Constructive analysis of periodic solutions with interval halving

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.

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 [1]. 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 [2, 3]. 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 [4–8]. We refer the reader to [9–20] 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 [9–12] and a smallness restriction of the type

is imposed, where K is the Lipschitz matrix and q_T depends on the period T. The improvement of condition (0.1) consists in maximising the value of the constant q_T.

In this paper, which is a continuation of [1], we provide a constructive approach to the study of solvability of the periodic problem (1.3), (1.4), where the analysis of convergence uses the interval halving technique. We shall see that, under fairly general assumptions, this idea allows one to replace (0.1) by the weaker condition

and, thus, significantly improve the convergence conditions established, in particular, in [6–9, 12]. 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 [1], which are stated in the course of exposition when appropriate.

The method that we are interested in deals with T-periodic solutions of a system of non-linear ordinary differential equations

where f:{\mathbb{R}}^{n+1}\to {\mathbb{R}}^n is a continuous function such that

for all z\in {\mathbb{R}}^n and t\in (-\mathrm{\infty },\mathrm{\infty }). Here, T is a given positive number. We restrict ourselves to considering continuously differentiable solutions of system (1.1) and, furthermore, instead of T-periodic solutions of (1.1), we shall always deal with the solutions u:[0,T]\to {\mathbb{R}}^n of the corresponding periodic boundary value problem on the bounded interval [0,T],

(1.3)

(1.4)

The passage to problem (1.3), (1.4) is justified by assumption (1.2).

Our main assumption is that f:[0,T]\times {\mathbb{R}}^n\to {\mathbb{R}}^n 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 {\mathbb{R}}^n. For the sake of simplicity, we assume that there exists a non-negative constant square matrix K of dimension n such that

for all \{x_1,x_2\}\subset D and t\in [0,T].

Here and below, the obvious notation |x|=col(|x_1|,|x_2|,\dots ,|x_n|) 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\in Q\} for any h={(h_i)}_{i=1}^n:Q\to {\mathbb{R}}^n, where Q\subset {\mathbb{R}}^m, m\le n, is defined as the column vector with the components max\{h_i(z):z\in Q\}, i=1,2,\dots ,n.

We fix an n\in \mathbb{N} and a bounded set D\subset {\mathbb{R}}^n. The following symbols are used in the sequel:

1. 1.

   1_n is the unit matrix of dimension n.

2. 2.

   r(K) is the maximal, in modulus, eigenvalue of a matrix K.

3. 3.

   Given a closed interval J\subseteq \mathbb{R}, we define the vector {\delta }_{J,D}(f) by setting

   {\delta }_{J,D}(f):=\underset{(t,z)\in J\times D}{max}f(t,z)-\underset{(t,z)\in J\times D}{min}f(t,z).

   (2.1)
4. 4.

   e_k, k=1,2,\dots ,n: see (10.5).

5. 5.

   ∂ Ω is the boundary of a domain Ω.

6. 6.

   {▷}_S: see Definition 10.1.

The notion of a set D(r) associated with D, which could have been called an inner r-neighbourhood of D, will often be used in what follows.

Definition 2.1 For any non-negative vector r\in {\mathbb{R}}^n, we put

where

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 {\varrho }_{\ast } be determined by the equality

We refer, e.g., to [12, 21] for the discussion of other ways of introducing the constant {\varrho }_{\ast } and for its meaning. What is important for us here is that {\varrho }_{\ast } is the constant appearing in Lemma 3.2. One can show by computation that

The method suggested by Samoilenko in [6, 7], originally called numerical-analytic method for the investigation of periodic solutions, was also referred to later as the method of periodic successive approximations [9–12]. Its scheme, which is described in a suitable for us form by Propositions 3.1 and 3.4 below, is quite simple and deals with the investigation of the parametrised equation

where z\in D is a parameter to be chosen later. For convenience of reference, we formulate the statements for the p-periodic problem

(3.2)

(3.3)

where g:[t_0,t_0+p]\times {\mathbb{R}}^n\to {\mathbb{R}}^n and t_0\in (-\mathrm{\infty },\mathrm{\infty }) is arbitrary but fixed.

Following [1], we now describe the original, unmodified, periodic successive approximations scheme for the p-periodic problem (3.2), (3.3) which we are going to modify and which is constructed as follows. With problem (3.2), (3.3), one associates the sequence of functions u_m(\cdot ,z), m\ge 0, defined according to the rule

for t\in [t_0,t_0+p] and m=1,2,\dots , where the vector z=col(z_1,z_2,\dots ,z_n) is regarded as a parameter, the value of which is to be determined later.

Proposition 3.1 ([[12], Theorem 3.17])

Let the function f satisfy the Lipschitz condition (1.5) with a matrix K for which the inequality

holds and, moreover,

Then, for any fixed z\in D(\frac{p}{4}{\delta }_{[t_0,t_0+p],D}(g)), the following assertions are true:

1. 1.

   Sequence (3.4) converges to a limit function

   u_{\mathrm{\infty }}(t,z)=\underset{m\to \mathrm{\infty }}{lim}{u}_m(t,z)

   (3.7)

uniformly in t\in [t_0,t_0+p].

1. 2.

   The limit function (3.7) satisfies the p-periodic boundary conditions

   u_{\mathrm{\infty }}(t_0,z)=u_{\mathrm{\infty }}(t_0+p,z).
2. 3.

