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Notes on interval halving procedure for periodic and two-point problems
Boundary Value Problems volume 2014, Article number: 164 (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 .
We consider the periodic boundary value problem
where , satisfies the Carathéodory conditions, and a solution is an absolutely continuous vector function satisfying (1) almost everywhere on . Our main assumption till the end of the paper is that there exist a certain matrix and a bounded closed set such that for a.e. . Here and below, given a square matrix with non-negative entries, stands for the set of functions satisfying the componentwise Lipschitz condition
for all and from . In (3) and all similar relations that will appear below, the symbols ≤ and are understood componentwise.
In its original form (see, e.g., ,  for references), the numerical-analytic approach that we are dealing with suggests one to look for a solution of (1), (2) among the limit functions of certain -parametric family of sequences possessing property (2) (see, e.g., , ). Given an arbitrary vector , consider the sequence of functions defined by the recurrence relation
with small enough and satisfying the condition
and for any compact and .
In (7), is the -core of defined as
for any non-negative vector , where
The main limitation of this approach is that, in order to guarantee the convergence, one has to assume a certain smallness of the eigenvalues of the matrix . It was shown, in particular, in  that the method based upon sequence (4) is applicable provided that
As the recent paper  shows, the limitation can be overcome by noticing that the quantity which is assumed be small enough is always proportional to the length of the interval. A natural interval halving technique then allows one to produce a version of the scheme where (11) is replaced by the condition
and, thus, weakened by half. A similar improvement is also achieved in relation to condition (7), which is replaced by the assumption that
Here, we modify the scheme of  so that its substantiation is simplified and, in particular, replace (7) by an assumption which is more transparent and, generally speaking, less restrictive. Indeed, the idea to start from a set where the nonlinearity is known to be Lipschitzian and look for its suitable subset that could potentially contain initial values of periodic solutions is somewhat unnatural because, in any case, it is the initial values that are of major interest, the regularity assumptions for the equation being only technical assumptions induced by the method. Instead of doing so, which used to be the case in  and in all the previous works, it is, however, more logical to choose a closed bounded set , where one expects to find initial values of the solution, and to assume that the nonlinearity is Lipschitzian on a suitable , with only as large as the method requires. It is not difficult to see that the argument of  then leads us to the choice , where is the -neighbourhood of in the sense that
where the symbol stands for the -neighbourhood of a vector (recall that the relations in (10) are componentwise). Besides its more natural character, the use of the pair of sets is also advantageous in contrast to because, geometrically, does not necessarily copy the shape of (see Figures 2 and 3 for examples where and the corresponding with gradually increasing are represented, respectively, by the blue and red regions). In fact, the operations of taking -core and -neighbourhood do not commute: the equality in the inclusion
is, in general, not true, whereas one obviously has
for any . The strict inclusion in (15) holds, in particular, in the example from Figure 4, where the points of the sets , and for several values of , , are plotted in red, blue and cyan, respectively. In that example, by choosing to be the red region, one should then widen it for the technical purposes related to the method up to the cyan one, and not the blue one. A comparison of (15) and (16) confirms the advantage of assuming conditions of type (6) on . Several examples of domains and the corresponding sets can be seen on Figure 5.
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
Thus, let us fix a closed bounded set , where the initial values of solutions of problem (1), (2) will be looked for. Without loss of generality, we shall choose to be convex. Let and be arbitrary vectors from . Let us put
and define the recurrence sequences of functions and , , according to the formulae
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).
The function has the property(23)
and is the unique solution of the initial value problem
The function has the property(27)
and is the unique solution of the initial value problem
The proposition stated above, which is an easy consequence of the definitions of the functions and , , suggests one to consider the function introduced according to the formula
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 .
Introduce the functions
If there exists a non-negative vectorwith the property
such thatfor a.e. with a certainand
then, for all fixed, the sequence (resp., ) converges to a limit function (resp., ) uniformly in (resp., ), and the following estimates hold:
for all and
(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)).
It should be mentioned that assumption (35), which, by Theorem 2, ensures the applicability of the iteration scheme based on formulae (19), (20), is twice as weak as assumption (11) for the original sequence (4). The same kind of improvement is achieved concerning the condition on the set where is Lipschitzian since, for the scheme without interval halving, one would require that
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])
Letandbe arbitrary functions such thatand. Then
for all , , and .
Letbe a vector satisfying relation (34). Then, for arbitraryand, the inclusions
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
Theorem 6 provides one a formal reduction of the periodic problem (1), (2) to the system of numerical equations (48) in the sense that the initial data of any solution of (1), (2) with properties (47) can be found from (48). Thus, under the conditions assumed, the question on solutions of the periodic boundary value problem (1), (2) can be replaced that of the system of numerical equations (48). A combination of Proposition 11 and Theorems 2, 6 then suggests one a scheme of investigation of the periodic boundary value problem (1), (2). The practical realisation of the scheme is based upon the so-called approximate determining functions
considered for a fixed value of and, thus, computable explicitly. Then, as in , the function
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.
for all and, thus, (52) is necessary and sufficient for to be a singular point of . Similarly to , the assumptions of the theorem then allow one to construct a non-degenerate deformation of into the vector field
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 a certain fixed, where
It is sufficient to apply Theorem 7 with the linear homotopy
for , , and use estimate (10.11) from . □
The next assertion is interesting especially because it is, in fact, based upon properties of the starting approximation and, thus, shows how a useful information can be obtained when no iterations have been carried out at all. Note that the zeroth approximation is very rough indeed in any case: the periodic solution is approximated by a piecewise linear function (see Figure 6).
