Notes on interval halving procedure for periodic and two-point problems
- András Rontó^{1}Email author,
- Miklós Rontó^{2} and
- Nataliya Shchobak^{3, 4}
https://doi.org/10.1186/s13661-014-0164-9
© Rontó et al.; licensee Springer 2014
Received: 18 March 2014
Accepted: 20 June 2014
Published: 26 September 2014
Abstract
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.
MSC: 34B15.
Keywords
1 Introduction
The present note is a continuation of [1] and deals with a constructive approach to the investigation of two-point boundary value problems. The approach is numerical-analytic [2], [3] 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 [4], [5] reminds one of the shooting method on first glance, but there are several essential differences [1].
for all ${z}_{1}$ and ${z}_{2}$ from $\mathrm{\Omega}$. In (3) and all similar relations that will appear below, the symbols ≤ and $|\cdot |$ are understood componentwise.
and ${\delta}_{J,V}(f):={max}_{(t,\xi )\in J\times V}f(t,\xi )-{min}_{(t,\xi )\in J\times V}f(t,\xi )$ for any compact $V\subseteq D$ and $J\subseteq [0,p]$.
Moreover, as is seen from condition (7), the set $D$ where (6) holds should be wide enough (in particular, such that $diamD\ge \frac{p}{2}{\delta}_{D}(f)$, with the natural componentwise definition of a vector-valued diameter of a set).
Clearly, the transition to (13) weakens (7) by half.
Using the $({\mathrm{\Omega}}_{\varrho},\mathrm{\Omega})$ 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 [1] 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).
The form of sequences (19), (20) is motivated by the following proposition.
Proposition 1
- 1.The function ${x}_{\mathrm{\infty}}(\cdot ,\xi ,\eta )$ has the property${x}_{\mathrm{\infty}}(\frac{p}{2},\xi ,\eta )-{x}_{\mathrm{\infty}}(0,\xi ,\eta )=\eta -\xi $(23)
- 2.The function ${y}_{\mathrm{\infty}}(\cdot ,\xi ,\eta )$ has the property${y}_{\mathrm{\infty}}(p,\xi ,\eta )-{y}_{\mathrm{\infty}}(\frac{p}{2},\xi ,\eta )=\xi -\eta $(27)
and, therefore, the function ${u}_{\mathrm{\infty}}(\cdot ,\xi ,\eta )$ is continuous on $[0,p]$ for any $(\xi ,\eta )\in {\mathrm{\Omega}}^{2}$. The use of this function, however, requires the knowledge of the fact that ${x}_{\mathrm{\infty}}(\cdot ,\xi ,\eta )$ and ${y}_{\mathrm{\infty}}(\cdot ,\xi ,\eta )$ are well defined for $(\xi ,\eta )\in {\mathrm{\Omega}}^{2}$.
Theorem 2
for all$t\in [p/2,p]$and$m\ge 3$.
(see [1], [3] for more details). The unpleasant side effect is, however, that estimates (36) and (37) under such a condition are established for $m$ 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 ${\gamma}_{0}-{\gamma}_{\ast}\approx 0.00727$.
Remark 3
It follows from [3], Lemma 3.12] that estimates (36) and (37) can be shown to hold for all $m\ge 0$ if the definition of functions ${\overline{\alpha}}_{1}$ and ${\overline{\overline{\alpha}}}_{1}$ is changed slightly (namely, the multiplier $10/9$ is added on the right-hand side of (32), (33)).
which is twice as strong as (34). In contrast to the related assumptions from [1] and earlier works, condition (34) is easier to verify because in order to do so one has only to find the value ${\delta}_{{\mathrm{\Omega}}_{\varrho}}(f)$, which is computed directly by estimating $f$. In addition, it is possible to estimate this value in certain cases where some further information on the behaviour of $f$ is known.
Comparing Theorem 2 with Theorems 6.1 and 6.3 of [1], where the values in (6.5) and (6.12) are computed over the entire domain where $f$ is Lipschitzian, we see that the values ${\delta}_{[0,p/2],{\mathrm{\Omega}}_{\varrho}}(f)$ and ${\delta}_{[p/2,p],{\mathrm{\Omega}}_{\varrho}}(f)$ in Theorem 2 are computed over ${\mathrm{\Omega}}_{\varrho}$ only.