   The function u_{\mathrm{\infty }}(\cdot ,z) is the unique solution of the Cauchy problem

   

   (3.8)

(3.9)

where

1. 4.

   Given an arbitrarily small positive ε, one can choose a number m_{\epsilon }\ge 1 such that the estimate

   |u_m(t,z)-u_{\mathrm{\infty }}(t,z)|\le \frac{1}{2}{\alpha }_{m_{\epsilon }}(t)K^{m_{\epsilon }-1}{({\varrho }_{\epsilon }pK)}^{m-m_{\epsilon }+1}{(1_n-{\varrho }_{\epsilon }pK)}^{-1}{\delta }_{[t_0,t_0+p],D}(g)

holds for all t\in [t_0,t_0+p] and m\ge m_{\epsilon }, where

Recall that, according to (2.2), condition (3.6) means the non-emptiness of the inner \frac{p}{4}{\delta }_{[t_0,t_0+p],D}(g)-neighbourhood of the set D, where {\delta }_{[t_0,t_0+p],D}(g) is the vector given by formula (2.1). This agrees with the natural supposition that, for an approximation technique to be applicable, the domain where the Lipschitz condition is assumed should be wide enough.

The proof of Proposition 3.1 is based on Lemma 3.2 formulated below, which provides an estimate for the sequence of functions {\alpha }_m, m\ge 0, given by the formula

where m\ge 1 and {\alpha }_0(t):=1, t\in [t_0,t_0+p]. We provide the formulation here for a clearer understanding of the constants appearing in the estimates.

Lemma 3.2 ([[16], Lemma 3])

For any \epsilon \in (0,+\mathrm{\infty }), one can specify an integer m_{\epsilon }\ge 1 such that

for all t\in [t_0,t_0+p] and m\ge m_{\epsilon }.

It should be noted that estimate (3.13) is optimal in the sense that ε can never be put equal to zero.

Remark 3.3 It follows from [[22], Lemma 4] that if \epsilon \ge {\epsilon }_0, where

then m_{\epsilon }=2 in Lemma 3.2 (here, of course, we think of m_{\epsilon } as of the least integer possessing the property indicated).

The assertion of Proposition 3.1 suggests a natural way to establish a relation between the p-periodic solutions of the given equation (1.3) and those of the perturbed equation (3.8) (or, equivalently, solutions of the initial value problem (3.8), (3.9)). Indeed, it turns out that, by choosing the value of z appropriately, one can use function (3.7) to construct a solution of the original periodic boundary value problem (1.3), (1.4).

Proposition 3.4 ([1, 12])

Let the assumptions of Proposition  3.1 hold. Then:

1. 1.

   Given a z\in D(\frac{p}{4}{\delta }_{[t_0,t_0+p],D}(g)), the function u_{\mathrm{\infty }}(\cdot ,z) is a solution of the p-periodic boundary value problem (3.2), (3.3) if and only if z is a root of the equation

   \mathrm{\Delta }(z)=0.

   (3.15)
2. 2.

   For any solution u(\cdot ) of problem (3.2), (3.3) with u(t_0)\in D(\frac{p}{4}{\delta }_{[t_0,t_0+p],D}(g)), there exists a z_0 such that u(\cdot )=u_{\mathrm{\infty }}(\cdot ,z_0).

The important assertion (2) means that equation (3.15), usually referred to as a determining equation, allows one to track all the solutions of the periodic boundary value problem (1.3), (1.4). In such a manner, the original infinite-dimensional problem is reduced to a system of n numerical equations.

The method thus consists of two parts, namely, the analytic part, when the integral equation (3.1) is dealt with by using the method of successive approximations (3.4), and the numerical one, which consists in finding a value of the unknown parameter from equation (3.15). This closely correlates with the idea of the Lyapunov-Schmidt reduction [2, 3].

The main obstacle for an efficient application of Proposition 3.4 is due to the fact that the function u_{\mathrm{\infty }}(\cdot ,z), z\in D(\frac{p}{4}{\delta }_{[t_0,t_0+p],D}(g)) and, therefore, the mapping \mathrm{\Delta }:D(\frac{p}{4}{\delta }_{[t_0,t_0+p],D}(g))\to {\mathbb{R}}^n are explicitly unknown. Nevertheless, it is possible to prove the existence of a solution on the basis of the properties of a certain iteration u_m(\cdot ,z) which is constructed explicitly for a certain fixed m. For this purpose, one studies the approximate determining system

where {\mathrm{\Delta }}_m:D(\frac{p}{4}{\delta }_{[t_0,t_0+p],D}(g))\to {\mathbb{R}}^n is defined by the formula

for z\in D(\frac{p}{4}{\delta }_{[t_0,t_0+p],D}(g)). This topic is discussed in detail, in particular, in [12], whereas a theorem of the kind specified, which corresponds to the scheme developed here, is proved in Section 9. Our main goal is to obtain a solvability theorem under assumptions weaker than those that would be needed when applying Proposition 3.1.

Indeed, in view of (2.5), assumption (3.5), which is essential for the proof of the uniform convergence of sequence (3.4), can be rewritten in the form

Inequality (3.17) can be treated either as a kind of upper bound for the Lipschitz matrix or as a smallness assumption on the period p, the latter interpretation presenting the scheme as particularly appropriate for the study of high-frequency oscillations.

