑧𝑏𝑎((𝑙𝑦)(𝑠)+𝑓(𝑠))(𝑠)𝑑𝑠.(5.7)
On the other hand, we already know that the function 𝑥(,𝑧,𝜆,𝜂), satisfies the two-point conditions (2.3) and is a solution of the initial value problem (5.1), (5.2) with 𝜇=𝜇𝑧,𝜆,𝜂, where the value 𝜇𝑧,𝜆,𝜂 is defined by formula (5.4). Consequently, 𝑥(𝑡,𝑧,𝜆,𝜂)=𝑧+𝑡𝑎𝑙𝑥(,𝑧,𝜆,𝜂)(𝑠)+𝑓(𝑠)𝑑𝑠+𝜇𝑧,𝜆,𝜂(𝑡𝑎),𝑡[𝑎,𝑏].(5.8)
Putting (𝑡)=𝑦(𝑡)𝑥(𝑡,𝑧,𝜆,𝜂),𝑡[𝑎,𝑏],(5.9) and taking (5.5), (5.8) into account, we obtain (𝑡)=𝑡𝑎(𝑙)(𝑠)𝑑𝑠+𝜇𝜇𝑧,𝜆,𝜂(𝑡𝑎),𝑡[𝑎,𝑏].(5.10)
Recalling the definition (5.4) of 𝜇𝑧,𝜆,𝜂 and using formula (5.7), we obtain 𝜇𝜇𝑧,𝜆,𝜂=1𝑏𝑎𝑏𝑎𝑙𝑥(,𝑧,𝜆,𝜂)𝑦(𝑠)𝑑𝑠=1𝑏𝑎𝑏𝑎(𝑙)(𝑠)𝑑𝑠,(5.11) and, therefore, equality (5.10) can be rewritten as (𝑡)=𝑡𝑎(𝑙)(𝑠)𝑑𝑠𝑡𝑎𝑏𝑎𝑏𝑎(𝑙)(𝑠)𝑑𝑠=1𝑡𝑎𝑏𝑎𝑡𝑎(𝑙)(𝑠)𝑑𝑠𝑡𝑎𝑏𝑎𝑏𝑡(𝑙)(𝑠)𝑑𝑠,𝑡[𝑎,𝑏].(5.12)
Applying Lemma 3.2 and recalling notation (4.6), we get ||(𝑡)||1𝑡𝑎𝑏𝑎𝑡𝑎𝐾𝑙(𝑠)𝑑𝑠+𝑡𝑎𝑏𝑎𝑏𝑡𝐾𝑙(𝑠)𝑑𝑠max𝜏[𝑎,𝑏]||(𝜏)||𝑄𝑙max𝜏[𝑎,𝑏]||(𝜏)||(5.13) for an arbitrary 𝑡[𝑎,𝑏]. By virtue of condition (4.7), inequality (5.13) implies that max𝜏[𝑎,𝑏]||(𝜏)||𝑄𝑚𝑙max𝜏[𝑎,𝑏]||(𝜏)||0(5.14)as 𝑚+. According to (5.9), this means that 𝑦 coincides with 𝑥(,𝑧,𝜆,𝜂), and, therefore, by (5.11), 𝜇=𝜇𝑧,𝜆,𝜂, which brings us to the desired conclusion.

We show that one can choose certain values of parameters 𝑧=𝑧,𝜆=𝜆,𝜂=𝜂for 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 Δ3𝑛𝑞𝑛 given by formula Δ(𝑧,𝜆,𝜂)=𝑔(𝑧,𝜆,𝜂)𝑏𝑎𝑙𝑥(,𝑧,𝜆,𝜂)(𝑠)+𝑓(𝑠)𝑑𝑠(5.15) with 𝑔(𝑧,𝜆,𝜂)=𝐷1𝑑𝐵𝜆+𝑁𝑞𝜂𝐷1𝐴+𝟙𝑛𝑧(5.16) 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 three-point boundary value problem (1.9), (1.10) if and only if the triplet 𝑧,𝜆,𝜂 satisfies the system of 3𝑛𝑞 algebraic equations Δ(𝑧,𝜆,𝜂)=0,(5.17)𝑒1𝑥(𝜉,𝑧,𝜆,𝜂)=𝜆1,𝑒2𝑥(𝜉,𝑧,𝜆,𝜂)=𝜆2,,𝑒𝑛𝑥(𝜉,𝑧,𝜆,𝜂)=𝜆𝑛,(5.18)𝑒𝑞+1𝑥(𝑏,𝑧,𝜆,𝜂)=𝜂1,𝑒𝑞+2𝑥(𝑏,𝑧,𝜆,𝜂)=𝜂2,,𝑒𝑞+𝑥(𝑏,𝑧,𝜆,𝜂)=𝜂𝑛𝑞.(5.19)

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).

