Notes on interval halving procedure for periodic and two-point problems
© Rontó et al.; licensee Springer 2014
Received: 18 March 2014
Accepted: 20 June 2014
Published: 26 September 2014
We continue our study of constructive numerical-analytic schemes of investigation of boundary problems. We simplify and improve the recently suggested interval halving technique allowing one to essentially weaken the convergence conditions.
The present note is a continuation of  and deals with a constructive approach to the investigation of two-point boundary value problems. The approach is numerical-analytic ,  in the sense that, although part of the computation is carried out analytically, the final stage of the method involves a numerical analysis of certain equations usually referred to as determining, or bifurcation, equations. This scheme of Lyapunov-Schmidt type ,  reminds one of the shooting method on first glance, but there are several essential differences .
for all and from . In (3) and all similar relations that will appear below, the symbols ≤ and are understood componentwise.
and for any compact and .
Using the setting, we further reformulate the scheme of the method further by removing certain unnecessary technicalities so that both the setting and the overall analysis are simplified. The new formulation, in particular, makes it particularly easy to adopt the approach to problems with two-point boundary conditions different from the periodic ones, which technique is also outlined in what follows.
2 Construction of iterations and proof of convergence
The considerations in  concerning the auxiliary parametrised problems (4.1), (4.2) and (4.3), (4.4) can be omitted. Clearly, (17) and (18) are the simplest choice of functions satisfying (21) and (22).
- 1.The function has the property(23)
- 2.The function has the property(27)
and, therefore, the function is continuous on for any . The use of this function, however, requires the knowledge of the fact that and are well defined for .
(see ,  for more details). The unpleasant side effect is, however, that estimates (36) and (37) under such a condition are established for sufficiently large only, which puts an obstacle in obtaining efficient solvability conditions in Corollary 8 below. This circumstance is not actually of primary importance since the very aim of the interval halving technique discussed here is to weaken assumption (35) by half and, in any case, the difference between the two conditions is quite insignificant because .
It follows from , Lemma 3.12] that estimates (36) and (37) can be shown to hold for all if the definition of functions and is changed slightly (namely, the multiplier is added on the right-hand side of (32), (33)).
which is twice as strong as (34). In contrast to the related assumptions from  and earlier works, condition (34) is easier to verify because in order to do so one has only to find the value , which is computed directly by estimating . In addition, it is possible to estimate this value in certain cases where some further information on the behaviour of is known.
Comparing Theorem 2 with Theorems 6.1 and 6.3 of , where the values in (6.5) and (6.12) are computed over the entire domain where is Lipschitzian, we see that the values and in Theorem 2 are computed over only.
The proof of Theorem 2 is carried out by a suitable modification of that of , Theorem 6.5] and is based upon the following lemmata.
(, Lemma 7.1])
for all , , and .
Let and be arbitrary vectors from . It is natural to argue by induction. Since is assumed to be convex, it follows from (17) that for any and for any , i.e., (43) and (44) are true for . Let us assume that (43) and (44) hold for a certain .
for . Since (43) and (44) are satisfied for , we see from (45), (46) that all the values of and are contained in a -neighbourhood of a point from , which means that (43) and (44) hold for . It now remains to use the arbitrariness of . □
The assertion of Theorem 2 is now obtained by replacing , Lemma 7.2] by Lemma 5 and arguing by analogy to the proof of Theorems 6.1 and 6.3 from . Furthermore, similarly to , using Proposition 1 and Theorem 2, one arrives at the following.
3 Constructive solvability analysis
for any , we can state the following.
for a certain fixedand there exists a continuous mappingwhich does not vanish onand is such that, . Then there exists a pairsuch that the functionis a solution of the periodic boundary value problem (1), (2) possessing properties (47).
It should be noted that the vector field is finite-dimensional and, thus, the degree involved in (55) is the Brower degree.
the singular points of which determine solutions of problem (1), (2) satisfying condition (47), and use the homotopy invariance of the degree. The remaining property in (47) is a consequence of Lemma 5. □
Let the binary relation be defined  for any as follows: functions and are said to satisfy the relation if and only if there exists a function such that at every point . Using this relation, one can formulate an efficient condition sufficient for the solvability of problem (1), (2).
for , , and use estimate (10.11) from . □
for any . Note that, unlike , the function depends on the phase variables only.
where and is given by (58). Arguing similarly to , Lemma 9.2] and Corollary 8 and taking Remark 3 into account, one can show that (63) ensures the non-degeneracy of homotopy (59). The required conclusion then follows from Theorem 7. □
Theorems of the kind specified above allow one to study the periodic problem (1), (2) following the lines of , . This analysis is constructive in the sense that the assumptions can be verified efficiently and the results of computation, regarded at first only as candidates for approximate solutions, simultaneously open a way to prove the solvability in a rigorous manner. As regards the computation of iterations themselves, it is helpful to apply suitable simplified versions of the algorithm, not discussed here, which are better adopted for use with computer algebra systems. The use of polynomial approximations under similar circumstances was considered, in particular, in .
for , which arises similarly to in the situation where no interval halving is carried out. In the latter case, one has the following statement, which is a reformulation of , Corollary 13.2].