Without assumption (3.17), Lemma 3.2 does not guarantee the convergence of sequence (3.4) when applied directly along the lines of the proof of Proposition 3.1. Nevertheless, it turns out that this limitation can be overcome and, by using a suitable parametrisation and modifying the scheme appropriately, one can always weaken the smallness condition (3.5) so that the constant on its right-hand side is doubled:

Note also that, although we have in mind to weaken mainly the smallness condition (3.17) guaranteeing the convergence of iterations, it turns out that the techniques suggested here for this purpose allow us to obtain a considerable improvement of condition (3.6) as well (Corollary 6.7).

Moreover, we shall see that, under the weaker condition (3.18), the modified scheme can be used to prove the existence of a periodic solution on the basis of results of computation (Theorem 10.2).

We should like to show that the approach described by Proposition 3.1 can also be used in the cases where the smallness condition (3.5), which guarantees the convergence, is violated. For this purpose, a natural trick based on the interval halving can be used, where the unmodified scheme, in a sense, should work twice. However, some care should be taken on the boundary conditions.

Indeed, from the first glance, one is tempted to implement halving in the sense that the original scheme should be applied for each of the resulting half-intervals, and thus sequence (3.4) would be constructed twice for problem (3.2), (3.3) with t_0=0, p=\frac{1}{2}T, g=f{|}_{[0,\frac{1}{2}T]\times {\mathbb{R}}^n} and t_0=\frac{1}{2}T, p=\frac{1}{2}T, g=f{|}_{[\frac{1}{2}T,T]\times {\mathbb{R}}^n}, respectively. This is impossible, however, because the boundary conditions on the half-intervals, with trivial exceptions, are never \frac{1}{2}T-periodic.

The correct halving scheme is obtained when, along with the periodic boundary value problem (1.3), (1.4), we consider two auxiliary problems

(4.1)

(4.2)

and

(4.3)

(4.4)

where \lambda =col({\lambda }_1,\dots ,{\lambda }_n) is a free parameter, the value of which is to be determined suitably from the argument related to gluing. The mutual disposition of the graphs of x and y satisfying, respectively, problems (4.1), (4.2) and (4.3), (4.4) is as shown on Figure 1.

Solutions of auxiliary problems on half-intervals. Possible solutions of the auxiliary two-point problems (4.1), (4.2) and (4.3), (4.4) for arbitrary λ. The gluing means that, in case of solvability, λ is chosen so that (4.9) holds.

Full size image

Our further reasoning related to problem (1.3), (1.4) uses the following simple observation. Let us put

Proposition 4.1 ([1])

Let x:[0,\frac{1}{2}T]\to {\mathbb{R}}^n and y:[\frac{1}{2}T,T]\to {\mathbb{R}}^n be solutions of problems (4.1), (4.2) and (4.3), (4.4), respectively, with a certain value of \lambda \in {\mathbb{R}}^n. Then the function

is a solution of the periodic problem boundary value problem (1.4) for the equation

Conversely, if a certain function u:[0,T]\to {\mathbb{R}}^n is a solution of problem (1.3), (1.4), then its restrictions x:=u{|}_{[0,\frac{1}{2}T]} and y:=u{|}_{[\frac{1}{2}T,T]} to the corresponding intervals satisfy, respectively, problems (4.1), (4.2) and (4.3), (4.4).

Remark 4.2 A solution of the functional differential equation (4.7) is understood in the Carathéodory sense, and a jump of u^{\prime } at \frac{1}{2}T is allowed. Note that function (4.6) is always continuous at \frac{1}{2}T.

The idea of Proposition 4.1 is, in fact, to rewrite the periodic boundary condition (1.4) in the form

which naturally leads us to the introduction of the parameter λ.

Proposition 4.1 allows one to treat the T-periodic problem (1.3), (1.4) as a kind of join of two independent two-point problems (4.1), (4.2) and (4.3), (4.4). Solving them independently and considering λ as an unknown parameter, one can then try to ‘glue’ their solutions together by choosing the value of λ so that (4.9) holds. The possibility of this gluing is equivalent to the solvability of the original problem. A rigorous formulation is contained in the following

Proposition 4.3 ([1])

Assume that x:[0,\frac{1}{2}T]\to {\mathbb{R}}^n and y:[\frac{1}{2}T,T]\to {\mathbb{R}}^n are solutions of problems (4.1), (4.2) and (4.3), (4.4), respectively, for a certain value of \lambda \in {\mathbb{R}}^n. Then the function u:[0,T]\to {\mathbb{R}}^n given by formula (4.6) is a solution of problem (1.3), (1.4) if and only if the equality

holds.

Conversely, if a certain u:[0,T]\to {\mathbb{R}}^n is a solution of problem (1.3), (1.4), then the functions x:=u{|}_{[0,\frac{1}{2}T]} and y:=u{|}_{[\frac{1}{2}T,T]} satisfy, respectively, problems (4.1), (4.2) and (4.3), (4.4).

