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

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2011, Article ID 326052, 22 pages doi:10.1155/2011/326052 Research Article On Nonseparated Three-Point Boundary Value Problems for Linear Functional Differential Equations A. Rontó1 and M. Rontó2 1 Institute of Mathematics, Academy of Sciences of the Czech Republic, 22 Žižkova St., 61662 Brno, Czech Republic 2 Department of Analysis, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary Correspondence should be addressed to A. Rontó, Received 20 January 2011; Accepted 27 April 2011 Academic Editor: Yuri V. Rogovchenko Copyright q 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. 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 2 Abstract and Applied Analysis 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 x t  ft, xt, Axa t ∈ a, b, 1.1 Dxb  d, 1.2 where x : a, b → Rn , −∞ < a < b < ∞, d ∈ Rn , det D /  0, the corresponding Cauchy problem for the modified parametrised system of integrodifferential equations has the form 12  x t  ft, xt  1  −1 D d − D−1 A b−a   1n z − 1 b−a b fs, xsds, a t ∈ a, b, 1.3 xa  z, where 1n is the unit matrix of dimension n and the parameter z ∈ Rn has the meaning of initial value of the solution at the point a. The expression  1  −1 D d − D−1 A b−a   1n z − 1 b−a b fs, xsds a 1.4 Abstract 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 xm 1 t, z : z t b a a 1 fs, xm s, zds − b−a   1n z ,  t − a  −1 D d − D−1 A b−a  fτ, xm τ, zdτ ds 1.5 m  0, 1, 2, . . . , where x0 t, z : z for all t ∈ a, b and z ∈ Rn , 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, z ∈ Rn , 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 equations  D−1 d − D−1 A b  1n z − fs, x∞ s, zds  0, 1.6 a 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 (...truncated)