With the given function involved in (1), we associate the function by putting
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 .
It is interesting to note that involved in Corollary 9 can be considered as a ‘halved’ analogue of the averaged map
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].
Let there exist somewith property (34). Let
and, , withsatisfying inequality (35). If
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.
The construction of such a scheme is based on the appropriate modification of the initial approximation, which will then depend on more parameters. Consider, e.g., the transition from to . Renaming the variables as in the former case for more convenience and denoting the initial approximation by , we rewrite (21), (22) in the form
Thus, the initial approximation in the corresponding iteration scheme with one division is the linear function joining the points , and . Extending this tree graph like notation to the case of two interval halvings () and arguing similarly, we arrive at the following equalities determining :
In other words, relations (68) mean that the function for depends on the array of parameters and is obtained by the linear interpolation of the points , , , , and . For , the structure of is completely analogous, the idea is clear from Table 1 and Figure 6: one simply draws a broken line joining the corresponding nodes. Once is constructed, the formulae for the subsequent approximations are derived automatically by rescaling the projection map to the corresponding subintervals (we do not need the corresponding explicit formulae here and, therefore, omit the details).
Fix a certain and consider the scheme with interval divisions. Fix an and construct for .
Solve the th approximate determining equations for , find a root , and put(69)
In the case the equation has multiple roots, (69) and the related analysis are repeated for each of them (one can study multiple solutions of the original problem in this way).
‘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)
compute the Lipschitz matrix for in , and verify the convergence condition
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.
In relation to the last remark, it is interesting to compare the approach discussed here with the Cesari method , which likewise provides one a way to reduce the periodic problem (1), (2) to a system of finitely many numerical equations. The idea of construction of the iterations there is based, in the notation of , on the use of the operator
in a suitable space of -periodic functions, where
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.
In particular, the method described above is rather easy to adopt for application to two-point boundary value problems different from the periodic ones. Indeed, consider the problem with linear two-point conditions where one of the coefficient matrices is non-singular. Without loss of generality, we can assume that the problem has the form
where is a square matrix of dimension (possibly, singular), , , and .
The transition from the periodic problem (1), (2) to problem (73), (74) is then surprisingly simple: one does not need but to adjust the functions and so that they satisfy the boundary condition (74). More precisely, let us fix a suitable and take arbitrary and in it. Introduce the sequences of functions and , , according to the same recurrence formulae as in (19), (20), where, instead of (17), (18), the functions and are given by the equalities
The function has the property(79)
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).
Based on Proposition 11, one can develop essentially the same techniques that have been indicated above for the periodic problem (1), (2). The main difference in the proofs is that, in addition to guaranteeing that the appropriate values should belong to , we also have to ensure that as well. The convergence of iterations is then guaranteed for all from the set and belonging to its subset defined by the relation
Clearly, is the union of all the subsets of invariant with respect to the transformation . For example, if is a set on the plane () containing the origin, then is the part of that is symmetric with respect to the diagonal passing through the first and the third quadrants (see Figure 7).
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.
The same remark as has been made above concerning Theorem 2 applies to Theorem 12: its assertion remains true if (35) is replaced by the inequality
with given by (38).
Theorem 12 is easily obtained by analogy to Theorem 2 for the periodic problem. The verification of the conditions of Theorem 12 is also pretty much similar to the latter case. One has to keep in mind that the techniques for the two-point problem (73), (74) are applicable for the values of parameters lying in , and not in the entire , which is the case in Theorem 2 (unless is the unit matrix and ). This circumstance has a natural explanation due to (77) and (78), whence one deduces that both and will eventually belong to one and the same set, which fact is then used in Lemma 5. If, for example, is equal to zero and
then the assertion of Theorem 12 is true only for the part of that is invariant under the rotation by counter-clockwise. In this way, e.g., Figure 5 is replaced by Figure 8 once the two-point problem (73), (74) with given by (82) is considered. Note that all the sets on Figures 5 and 8 contain the origin, and the yellow regions on the latter one indicate the points from that cannot be regarded as candidates for initial values of the solution in question.
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The work was supported in part by RVO: 67985840 (A Rontó) and the SAIA National Scholarship Programme of the Slovak Republic (N Shchobak).
The authors declare that they have no competing interests.
All the authors contributed equally to the final version of this work and approved its present form.
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About this article
- periodic solution
- two-point problem
- Lyapunov-Schmidt reduction
- determining equation
- periodic successive approximations
- numerical-analytic method
- Cesari method
- interval halving