Introduce the functions {\overline{\alpha }}_m:[0,\frac{1}{2}T]\to [0,+\mathrm{\infty }) and {\overline{\overline{\alpha }}}_m:[\frac{1}{2}T,T]\to [0,+\mathrm{\infty }), m\ge 0, by putting {\overline{\alpha }}_0\equiv 1, {\overline{\overline{\alpha }}}_0\equiv 1,

for t\in [0,\frac{1}{2}T], and

for t\in [\frac{1}{2}T,T]. In particular, we have

and

Functions (4.10) and (4.11), which are, in fact, appropriately scaled versions of (3.12), are involved in the estimates given in the sequel.

As Proposition 4.3 suggests, our approach to the T-periodic problem (1.3), (1.4) requires that we first study the auxiliary problems (4.1), (4.2) and (4.3), (4.4) separately, for which purpose appropriate iteration processes will be introduced. Let us start by considering problem (4.1), (4.2). Following [1], we set

and define the recurrence sequence of functions X_m:[0,\frac{1}{2}T]\times {\mathbb{R}}^{2n}\to {\mathbb{R}}^n, m=0,1,\dots , by putting

(5.2)

for all m=1,2,\dots , \xi \in {\mathbb{R}}^n and \lambda \in {\mathbb{R}}^n. In a similar manner, for the parametrised problem (4.3), (4.4) on the interval [\frac{1}{2}T,T], we introduce the sequence of functions Y_m:[\frac{1}{2}T,T]\times {\mathbb{R}}^{2n}\to {\mathbb{R}}^n, m\ge 0, according to the formulae

(5.3)

(5.4)

for all η and λ from {\mathbb{R}}^n.

The recurrence sequences determined by equalities (5.1), (5.2) and (5.3), (5.4) arise in a natural way when boundary value problems of type (4.1), (4.1) and (4.3), (4.4) are considered. It is not difficult to verify that formulae (5.1), (5.2) and (5.3), (5.4) are particular cases of those corresponding the iteration scheme for two-point boundary value problems (see, e.g., [23]). One can also derive these formulae directly from Proposition 3.1 by carrying out, respectively, the substitutions x(t)=u(t)-2t{T}^{-1}\lambda, t\in [0,\frac{1}{2}T], and y(t)=u(t)+(2t{T}^{-1}-1)\lambda, t\in [\frac{1}{2}T,T], after which one arrives at parametrised \frac{1}{2}T-periodic boundary value problems on the corresponding half-intervals.

It is important to note that all the members of the sequences X_m(\cdot ,\xi ,\lambda ), m\ge 0, and Y_m(\cdot ,\xi ,\lambda ), m\ge 0, satisfy, respectively, conditions (4.2) and (4.4).

Lemma 5.1 For any \{\xi ,\eta ,\lambda \}\subset {\mathbb{R}}^n and m\ge 0, the functions X_m(\cdot ,\xi ,\lambda ) and Y_m(\cdot ,\eta ,\lambda ) satisfy the boundary conditions

(5.5)

(5.6)

Now recall that the vector λ, which is involved in all the above-stated relations, is the ‘gluing’ parameter determining the pair of auxiliary boundary value problems (4.1), (4.2) and (4.3), (4.4), for which a continuous join described by Proposition 4.3 is possible. In this relation, the following property is important.

Lemma 5.2 Let m\ge 0 be arbitrary. Then the equality

holds if and only if

Proof Indeed, it follows directly from (5.1) and (5.3) that X_0(\frac{1}{2}T,\xi ,\lambda )=\xi +\lambda and Y_0(\frac{1}{2}T,\eta ,\lambda )=\eta, whence the assertion is obvious for m=0. Similarly, if m\ge 1, then, according to (5.2) and (5.4), we have X_m(\frac{1}{2}T,\xi ,\lambda )=\xi +\lambda and Y_m(\frac{1}{2}T,\eta ,\lambda )=\eta and, consequently, relation (5.7) is equivalent to (5.8) for any m. □

Let us now pass to the construction of the iteration scheme for the original T-periodic problem (1.3), (1.4). The sequences X_m:[0,\frac{1}{2}T]\times {\mathbb{R}}^{2n}\to {\mathbb{R}}^n and Y_m:[\frac{1}{2}T,T]\times {\mathbb{R}}^{2n}\to {\mathbb{R}}^n, m\ge 0, from the preceding section will be used for this purpose. We shall see that, for this purpose, the graphs of the respective members of the last named sequences should be glued together in the sense of Lemma 5.2. Namely, we put

(6.1)

(6.2)

for any m=0,1,\dots . Functions (6.1) and (6.2) will be considered only for those values of ξ and η that are located, in a sense, sufficiently far from the boundary of the domain D. More precisely, we consider (\xi ,\eta ) from the set G_D(r), which, for any non-negative vector r, is defined by the equality

Recall that we use notation (2.3). In other words, a couple of vectors (\xi ,\eta ) belongs to G_D(r) if and only if every convex combination of ξ and η lies in D together with its r-neighbourhood. The inclusion (\xi ,\eta )\in G_D(r) implies, in particular, that B(\xi ,r)\subset D and B(\eta ,r)\subset D, i.e., the vectors ξ and η both belong to the set D(r) defined by formula (2.2). It is also obvious from (6.3) that G_D(r)\subset D^2 for any r.

The following statement shows that sequence (6.1) is uniformly convergent and its limit is a solution of a certain perturbed problem for all (\xi ,\eta ) which are admissible in the sense that (\xi ,\eta )\in G_D(r) with r sufficiently large.

