Constructive analysis of periodic solutions with interval halving

Rontó et al. Boundary Value Problems 2013, 2013:57 http://www.boundaryvalueproblems.com/content/2013/1/57 RESEARCH Open Access Constructive analysis of periodic solutions with interval halving András Rontó1* , Miklós Rontó2 and Nataliya Shchobak3 * Correspondence: Institute of Mathematics, Academy of Sciences of Czech Republic, 22 Žižkova St., Brno, 616 62, Czech Republic Full list of author information is available at the end of the article 1 Abstract For a constructive analysis of the periodic boundary value problem for systems of non-linear non-autonomous ordinary differential equations, a numerical-analytic approach is developed, which allows one to both study the solvability and construct approximations to the solution. An interval halving technique, by using which one can weaken significantly the conditions required to guarantee the convergence, is introduced. The main assumption on the equation is that the non-linearity is locally Lipschitzian. An existence theorem based on properties of approximations is proved. A relation to Mawhin’s continuation theorem is indicated. MSC: 34B15 Keywords: periodic solution; periodic boundary value problem; parametrisation; periodic successive approximations; numerical-analytic method; interval halving; existence; continuation; Mawhin’s theorem Introduction In this paper, we shall develop a numerical-analytic approach to the analysis of periodic solutions of systems of non-autonomous ordinary differential equations using the idea introduced in []. The method is numerical-analytic in the sense that its realisation consists of two stages concerning, respectively, an explicit construction of certain equations and their numerical analysis and is close in the spirit to the Lyapunov-Schmidt reductions [, ]. However, neither a small parameter nor an implicit function argument is used. We focus on numerical-analytic schemes based upon successive approximations. In the context of the theory of non-linear oscillations, such types of methods were apparently first developed in [–]. We refer the reader to [–] for the related bibliography. For a boundary value problem, the numerical-analytic approach usually replaces the problem by a family of initial value problems for a suitably perturbed system containing a vector parameter which most often has the meaning of the initial value of the solution. The solution of the Cauchy problem for the perturbed system is sought for in an analytic form by successive approximations, whereas the numerical value of the parameter is determined later from the so-called determining equations. In order to guarantee the convergence, a kind of the Lipschitz condition is usually assumed [–] and a smallness restriction of the type r(K) ≤ qT (.) © 2013 Rontó et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Rontó et al. Boundary Value Problems 2013, 2013:57 http://www.boundaryvalueproblems.com/content/2013/1/57 Page 2 of 34 is imposed, where K is the Lipschitz matrix and qT depends on the period T. The improvement of condition (.) consists in maximising the value of the constant qT . In this paper, which is a continuation of [], we provide a constructive approach to the study of solvability of the periodic problem (.), (.), where the analysis of convergence uses the interval halving technique. We shall see that, under fairly general assumptions, this idea allows one to replace (.) by the weaker condition r(K) ≤ qT (.) and, thus, significantly improve the convergence conditions established, in particular, in [–, ]. The restriction imposed on the width of the domain is likewise improved. Finally, an existence theorem based upon the properties of approximate solutions is proved. The proofs use a number of technical facts from [], which are stated in the course of exposition when appropriate. 1 Problem setting and basic assumptions The method that we are interested in deals with T-periodic solutions of a system of nonlinear ordinary differential equations   u (t) = f t, u(t) , t ∈ (–∞, ∞), (.) where f : Rn+ → Rn is a continuous function such that f (t, z) = f (t + T, z) (.) for all z ∈ Rn and t ∈ (–∞, ∞). Here, T is a given positive number. We restrict ourselves to considering continuously differentiable solutions of system (.) and, furthermore, instead of T-periodic solutions of (.), we shall always deal with the solutions u : [, T] → Rn of the corresponding periodic boundary value problem on the bounded interval [, T],   u (t) = f t, u(t) , t ∈ [, T], u() = u(T). (.) (.) The passage to problem (.), (.) is justified by assumption (.). Our main assumption is that f : [, T] × Rn → Rn is Lipschitzian with respect to the space variable in a certain bounded set D, which is the closure of a bounded and connected domain in Rn . For the sake of simplicity, we assume that there exists a non-negative constant square matrix K of dimension n such that   f (t, x ) – f (t, x ) ≤ K|x – x | (.) for all {x , x } ⊂ D and t ∈ [, T]. Here and below, the obvious notation |x| = col(|x |, |x |, . . . , |xn |) is used, and the inequalities between vectors are understood componentwise. The same convention is adopted implicitly for the operations ‘max’ and ‘min’ so that, e.g., max{h(z) : z ∈ Q} for any h = Rontó et al. Boundary Value Problems 2013, 2013:57 http://www.boundaryvalueproblems.com/content/2013/1/57 Page 3 of 34 (hi )ni= : Q → Rn , where Q ⊂ Rm , m ≤ n, is defined as the column vector with the components max{hi (z) : z ∈ Q}, i = , , . . . , n. 2 Notation and symbols We fix an n ∈ N and a bounded set D ⊂ Rn . The following symbols are used in the sequel: . n is the unit matrix of dimension n. . r(K) is the maximal, in modulus, eigenvalue of a matrix K . . Given a closed interval J ⊆ R, we define the vector δJ,D (f ) by setting δJ,D (f ) := max f (t, z) – min f (t, z). (.) (t,z)∈J×D (t,z)∈J×D . ek , k = , , . . . , n: see (.). . ∂ is the boundary of a domain . . S : see Definition .. The notion of a set D(r) associated with D, which could have been called an inner rneighbourhood of D, will often be used in what follows. Definition . For any non-negative vector r ∈ Rn , we put   D(r) := z ∈ D : B(z, r) ⊂ D , (.) where   B(z, r) := ξ ∈ Rn : |ξ – z| ≤ r . (.) One of the assumptions to be used below means that the inner r-neighbourhood of D is non-empty for r sufficiently large. Finally, let the positive number ∗ be determined by the equality       – q>:q = exp τ (τ – )q dτ . ∗– = inf (.)  We refer, e.g., to [, ] for the discussion of other ways of introducing the constant ∗ and for its meaning. What is important for us here is that ∗ is t (...truncated)