The proof of Theorem 2 is carried out by a suitable modification of that of [1], Theorem 6.5] and is based upon the following lemmata.
Lemma 4
([1], Lemma 7.1])
for$t\in [p/2,p]$.
for all $t\in [\frac{i}{2}p,\frac{1}{2}(i+1)p]$, $i\in \{0,1\}$, and $v\in C([\frac{i}{2}p,\frac{1}{2}(i+1)p],{\mathbb{R}}^{n})$.
Lemma 5
hold.
Proof
Let $\xi $ and $\eta $ be arbitrary vectors from $\mathrm{\Omega}$. It is natural to argue by induction. Since $\mathrm{\Omega}$ is assumed to be convex, it follows from (17) that ${x}_{0}(t,\xi ,\eta )\in \mathrm{\Omega}$ for any $t\in [0,p/2]$ and ${y}_{0}(t,\xi ,\eta )\in \mathrm{\Omega}$ for any $t\in [p/2,p]$, i.e., (43) and (44) are true for $m=0$. Let us assume that (43) and (44) hold for a certain $m={m}_{0}$.
for $t\in [p/2,p]$. Since (43) and (44) are satisfied for $m=0$, we see from (45), (46) that all the values of ${x}_{{m}_{0}+1}(\cdot ,\xi ,\eta )$ and ${y}_{{m}_{0}+1}(\cdot ,\xi ,\eta )$ are contained in a $\varrho $-neighbourhood of a point from $\mathrm{\Omega}$, which means that (43) and (44) hold for $m={m}_{0}+1$. It now remains to use the arbitrariness of ${m}_{0}$. □
The assertion of Theorem 2 is now obtained by replacing [1], Lemma 7.2] by Lemma 5 and arguing by analogy to the proof of Theorems 6.1 and 6.3 from [1]. Furthermore, similarly to [1], using Proposition 1 and Theorem 2, one arrives at the following.
Theorem 6
Recall that the functions $\mathrm{\Xi}:{\mathrm{\Omega}}^{2}\to {\mathbb{R}}^{n}$ and $\mathrm{H}:{\mathrm{\Omega}}^{2}\to {\mathbb{R}}^{n}$ are defined according to equalities (26) and (30), and the latter equalities make sense in view of Theorem 2.
3 Constructive solvability analysis
for any $(\xi ,\eta )\in {\mathrm{\Omega}}^{2}$, we can state the following.
Theorem 7
for a certain fixed$m\ge 0$and there exists a continuous mapping$Q:[0,1]\times {\mathrm{\Omega}}^{2}$which does not vanish on$(0,1)\times \partial {\mathrm{\Omega}}^{2}$and is such that$Q(0,\cdot )={\mathrm{\Phi}}_{m}$, $Q(1,\cdot )={\mathrm{\Phi}}_{\mathrm{\infty}}$. Then there exists a pair$({\xi}^{\ast},{\eta}^{\ast})\in {\mathrm{\Omega}}^{2}$such that the function$u:={u}_{\mathrm{\infty}}(\cdot ,{\xi}^{\ast},{\eta}^{\ast})$is a solution of the periodic boundary value problem (1), (2) possessing properties (47).
It should be noted that the vector field ${\mathrm{\Phi}}_{m}$ is finite-dimensional and, thus, the degree involved in (55) is the Brower degree.
Proof
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 ${\u25b7}_{S}$ be defined [3] for any $S\subset {\mathbb{R}}^{2n}$ as follows: functions $g={({g}_{i})}_{i=1}^{2n}:{\mathbb{R}}^{2n}\to {\mathbb{R}}^{2n}$ and $h={({h}_{i})}_{i=1}^{2n}:{\mathbb{R}}^{2n}\to {\mathbb{R}}^{2n}$ are said to satisfy the relation $g{\u25b7}_{S}h$ if and only if there exists a function $\nu :S\to \{1,2,\dots ,2n\}$ such that ${g}_{\nu (z)}(z)>{h}_{\nu (z)}(z)$ at every point $z\in S$. Using this relation, one can formulate an efficient condition sufficient for the solvability of problem (1), (2).