Theorem 6.1 Let the vector-function f:[0,T]\times D\to {\mathbb{R}}^n satisfy the Lipschitz condition (1.5) on the set D with a matrix K such that

Moreover, assume that

Then, for an arbitrary pair of vectors (\xi ,\eta )\in G_D(\frac{T}{8}{\delta }_{[0,\frac{1}{2}T],D}(f)):

1. 1.

   The uniform, in t\in [0,\frac{1}{2}T], limit

   \underset{m\to \mathrm{\infty }}{lim}{x}_m(t,\xi ,\eta )=:x_{\mathrm{\infty }}(t,\xi ,\eta )

   (6.6)

exists and, moreover,

1. 2.

   The function x_{\mathrm{\infty }}(\cdot ,\xi ,\eta ) is the unique solution of the Cauchy problem

   

   (6.8)

(6.9)

where

1. 3.

   Given an arbitrarily small positive ε, one can specify a number m_{\epsilon }\ge 1 such that

   

   (6.11)

for all t\in [0,\frac{1}{2}T] and m\ge m_{\epsilon }, where {\varrho }_{\epsilon } is given by (3.11).

Recall that the constant {\varrho }_{\ast } involved in condition (6.4) is given by equality (2.4), while the vector {\delta }_{[0,\frac{1}{2}T],D}(f) arising in (6.5) is defined according to (2.1).

Remark 6.2 The error estimate (6.11) may look inconvenient because it is guaranteed starting from a sufficiently large iteration number, m_{\epsilon }, depending on the value of ε which can be arbitrarily small. It is, however, quite transparent when the required constant is not ‘too close’ to {\varrho }_{\ast } (i.e., if ε is not ‘too small’). More precisely, in view of Remark 3.3, m_{\epsilon }=2 for \epsilon \ge {\epsilon }_0, where

is given by formula (3.14). Consequently, inequality (6.11) with \epsilon \ge {\epsilon }_0 holds for an arbitrary value of m\ge 2.

By analogy with Theorem 6.1, under similar conditions, we can establish the uniform convergence of sequence (6.2). Namely, the following statement holds.

Theorem 6.3 Assume that the vector-function f satisfies conditions (1.5), (6.4) and, moreover,

Then, for all fixed (\xi ,\eta )\in G_D(\frac{T}{8}{\delta }_{[\frac{1}{2}T,T],D}(f)):

1. 1.

   The uniform, in t\in [\frac{1}{2}T,T], limit

   \underset{m\to \mathrm{\infty }}{lim}{y}_m(t,\xi ,\eta )=:y_{\mathrm{\infty }}(t,\xi ,\eta )

   (6.13)

exists and, moreover,

1. 2.

   The function y_{\mathrm{\infty }}(\cdot ,\xi ,\eta ) is the unique solution of the Cauchy problem

   

   (6.15)

(6.16)

where

1. 3.

   For an arbitrarily small positive ε, one can find a number m_{\epsilon }\ge 1 such that

   

   (6.18)

for all t\in [\frac{1}{2}T,T] and m\ge m_{\epsilon }, where {\varrho }_{\epsilon } is given by (3.11).

Remark 6.4 Similarly to Remark 6.2, one can conclude that the validity of estimate (6.18) is ensured for all m\ge 1 provided that \epsilon \ge {\epsilon }_0 with {\epsilon }_0 given by formula (3.14).

Theorems 6.1 and 6.3 are improved versions of Theorems 1 and 2 from [1], and their proofs follow the lines of those given therein. The main difference here is the use of Lemma 7.2 in order to guarantee that the values of the iterations do not escape from D. The rest of the argument is pretty similar to that of [1], and we omit it.

Note that the assumptions of Theorems 6.1 and 6.3 differ from each other in conditions (6.5) and (6.12) only. Therefore, by putting

we arrive immediately at the following statement summarising the last two theorems.

Theorem 6.5 Assume that the function f satisfies the Lipschitz condition (1.5) in D with K satisfying relation (6.4) and, moreover, D is such that

Then, for any (\xi ,\eta )\in G_D(r_D(f)), the assertions of Theorems 6.1 and  6.3 hold.

Recall that D is the main domain where the Lipschitz condition (1.5) is assumed, whereas G_D(r_D(f)) is the subset of D^2 defined according to (6.3). The latter set is, in a sense, a two-dimensional analogue of D(r_D(f)) and, as has already been noted above, the inclusion

is true. By virtue of (6.21), assumption (6.20) implies in particular that

which is a condition of type (3.6) appearing in Proposition 3.1 (see Figure 2). It turns out that, in the case of a convex domain, condition (6.20) can always be replaced by (6.22). Indeed, the following statement holds.

The set\mathit{D}\mathbf{(}{\mathit{r}}_{\mathit{D}}\mathbf{(}\mathit{f}\mathbf{)}\mathbf{)}. A schematic picture of the set D(r_D(f)) appearing in condition (6.22) with r_D(f) given by equality (6.19).

