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International Journal of Differential Equations
Volume 2011 (2011), Article ID 612041, 20 pages
http://dx.doi.org/10.1155/2011/612041
Research Article

A Topological Approach to Bend-Twist Maps with Applications

Department of Mathematics and Computer Science, University of Udine, via delle Scienze 206, 33100 Udine, Italy

Received 26 May 2011; Accepted 21 July 2011

Academic Editor: Leonid Berezansky

Copyright © 2011 Anna Pascoletti and Fabio Zanolin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we reconsider, in a purely topological framework, the concept of bend-twist map previously studied in the analytic setting by Tongren Ding in (2007). We obtain some results about the existence and multiplicity of fixed points which are related to the classical Poincaré-Birkhoff twist theorem for area-preserving maps of the annulus; however, in our approach, like in Ding (2007), we do not require measure-preserving conditions. This makes our theorems in principle applicable to nonconservative planar systems. Some of our results are also stable for small perturbations. Possible applications of the fixed point theorems for topological bend-twist maps are outlined in the last section.

1. Introduction, Basic Setting and Preliminary Results

The investigation of twist maps defined on annular domains can be considered as a relevant topic in the study of dynamical systems in two-dimensional manifolds. Twist maps naturally appear in a broad number of situations, and thus they have been widely considered both from the theoretical point of view and for their significance in various applications which range from celestial mechanics to fluid dynamics.

One of the most classical examples of a fixed point theorem concerning twist maps on an annulus is the celebrated Poincaré-Birkhoff “twist theorem,” also known as the “Poincaré last geometric theorem.” It asserts the existence of at least two fixed points for an area-preserving homeomorphism of a closed planar annulus () onto itself which leaves the inner boundary and the outer boundary invariant and rotates and in the opposite sense (this is the so-called twist condition at the boundary). The Poincaré-Birkhoff fixed point theorem was stated (and proved in some special cases) by Poincaré [1] in 1912, the year of his death. In 1913 [2], Birkhoff, with an ingenious application of the index of a vector field along a curve, gave a proof of the existence of a fixed point (see also [3]). A complete description of Birkhoff’s approach, with also the explanation how to obtain a second fixed point, can be found in the expository article by Brown and Neumann [4]. The history of the “twist” theorem and its generalizations and developments is quite interesting but impossible to summarize in few lines. After about hundred years of studies on this topic, some controversial “proofs” of its extensions have been settled only recently. We refer the interested reader to [5] where the part of the story concerning the efforts of avoiding the condition of boundary invariance is described. In this connection, we also recommend the recent works by Martins and Ureña [6] and by Le Calvez and Wang [7] as well as the references therein.

Our study for the present paper is motivated by a recent approach considered by Ding in [8, Chapter 7] for the proof of the Poincaré-Birkhoff theorem for analytic functions. In the same chapter, a concept of bend-twist map is introduced. Roughly speaking, analytic bend-twist maps are those analytic twist maps in which the radial displacement changes its sign on a Jordan closed curve which is noncontractible in the annulus and where the angular displacement vanishes. Our goal is to extend Ding’s definition to a pure topological setting and obtain some fixed point theorems for continuous bend-twist maps. Our results do not require any regularity on the maps involved. Moreover, we do not assume hypotheses like homeomorphism, area preserving, or invariance of the boundaries, and, as an additional feature, some of our results are stable under small perturbations. These facts, in principle, suggest the possibility to produce some new applications to planar differential systems which are not conservative. Our main existence theorem (see Theorem 2.7) follows from the Borsuk separation theorem and Alexander’s lemma which we have extensively applied in a recent paper [9]. Our result partially extends Ding’s theorem to the nonanalytic setting. The main difference between Theorem 2.7 and the corresponding theorem in [8] lies on the fact that we obtain at least one fixed point, whence two fixed points are given in [8]. On the other hand, we show, by a simple example, that only one fixed point may occur in some situations. Both in our case and in Ding’s, the main hypothesis for the bend-twist theorem is a rather abstract one. Hence some more applicable corollaries, in the line of [8], are provided (see Theorem 2.9 and Corollary 2.10). In a final section we outline an application of our results to the periodic problem for some nonlinear ordinary differential equations.

We end this introduction with some definition and basic results which will be useful in the subsequent sections.

Definition 1.1. A topological space is called a topological annulus if it is homeomorphic to a planar annulus with defined as in (1.1).