Proposition 5.3. Assume condition (4.7). Then, for any (𝑧𝑗,𝜆𝑗,𝜂𝑗), 𝑗=0,1, the estimate max𝑡[𝑎,𝑏]||𝑥𝑡,𝑧0,𝜆0,𝜂0𝑥𝑡,𝑧1,𝜆1,𝜂1||𝟙𝑛𝑄𝑙1v𝑧0,𝜆0,𝜂0,𝑧1,𝜆1,𝜂1(5.20)holds, where 𝑣𝑧0,𝜆0,𝜂0,𝑧1,𝜆1,𝜂1=max𝑡[𝑎,𝑏]||𝜑𝑧0,𝜆0,𝜂0(𝑡)𝜑𝑧1,𝜆1,𝜂1(𝑡)||.(5.21)

Proof. Let us fix two arbitrary triplets (𝑧𝑗,𝜆𝑗,𝜂𝑗), 𝑗=0,1, and put 𝑢𝑚(𝑡)=𝑥𝑚𝑡,𝑧0,𝜆0,𝜂0𝑥𝑚𝑡,𝑧1,𝜆1,𝜂1,𝑡[𝑎,𝑏].(5.22) Consider the sequence of vectors 𝑐𝑚, 𝑚=0,1,, determined by the recurrence relation 𝑐𝑚=𝑐0+𝑄𝑙𝑐𝑚1,𝑚1,(5.23)with 𝑐0=𝑣𝑧0,𝜆0,𝜂0,𝑧1,𝜆1,𝜂1.(5.24)
Let us show that max𝑡[𝑎,𝑏]||𝑢𝑚(𝑡)||𝑐𝑚(5.25)for all 𝑚0. Indeed, estimate (5.25) is obvious for 𝑚=0. Assume that max𝑡[𝑎,𝑏]||𝑢𝑚1(𝑡)||𝑐𝑚1.(5.26)
It follows immediately from (4.1) that 𝑢𝑚(𝑡)=𝜑𝑧0,𝜆0,𝜂0(𝑡)𝜑𝑧1,𝜆1,𝜂1(𝑡)+𝑡𝑎𝑙𝑢𝑚1(𝑠)𝑑𝑠𝑡𝑎𝑏𝑎𝑏𝑎𝑙𝑢𝑚1(𝑠)𝑑𝑠=𝜑𝑧0,𝜆0,𝜂0(𝑡)𝜑𝑧1,𝜆1,𝜂1(𝑡)+1𝑡𝑎𝑏𝑎𝑡𝑎𝑙𝑢𝑚1(𝑠)𝑑𝑠𝑡𝑎𝑏𝑎𝑏𝑡𝑙𝑢𝑚1(𝑠)𝑑𝑠,(5.27)whence, by virtue of (5.21), estimate (3.7) to Lemma 3.2, and assumption (5.26), ||𝑢𝑚(𝑡)||||𝜑𝑧0,𝜆0,𝜂0(𝑡)𝜑𝑧1,𝜆1,𝜂1(𝑡)||+1𝑡𝑎𝑏𝑎𝑡𝑎||𝑙𝑢𝑚1(𝑠)||𝑑𝑠+𝑡𝑎𝑏𝑎𝑏𝑡||𝑙𝑢𝑚1(𝑠)||𝑑𝑠𝑣𝑧0,𝜆0,𝜂0,𝑧1,𝜆1,𝜂1+1𝑡𝑎𝑏𝑎𝑡𝑎𝐾𝑙(𝑠)𝑑𝑠max𝑡[𝑎,𝑏]||𝑢𝑚1(𝑡)||+𝑡𝑎𝑏𝑎𝑏𝑡𝐾𝑙(𝑠)𝑑𝑠max𝑡[𝑎,𝑏]||𝑢𝑚1(𝑡)||𝑣𝑧0,𝜆0,𝜂0,𝑧1,𝜆1,𝜂1+1𝑡𝑎𝑏𝑎𝑡𝑎𝐾𝑙(𝑠)𝑑𝑠+𝑡𝑎𝑏𝑎𝑏𝑡𝐾𝑙(𝑠)𝑑𝑠𝑐𝑚1𝑣𝑧0,𝜆0,𝜂0,𝑧1,𝜆1,𝜂1+𝑄𝑙𝑐𝑚1,(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 max𝑡[𝑎,𝑏]||𝑢𝑚(𝑡)||𝑐0+𝑄𝑙𝑐𝑚1=𝑐0+𝑄𝑙𝑐0+𝑄2𝑙𝑐𝑚2==𝑚1𝑘=0𝑄𝑘𝑙𝑐0+𝑄𝑚𝑙𝑐0.(5.29)
Due to assumption (4.7), lim𝑚+𝑄𝑚𝑙=0. Therefore, passing to the limit in (5.29) as 𝑚+ and recalling notation (5.22), we obtain the estimate max𝑡[𝑎,𝑏]||𝑥𝑡,𝑧0,𝜆0,𝜂0𝑥𝑡,𝑧1,𝜆1,𝜂1||+𝑘=0𝑄𝑘𝑙𝑐0=𝟙𝑛𝑄𝑙1𝑐0,(5.30) which, in view of (5.24), coincides with (5.20).