Corollary 8
Then there exists a pair$({\xi}^{\ast},{\eta}^{\ast})\in {\mathrm{\Omega}}^{2}$such that$u:={u}_{\mathrm{\infty}}(\cdot ,{\xi}^{\ast},{\eta}^{\ast})$is a solution of problem (1), (2) possessing properties (47).
Proof
for $(\xi ,\eta )\in {\mathrm{\Omega}}^{2}$, $\theta \in [0,1]$, and use estimate (10.11) from [1]. □
Recall that ${\gamma}_{0}$ in (58) is given by (12). It is important to emphasise that conditions of Corollary 8 are assumed for a fixed$m$, and all the values depending on it are evaluated in finitely many steps.
for any $(\xi ,\eta )\in {\mathrm{\Omega}}^{2}$. Note that, unlike $f$, the function ${f}^{\mathrm{\#}}$ depends on the phase variables only.
Corollary 9
Then the$p$-periodic problem (1), (2) has at least one solution$u(\cdot )$which possesses properties (47).
Proof
where ${\tilde{M}}_{0}:=(10/9){M}_{0}$ and ${M}_{0}$ is given by (58). Arguing similarly to [1], 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. □
4 Discussion
Theorems of the kind specified above allow one to study the periodic problem (1), (2) following the lines of [1], [3]. 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 [7].
for $x\in \mathrm{\Omega}$, which arises similarly to ${f}^{\mathrm{\#}}$ in the situation where no interval halving is carried out. In the latter case, one has the following statement, which is a reformulation of [1], Corollary 13.2].
Corollary 10
then the$p$-periodic problem (1), (2) has a solution$u(\cdot )$with properties (47).
Assumption (65) with $\overline{f}$ given by (64) arises frequently in topological continuation theorems where the homotopy to the averaged equation is considered (see, e.g., [8], [9]).
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 ${k}_{0}$ interval halvings, one ultimately arrives a system of ${2}^{{k}_{0}}$ determining equation with respect to ${2}^{{k}_{0}}$ 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
${\mathit{k}}_{\mathbf{0}}$ | Variables in the determining equations |
---|---|
0 | ξ |
1 | ${\xi}_{-1}$, ${\xi}_{1}$ |
2 | ${\xi}_{-1,-1}$, ${\xi}_{-1,1}$, ${\xi}_{1,-1}$, ${\xi}_{1,1}$ |
3 | ${\xi}_{-1,-1,-1}$, ${\xi}_{-1,-1,1}$, ${\xi}_{-1,1,-1}$, ${\xi}_{-1,1,1}$, ${\xi}_{1,-1,-1}$, ${\xi}_{1,-1,1}$, ${\xi}_{1,1,-1}$, ${\xi}_{1,1,1}$ |
… | … |
- 1.
Fix a certain ${k}_{0}$ and consider the scheme with ${k}_{0}$ interval divisions. Fix an ${m}_{0}$ and construct ${u}_{m}(\cdot ,\xi )$ for $m=0,1,\dots ,{m}_{0}$.
- 2.Solve the $m$th approximate determining equations for $\xi $, find a root ${\xi}^{[m]}$, and put${U}_{m}(t):={u}_{m}(t,{\xi}^{[m]}),\phantom{\rule{1em}{0ex}}t\in [0,p],m=0,1,\dots ,{m}_{0}.$(69)
- 3.‘Check’ the behaviour of the functions ${U}_{0},{U}_{1},\dots ,{U}_{{m}_{0}}$ (the heuristic step). If promising (i.e., there are some signs of convergence), choose a suitable $\mathrm{\Omega}$ containing the graph of ${U}_{{m}_{0}}$, find a $\varrho $ from the condition$\varrho \ge \frac{p}{{2}^{{k}_{0}+2}}{\delta}_{{\mathrm{\Omega}}_{\varrho}}(f),$(70)
- 4.