Full size image

Lemma 6.6 If the domain D is convex, then the corresponding set G_D(r_D(f)) has the form

Proof In view of (6.21), it is sufficient to show that

Indeed, let us put r:=r_D(f) (the assertion is, of course, true for any non-negative vector r, but the present formulation is sufficient for our purposes) and assume that, on the contrary, inclusion (6.23) does not hold. Then one can specify some ξ and η such that

(6.24)

(6.25)

According to definition (6.3), relation (6.25) means the existence of certain {\theta }_0\in [0,1] and z\in {\mathbb{R}}^n such that

Let us put h:=z-(1-{\theta }_0)\xi -{\theta }_0\eta. Then, in view of (6.26), we have

Furthermore, it is obvious that

and, consequently, z is a convex combination of \xi +h and \eta +h. By virtue of (2.2), (6.24) and (6.27), both vectors \xi +h and \eta +h belong to D and, therefore, so does z because (6.28) holds and the set D is convex. However, this contradicts relation (6.26). Thus, inclusion (6.23) holds, and our lemma is proved. □

By virtue of Lemma 6.6, the assertion of Theorem 6.5 for f Lipschitzian in a convex domain can be reformulated as follows.

Corollary 6.7 Let f satisfy conditions (1.5) and (6.4). If, moreover, the domain D is convex and (6.22) holds, then, for any ξ and η from D(r_D(f)), all the assertions of Theorems 6.1 and  6.3 hold.

The convexity assumption on D is rather natural and, in fact, the domain where the Lipschitz condition for the non-linearity is verified most frequently has the form of a ball (in our case, where the inequalities between vectors are understood componentwise, it is an n-dimensional rectangular parallelepiped).

We note that the smallness assumption (6.4), which guarantees the convergence of iterations in Corollary 6.7, is twice as weak as the corresponding condition (3.5) of Proposition 3.1:

Furthermore, it is rather interesting to observe that the condition on inner neighbourhoods also becomes less restrictive after the interval halving has been carried out. Indeed, it is clear from (2.1) and (6.19) that, for condition (6.22) of Corollary 6.7 to be satisfied, it would be sufficient if

whereas, at the same time, assumption (3.6) of Proposition 3.1 would require the relation

The radius of the inner neighbourhood in (6.30) is less by half. Comparing (6.4) and (6.30) with the corresponding conditions (6.29) and (6.31) arising in Proposition 3.1, we conclude that the idea of interval halving described above thus allows us to improve the original scheme of periodic successive approximations in both directions.

Theorem 6.5 suggests that the iteration sequences (5.2) and (5.4) can be used to construct the solutions of auxiliary problems (4.1), (4.2) and (4.3), (4.4) and ultimately of the original problem (1.3), (1.4). A further analysis, which will lead us to an existence theorem, involves determining equations. Before continuing, we give some auxiliary statements.

Several technical lemmata given below are needed in the proof of Theorems 6.1 and 6.3. We implicitly assume in the formulations that condition (6.20) is satisfied.

Given arbitrary i\in \{0,1\} and v\in C([\frac{1}{2}iT,\frac{1}{2}(i+1)T],{\mathbb{R}}^n), put

for all t\in [\frac{1}{2}iT,\frac{1}{2}(i+1)T]. The linear mapping P_i, which obviously transforms the space C([\frac{1}{2}iT,\frac{1}{2}(i+1)T],{\mathbb{R}}^n) to itself, is in fact a scaled version of the corresponding projection operator used rather frequently in studies of the periodic boundary problem (see, e.g., [12]). In our case, properties of this mapping are used when estimating the values of the Nemytskii operator generated by the function f involved in equation (1.3).

Lemma 7.1 Let x:[0,\frac{1}{2}T]\to {\mathbb{R}}^n and y:[\frac{1}{2}T,T]\to {\mathbb{R}}^n be arbitrary functions such that \{x(t):t\in [0,\frac{1}{2}T]\}\subset D and \{y(t):t\in [\frac{1}{2}T,T]\}\subset D. Then:

1. 1.

   For t\in [0,\frac{1}{2}T],

   \begin{array}{rl}\left|P_{0}f(\cdot ,x(\cdot ))\right|(t)& \le \frac{1}{2}{\overline{\alpha }}_1(t){\delta }_{[0,\frac{1}{2}T],D}(f)\\ \le \frac{T}{8}{\delta }_{[0,\frac{1}{2}T],D}(f).\end{array}

   (7.2)
2. 2.

   For t\in [\frac{1}{2}T,T],

   \begin{array}{rl}\left|P_{1}f(\cdot ,y(\cdot ))\right|(t)& \le \frac{1}{2}{\overline{\overline{\alpha }}}_1(t){\delta }_{[\frac{1}{2}T,T],D}(f)\\ \le \frac{T}{8}{\delta }_{[\frac{1}{2}T,T],D}(f).\end{array}

   (7.3)

Recall that {\overline{\alpha }}_1 and {\overline{\overline{\alpha }}}_1 are functions (4.12), (4.13), and the vectors {\delta }_{[0,\frac{1}{2}T],D}(f), {\delta }_{[\frac{1}{2}T,T],D}(f) are defined according to (2.1). The proof of Lemma 7.1 is almost a literal repetition of that of [[1], Lemma 7] and uses the estimate obtained in [[22], Lemma 3].

Lemma 7.2 For arbitrary m\ge 0 and (\xi ,\eta )\in G_D(r_D(f)), the inclusions

and

hold.

