Constructive analysis of periodic solutions with interval halving

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.

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 $q_T$.

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.

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).

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 $max\{h_i(z):z\in Q\}$, $i=1,2,\dots ,n$.

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.

One of the assumptions to be used below means that the inner r-neighbourhood of D is non-empty for r sufficiently large.

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.

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])

(3.9)

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.

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 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.

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])

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].

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.

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.

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.

Proposition 4.1 ([1])

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 $\frac{1}{2}T$.

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])

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).

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.

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.

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)

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.

(6.9)

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).

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.

(6.16)

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.

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

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

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.

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)

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).

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.

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).

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].

hold.

Proof

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 $m=0$.

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).

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

(7.15)

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.

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.

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.

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.