Verify conditions of the existence theorem for $\mathrm{\Omega}$ and ${m}_{0}$. If not satisfied, or if the precision of ${U}_{{m}_{0}}$ is insufficient, pass to $m={m}_{0}+1$ and study ${U}_{{m}_{0}+1}$. Otherwise the algorithm stops, and the outcome is:
- (a)
- (b)
$\mathrm{\exists}({\xi}_{\ast},{\eta}_{\ast})\in {\mathrm{\Omega}}^{2}$: $u(\cdot )={u}_{\mathrm{\infty}}(\cdot ,{\xi}_{\ast},{\eta}_{\ast})$;
- (c)
the space localisation of the graph of $u$ is described by properties (47).
Note the role of interval divisions in the algorithm: for $K$ not satisfying the smallness condition (11) and ${k}_{0}=0$ (i.e., when ${u}_{m}$ 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 ${k}_{0}$ is chosen to be large enough.
$m$ is fixed, and ${P}_{m}$ stands for the $m$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 [11]). The number of resulting determining equations therewith depends on the Lipschitz constant of $f$ as well (in fact, it grows with $m$, the convergence being guaranteed by suitable properties of ${H}_{m}$ for $m$ 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 $A$ is a square matrix of dimension $n$ (possibly, singular), $c\in {\mathbb{R}}^{n}$, $f:[0,p]\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$, and $p\in (0,\mathrm{\infty})$.
which reduce to (17), (18) if $A$ is the unit matrix and $c=0$. Then, similarly to Proposition 1, it is not difficult to prove the following.
Proposition 11
- 1.The function ${x}_{\mathrm{\infty}}(\cdot ,\xi ,\eta )$ has the property${x}_{\mathrm{\infty}}(\frac{p}{2},\xi ,\eta )-A{x}_{\mathrm{\infty}}(0,\xi ,\eta )=\eta -A\xi $(79)
- 2.The function ${y}_{\mathrm{\infty}}(\cdot ,\xi ,\eta )$ has the property${y}_{\mathrm{\infty}}(p,\xi ,\eta )-A{y}_{\mathrm{\infty}}(\frac{p}{2},\xi ,\eta )=A(\xi -\eta )+c$(80)
and is the unique solution of the initial value problem (28), (29) with$\mathrm{H}$given by (30).
We see from Proposition 11 that properties of sequences ${x}_{m}(\cdot ,\xi ,\eta )$, ${y}_{m}(\cdot ,\xi ,\eta )$, $m\ge 0$, constructed for problem (73), (74) are rather similar to those for the periodic problem (1), (2) (in particular, the definition of functions $\mathrm{\Xi}$ and $\mathrm{H}$ 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 ${x}_{0}(\cdot ,\xi ,\eta )$ and ${y}_{0}(\cdot ,\xi ,\eta )$. As a result, the corresponding limit functions satisfy the boundary conditions (79), (80).
Theorem 12
Let there exist a non-negative vector$\varrho $with property (34) such that$f(t,\cdot )\in {Lip}_{K}({\mathrm{\Omega}}_{\varrho})$for a.e. $t\in [0,p]$with a certain matrix$K$satisfying inequality (35). Then, for all fixed$(\xi ,\eta )\in {S}_{A,c}(\mathrm{\Omega})\times \mathrm{\Omega}$, the sequences$\{{x}_{m}(\cdot ,\xi ,\eta ):m\ge 0\}$and$\{{y}_{m}(\cdot ,\xi ,\eta ):m\ge 0\}$with${x}_{0}(\cdot ,\xi ,\eta )$, ${y}_{0}(\cdot ,\xi ,\eta )$defined by (75) and (76) converge uniformly on the corresponding intervals and, moreover, estimates (36), (37) hold.
with ${\gamma}_{\ast}$ given by (38).
Declarations
Acknowledgements
The work was supported in part by RVO: 67985840 (A Rontó) and the SAIA National Scholarship Programme of the Slovak Republic (N Shchobak).
Authors’ Affiliations
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