Proof

Let us fix an arbitrary pair of vectors

and prove, e.g., relation (7.4). We shall argue by induction. Indeed, in view of (5.1),

for t\in [0,\frac{1}{2}T]. This means that, at every point t from [0,\frac{1}{2}T], the value of x_0(t,\xi ,\eta ) is a convex combination of ξ and η. Recalling definition (6.3) of the set G_D(r_D(f)) and using assumption (7.6), we conclude that all the values of the function X_0(\cdot ,\xi ,\eta -\xi ) lie in D, i.e., (7.4) holds with m=0.

Assume now that

for a certain value of m and show that the inclusion

holds as well. Indeed, considering (5.2) and recalling notation (7.1), we conclude that, for all m, the identity

holds for any t\in [0,\frac{1}{2}T]. Since the validity of inclusion (7.4) has been assumed, we see that inequality (7.2) of Lemma 7.1 can be applied and, therefore, identity (7.10) yields

for all t\in [0,\frac{1}{2}T]. It follows from (7.11) that, at every point t\in [0,\frac{1}{2}T], the value X_{m+1}(t,\xi ,\eta -\xi ) lies in the \frac{T}{8}{\delta }_{[0,\frac{1}{2}T],D}(f)-neighbourhood of a convex combination of the vectors ξ and η. Since ξ and η satisfy (7.6) and, by (6.19), r_D(f)\ge \frac{T}{8}{\delta }_{[0,\frac{1}{2}T],D}(f), it follows from definition (6.3) of the set G_D(r_D(f)) that all the values of the function X_{m+1}(\cdot ,\xi ,\eta -\xi ) belong to D, i.e., (7.9) holds. Thus, inclusion (7.4) is true for all m\ge 0. Recalling notation (6.1), we arrive immediately to (7.4).

Relation (7.5) is proved by analogy. Indeed, it follows from (5.3) that

where \theta (t):=2(1-t{T}^{-1}) for any t\in [\frac{1}{2}T,T]. Since, obviously, 0\le \theta (t)\le 1 for all t\in [\frac{1}{2}T,T], identity (7.12) and assumption (7.6) guarantee that the function Y_0(\cdot ,\eta ,\eta -\xi ) has values in D. Let us assume that, for a certain m,

and show that

By virtue of (5.4), for any t\in [\frac{1}{2}T,T], we have

(7.15)

with the same definition of \theta (\cdot ) as in (7.12). According to assumption (7.13), the function Y_m(\cdot ,\xi ,\eta -\xi ) has values in D. Therefore, using equality (7.15) and estimate (7.3) of Lemma 7.1, we obtain

for all t\in [\frac{1}{2}T,T]. Since \theta :[\frac{1}{2}T,T]\to [0,1], inequality (7.16) implies that all the values of the function Y_{m+1}(\cdot ,\eta ,\eta -\xi ) belong to the \frac{T}{8}{\delta }_{[\frac{1}{2}T,T],D}(f)-neighbourhood of a convex combination of ξ and η. Recalling now (6.3) and (6.19) and using assumption (7.6), we arrive at (7.14). Consequently, inclusion (7.13) holds for all m, and (7.5) follows immediately from (6.2) and (7.13). The lemma is proved. □

Finally, the corresponding assertions of Theorems 6.1 and 6.3 lead us immediately to the following statement.

Lemma 7.3 Under the assumptions of Theorem  6.5, the inclusions

and

hold true for any (\xi ,\eta )\in G_D(r_D(f)).

The proof of Lemma 7.3 consists in passing to the limit in (7.4) and (7.5) as m\to +\mathrm{\infty }, the possibility of which is ensured by Theorem 6.5.

The techniques based on the original periodic successive approximations (3.4), the applicability of which is guaranteed by Proposition 3.1, lead one to the necessary and sufficient conditions for the solvability formulated in terms of determining equations (3.15) of Proposition 3.4. A certain analogue of the last mentioned statement should also be established for our new version of the method, with iterations constructed using the interval halving procedure, for the resulting scheme to be logically complete. It is natural to expect that the limit functions of the iterations on the half-intervals will help one to formulate criteria of solvability of the original problem, and, in fact, it turns out that it is the functions \mathrm{\Xi }:G_D(r_D(f))\to {\mathbb{R}}^n and \mathrm{H}:G_D(r_D(f))\to {\mathbb{R}}^n defined according to equalities (6.10) and (6.17) that provide such a characterisation.

Indeed, Theorems 6.1 and 6.3 guarantee that, under the conditions assumed, the functions x_{\mathrm{\infty }}(\cdot ,\xi ,\eta ):[0,\frac{1}{2}T]\to {\mathbb{R}}^n and y_{\mathrm{\infty }}(\cdot ,\eta ,\eta ):[\frac{1}{2}T,T]\to {\mathbb{R}}^n are well defined for all (\xi ,\eta )\in G_D(r_D(f)). Therefore, by putting

we obtain a function u_{\mathrm{\infty }}(\cdot ,\xi ,\eta ):[0,T]\to {\mathbb{R}}^n, which is well defined for the same values of (\xi ,\eta )\in G_D(r_D(f)). This function is obviously continuous.

The following theorem, which is a modified version of [[1], Theorem 4], establishes a relation of this function to the original periodic problem (1.3), (1.4) in terms of the zeroes of Ξ and H.