Now we establish some properties of the “determining function” Δ3𝑛𝑞𝑛 given by equality (5.15).

Proposition 5.4. Under condition (3.10), formula (5.15) determines a well-defined function Δ3𝑛𝑞𝑛, which, moreover, satisfies the estimate ||Δ𝑧0,𝜆0,𝜂0Δ𝑧1,𝜆1,𝜂1||||𝐺[𝑧0𝑧1,𝜆0𝜆1,𝜂0𝜂1]||+𝑅𝑙max𝑡[𝑎,𝑏]|||𝑧0𝑧1+𝑡𝑎𝑏𝑎𝐺[𝑧0𝑧1,𝜆0𝜆1,𝜂0𝜂1]|||,(5.31)for all (𝑧𝑗,𝜆𝑗,𝜂𝑗), 𝑗=0,1, where the (𝑛×𝑛)-matrices 𝐺 and 𝑅𝑙 are defined by the equalities 𝐺=𝐷1𝐴+𝐷,𝐵,𝑁𝑞,𝑅𝑙=𝑏𝑎𝐾𝑙(𝑠)𝑑𝑠𝟙𝑛𝑄𝑙1.(5.32)

Proof. According to the definition (5.15) of Δ, we have Δ𝑧0,𝜆0,𝜂0Δ𝑧1,𝜆1,𝜂1=𝑔𝑧0,𝜆0,𝜂0𝑔𝑧1,𝜆1,𝜂1𝑏𝑎𝑙𝑥,𝑧0,𝜆0,𝜂0𝑥,𝑧1,𝜆1,𝜂1(𝑠)𝑑𝑠,(5.33)whence, due to Lemma 3.2, ||Δ𝑧0,𝜆0,𝜂0Δ𝑧1,𝜆1,𝜂1||||𝑔𝑧0,𝜆0,𝜂0𝑔𝑧1,𝜆1,𝜂1||+𝑏𝑎||𝑙𝑥,𝑧0,𝜆0,𝜂0𝑥,𝑧1,𝜆1,𝜂1(𝑠)||𝑑𝑠||𝑔𝑧0,𝜆0,𝜂0𝑔𝑧1,𝜆1,𝜂1||+𝑏𝑎𝐾𝑙(𝑠)𝑑𝑠max𝜏[𝑎,𝑏]||𝑥𝜏,𝑧0,𝜆0,𝜂0𝑥𝜏,𝑧1,𝜆1,𝜂1(𝑠)||.(5.34)
Using Proposition 5.3, we find ||Δ𝑧0,𝜆0,𝜂0Δ𝑧1,𝜆1,𝜂1||||𝑔𝑧0,𝜆0,𝜂0𝑔𝑧1,𝜆1,𝜂1||+𝑏𝑎𝐾𝑙(𝑠)𝑑𝑠𝟙𝑛𝑄𝑙1𝑣𝑧0,𝜆0,𝜂0,𝑧1,𝜆1,𝜂1.(5.35)
On the other hand, recalling (4.2) and (5.21), we get 𝑣𝑧0,𝜆0,𝜂0,𝑧1,𝜆1,𝜂1=max𝑡[𝑎,𝑏]|||𝑧0𝑧1+𝑡𝑎𝑏𝑎𝑔𝑧0,𝜆0,𝜂0𝑔𝑧1,𝜆1,𝜂1|||.(5.36)
It follows immediately from (5.16) that 𝑔𝑧0,𝜆0,𝜂0𝑔𝑧1,𝜆1,𝜂1=𝐷1𝐵𝜆0𝜆1𝐷1𝑁𝑞𝜂0𝜂1𝐷1𝐴+𝟙𝑛𝑧0𝑧1=𝐷1𝐵𝜆0𝜆1+𝑁𝑞𝜂0𝜂1+(𝐴+𝐷)𝑧0𝑧1=𝐷1𝐴+𝐷,𝐵,𝑁𝑞𝑧0𝑧1𝜆0𝜆1𝜂0𝜂1.(5.37)