It should also be noted that, as a natural extension of the above said, one can consider a scheme with multiple interval divisions. Although the addition of intermediate nodes increases the number of equations to be solved numerically (at interval halvings, one ultimately arrives a system of determining equation with respect to variables), the important gain is the ability to apply the method regardless of the value of the Lipschitz constant.
Variables involved in the determining equations for the respective number of interval halvings
Variables in the determining equations
, , ,
, , , , , , ,
Fix a certain and consider the scheme with interval divisions. Fix an and construct for .
- 2.Solve the th approximate determining equations for , find a root , and put(69)
- 3.‘Check’ the behaviour of the functions (the heuristic step). If promising (i.e., there are some signs of convergence), choose a suitable containing the graph of , find a from the condition(70)
Verify conditions of the existence theorem for and . If not satisfied, or if the precision of is insufficient, pass to and study . Otherwise the algorithm stops, and the outcome is:
the space localisation of the graph of is described by properties (47).
Note the role of interval divisions in the algorithm: for not satisfying the smallness condition (11) and (i.e., when is constructed according to (4) without any interval divisions), the algorithm would stop at step 3 without any result. However, it is obvious that (70) and (71) are both satisfied if is chosen to be large enough.
is fixed, and stands for the th partial sum of the Fourier series of the corresponding function. There are visible similarities between the two approaches and, most importantly, the scheme of Cesari is also proved to be applicable regardless on the smallness of the Lipschitz constant (see ). The number of resulting determining equations therewith depends on the Lipschitz constant of as well (in fact, it grows with , the convergence being guaranteed by suitable properties of for large enough), which reminds us of Table 1 in our case. The approach presented in this note, in our opinion, has the advantage that, firstly, the computation of iterations is significantly simpler (apart of the integral mean, one does not need to compute any higher order terms in the Fourier expansion) and, secondly, it can be used for other problems as well, whereas, due to the nature of formula (72), the use of Cesari’s scheme is limited to periodic functions.
where is a square matrix of dimension (possibly, singular), , , and .
- 1.The function has the property(79)
- 2.The function has the property(80)
We see from Proposition 11 that properties of sequences , , , constructed for problem (73), (74) are rather similar to those for the periodic problem (1), (2) (in particular, the definition of functions and is the same as in Proposition 1). In both cases, the iteration is carried out according to formulae (19), (20), the only difference being in equalities (75), (76) for and . As a result, the corresponding limit functions satisfy the boundary conditions (79), (80).
Let there exist a non-negative vectorwith property (34) such thatfor a.e. with a certain matrixsatisfying inequality (35). Then, for all fixed, the sequencesandwith, defined by (75) and (76) converge uniformly on the corresponding intervals and, moreover, estimates (36), (37) hold.
with given by (38).
The work was supported in part by RVO: 67985840 (A Rontó) and the SAIA National Scholarship Programme of the Slovak Republic (N Shchobak).
- Rontó A, Rontó M, Shchobak N: Constructive analysis of periodic solutions with interval halving. Bound. Value Probl. 2013., 2013: Article ID 57 10.1186/1687-2770-2013-57Google Scholar
- Samoilenko AM, Rontó NI: Numerical-Analytic Methods in the Theory of Boundary-Value Problems for Ordinary Differential Equations. Naukova Dumka, Kiev; 1992.Google Scholar
- Rontó A, Rontó M: Successive approximation techniques in non-linear boundary value problems for ordinary differential equations. In Handbook of Differential Equations: Ordinary Differential Equations. Elsevier, Amsterdam; 2008:441-592.Google Scholar
- Nirenberg L: Topics in Nonlinear Functional Analysis. Am. Math. Soc., New York; 1974.Google Scholar
- Gaines RE, Mawhin JL: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin; 1977.Google Scholar
- Rontó M, Samoilenko AM: Numerical-Analytic Methods in the Theory of Boundary-Value Problems. World Scientific, River Edge; 2000.View ArticleGoogle Scholar
- Rontó A, Rontó M, Holubová G, Nečesal P: Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions. Bound. Value Probl. 2011., 2011: Article ID 58 10.1186/1687-2770-2011-58Google Scholar
- Mawhin J: Topological Degree Methods in Nonlinear Boundary Value Problems. Am. Math. Soc., Providence; 1979.Google Scholar
- Capietto A, Mawhin J, Zanolin F: A continuation approach to superlinear periodic boundary value problems. J. Differ. Equ. 1990, 88(2):347-395. 10.1016/0022-0396(90)90102-UMathSciNetView ArticleGoogle Scholar
- Cesari L: Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. 3rd edition. Springer, New York; 1971.View ArticleGoogle Scholar
- Knobloch H-W: Remarks on a paper of L. Cesari on functional analysis and nonlinear differential equations. Mich. Math. J. 1963, 10: 417-430. 10.1307/mmj/1028998978MathSciNetView ArticleGoogle Scholar
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