Theorem 8.1 Let f satisfy the Lipschitz condition (1.5) with a matrix K such that (6.4) holds. Furthermore, assume that D has property (6.20). Then:

1. 1.

   The function u_{\mathrm{\infty }}(\cdot ,\xi ,\eta ):[0,T]\to {\mathbb{R}}^n defined by (8.1) is a solution of the periodic boundary value problem (1.3), (1.4) if and only if the pair (\xi ,\eta ) satisfies the system of 2n equations

   \begin{array}{r}\mathrm{\Xi }(\xi ,\eta )=0,\\ \mathrm{H}(\xi ,\eta )=0.\end{array}

   (8.2)
2. 2.

   For every solution u(\cdot ) of problem (1.3), (1.4) with (u(0),u(\frac{1}{2}T))\in G_D(r_D(f)), there exists a pair ({\xi }_0,{\eta }_0) such that u(\cdot )=u_{\mathrm{\infty }}(\cdot ,{\xi }_0,{\eta }_0).

Equations (8.2) are usually referred to as determining or bifurcation equations [3, 12] because their roots determine solutions of the original problem. The variables involved in system (8.2) admit a natural interpretation: ξ means the value of the solution at 0, whereas η is responsible for its value at \frac{1}{2}T. We can observe the main difference between the unmodified periodic successive approximations (Proposition 3.1) and a similar scheme obtained after the interval halving (Theorem 6.5): the convergence condition is twice as weak but, instead of n numerical equations (3.15) of Proposition 3.4, we need to solve 2n equations (8.2) of Theorem 8.1.

A constructive solvability analysis involves a natural concept of approximate determining equations, which is discussed below.

Although Theorem 8.1 provides a theoretical answer to the question on the construction of a solution of the periodic problem (1.3), (1.4), its application faces difficulties due to the fact that the explicit form of the functions \mathrm{\Xi }:G_D(r_D(f))\to {\mathbb{R}}^n and \mathrm{H}:G_D(r_D(f))\to {\mathbb{R}}^n appearing in (8.2) is usually unknown. This complication can be overcome by using the functions

and

for a fixed m, which will lead one to the so-called approximate determining equations. More precisely, similarly to [12, 24], it can be shown that, under certain natural assumptions, one can replace the exact determining system (8.2) by its approximate analogue

Note that, unlike system (8.2), the m th approximate determining system (9.3) contains only terms involving the functions x_m:[0,\frac{1}{2}T]\times G_D(r_D(f))\to {\mathbb{R}}^n and y_m:[\frac{1}{2}T,T]\times G_D(r_D(f))\to {\mathbb{R}}^n and, thus, known explicitly.

It is natural to expect that approximations to the unknown solution of (1.3), (1.4) can be obtained by using the function u_m(\cdot ,\xi ,\eta ):[0,T]\to {\mathbb{R}}^n,

which is an ‘approximate’ version of (8.1) well defined for all t\in [0,T] and (\xi ,\eta )\in G_D(r_D(f)).

The piecewise character of the definition of function (9.4) does not affect the properties that a potential approximation obtained from it should possess. Indeed,

Proposition 9.1 If ξ and η satisfy equations (9.3) for a certain m, then the function u_{m+1}(\cdot ,\xi ,\eta ) determined by equality (9.4) is continuously differentiable on [0,T].

Proof It follows immediately from (5.2), (5.4) and (9.4) that

and

Recall that, by virtue of (5.4) and (6.2),

Then, in view of (9.1) and (9.2), it follows from (9.3), (9.5) and (9.6) that

and, therefore, u_{m+1}^{\prime }(\cdot ,\xi ,\eta ) is continuous at \frac{1}{2}T. The continuous differentiability of the function u_{m+1}(\cdot ,\xi ,\eta ) at other points is obvious from its definition. □

In order to prove a statement on the solvability of problem (1.3), (1.4), we need some estimates of the functions {\mathrm{\Xi }}_m:G_D(r_D(f))\to {\mathbb{R}}^n and {\mathrm{H}}_m:G_D(r_D(f))\to {\mathbb{R}}^n, m=0,1,\dots , defined by (9.1) and (9.2).

Lemma 9.2 Assume that (6.20) holds. Let f satisfy the Lipschitz condition (1.5) with a matrix K such that

Then the estimates

and

hold for any values of (\xi ,\eta )\in G_D(r_D(f)) and m\ge 2.

Proof Let us fix arbitrary (\xi ,\eta )\in G_D(r_D(f)) and m\ge 2. Recalling (6.10) and (9.1), we obtain

By Lemma 7.3, the function x_{\mathrm{\infty }}(\cdot ,\xi ,\eta ):[0,\frac{1}{2}T]\to {\mathbb{R}}^n has values in D and, therefore, the Lipschitz condition (1.5) can be used in (9.10). Then, applying estimate (6.11) of Theorem 6.1 with \epsilon ={\epsilon }_0, where {\epsilon }_0\approx 0.00727 is given by (3.14), we obtain

(9.11)

Recall now that, in view of Remark 6.2 and relations (3.11) and (3.14), one has

and, therefore, (9.11) can be rewritten in the form

(9.13)

Furthermore, it follows from (4.12) and (4.10) that the function {\overline{\alpha }}_2 has the form