Therefore, (5.35) and (5.36) yield the estimate ||Δ𝑧0,𝜆0,𝜂0Δ𝑧1,𝜆1,𝜂1||||||||||𝐷1𝐴+𝐷,𝐵,𝑁𝑞𝑧0𝑧1𝜆0𝜆1𝜂0𝜂1||||||||+𝑏𝑎𝐾𝑙(𝑠)𝑑𝑠𝟙𝑛𝑄𝑙1max𝑡[𝑎,𝑏]||||||||𝑧0𝑧1+𝑡𝑎𝑏𝑎𝐷1𝐴+𝐷,𝐵,𝑁𝑞𝑧0𝑧1𝜆0𝜆1𝜂0𝜂1||||||||,(5.38)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 𝑧𝑛, according to (4.1), we analytically construct the sequence of functions 𝑥𝑚(,𝑧,𝜆,𝜂) depending on the parameters 𝑧,𝜆,𝜂 and satisfying the auxiliary two-point boundary condition (2.3).(2)We find the limit 𝑥(,𝑧,𝜆,𝜂) of the sequence 𝑥𝑚(,𝑧,𝜆,𝜂) satisfying (2.3).(3)We construct the algebraic determining system (5.17), (5.18), and (5.19) with respect 3𝑛𝑞 scalar variables.(4)Using a suitable numerical method, we (approximately) find a root 𝑧𝑛,𝜆𝑛,𝜂𝑛𝑞(6.1)of the determining system (5.17), (5.18), and (5.19).(5)Substituting values (6.1) into 𝑥(,𝑧,𝜆,𝜂), we obtain a solution of the original three-point boundary value problem (1.9), (1.10) in the form 𝑥(𝑡)=𝑥𝑡,𝑧,𝜆,𝜂,𝑡[𝑎,𝑏].(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 𝑚1, construct the corresponding function 𝑥𝑚(,𝑧,𝜆,𝜂) according to relation (4.1), and define the function Δ𝑚3𝑛𝑞𝑛 by putting Δ𝑚(𝑧,𝜆,𝜂)=𝐷1𝑑𝐵𝜆+𝑁𝑞𝜂𝐷1𝐴+𝟙𝑛𝑧𝑏𝑎𝑙𝑥𝑚(,𝑧,𝜆,𝜂)(𝑠)+𝑓(𝑠)𝑑𝑠,(6.4)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 systemΔ𝑚(𝑧,𝜆,𝜂)=0,𝑒1𝑥𝑚(𝜉,𝑧,𝜆,𝜂)=𝜆1,𝑒2𝑥𝑚(𝜉,𝑧,𝜆,𝜂)=𝜆2,,𝑒𝑛𝑥𝑚(𝜉,𝑧,𝜆,𝜂)=𝜆𝑛,𝑒𝑞+1𝑥𝑚(𝑏,𝑧,𝜆,𝜂)=𝜂1,,𝑒𝑛𝑥𝑚(𝑏,𝑧,𝜆,𝜂)=𝜂𝑛𝑞,(6.5)where 𝑒𝑖, 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]).

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.

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