Let be a topological annulus, and let be a homeomorphism. As a consequence of Schoenflies’s theorem [10], the set is independent of the choice of the homeomorphism . We call the set the contour of and denote it by . Clearly, for a topological annulus embedded in , the contour of coincides with the boundary of . The contour of consists of two connected components which are closed arcs (Jordan curves) since they are homeomorphic to . We call such closed arcs and . For a planarly embedded topological annulus, they could be chosen as the inner and the outer boundaries of the annulus. In such a special case, the bounded component of turns out to be an open simply connected set with and homeomorphic to the closed unit disc (this can be proved by means of the Jordan-Schoenflies theorem [10]). On the other hand, in the general setting, speaking of inner and outer boundaries is meaningless; yet we keep this terminology. Finally, we define the interior of as Slightly modifying an analogous definition of Berarducci et al. [11, Definition 2.1] we give the following.

Definition 1.2. Let be a topological space and let be two nonempty disjoint sets. Let also . We say that S cuts the paths between and if , for every continuous map (from now on, a path) such that and . The notation is used.

In order to simplify the presentation, we write to express the fact that cuts the paths between and . To make Definition 1.2 meaningful, we implicitly assume that there exists at least a path in connecting with (otherwise, we could take , or any subset of ). Such assumption will always be satisfied in the sequel. Clearly, if a set satisfies the cutting property of Definition 1.2, then also its closure cl cuts the paths between and . Therefore, we usually assume closed.

Let be an arcwise connected topological space. We say that a set is essentially embedded in if the inclusion is not homotopic to a constant map in .

The next result is a corollary of the Borsuk separation theorem [12, Theorem 6-47] adapted to our context. For a detailed proof, see also [9].

Lemma 1.3. Let be a topological annulus, and let be a closed set. Then is essentially embedded in if and only if .

For our applications to bend-twist maps we also need a more refined version of the above result which reads as follows.

Lemma 1.4. Let be a topological annulus and let be a closed set such that . Then there exists a compact, connected set such that (and, therefore, is essentially embedded in ).

Proof. First of all we claim that there exists a closed set such that , with minimal with respect to the cutting property. This follows from a standard application of Zorn’s lemma (see [9] for the details). Suppose, by contradiction, that is not connected, and let be two closed nonempty disjoint sets with . Since is minimal and , are proper subsets of , there exist two paths , in which connects to and such that avoids (for ). Then, by Alexander’s lemma [13] there exists a path with and with , contradicting the assumption that . The continuum is also essentially embedded in by Lemma 1.3.

In the sequel we denote by the standard covering projection of onto , defined by the polar coordinates .

2. Bend-Twist Maps

In this section, we reconsider, in a purely topological framework, the concept of bend-twist map introduced by Ding in [8]. The basic setting in [8] considers a pair of starlike planar annuli , with and a continuous map . Without loss of generality (via a translation of the origin), one can always assume that belongs to the open set . Accordingly, our basic setting will be the following.

Let be a topological annulus (embedded in the plane) with . Passing to the covering space , we have that the inner and outer boundaries of are lifted to the lines and which are periodic in the sense that if and only if , for . In [8] the boundaries are assumed to be starlike, that is, both and are graphs of -periodic functions , (for ) with , for all . The condition about the strictly star-shaped boundaries of is crucial for entering in the setting of the Poincaré-Birkhoff theorem (see [68, 14]). However, we will not assume it unless when explicitly required.

Let be a continuous map. By the theory of covering projections [15] there exists a (continuous) lifting of defined on such that By definition, given a lifting of , all the other liftings of are of the form for some . We assume that can be expressed as where , are continuous real-valued functions defined on and -periodic in the -variable. We also introduce an auxiliary function giving the radial displacement Observe that, instead of using the polar coordinates, we can equivalently express on the points of as Note also that the angular displacement can be referred directly to the points of , since is the same for any . This allows to define as for .

In some applications (for instance, to some planar maps associated to ordinary differential equations), the number represents a rotation number associated to a given trajectory departing from the point . In particular, observe that any solution of the system determines a fixed point of the map . Such a fixed point is “tagged” with the integer . This is an important information associated to in the sense that, once we have fixed in order to express as in (2.3), then solutions of (2.6) for different values of determine different fixed points of . In other words, if and are solutions of (2.6) for and , respectively, with then In fact, if, by contradiction, , then and for some . Hence, by the -periodicity of , we have a contradiction.

Conversely, one can easily check that any fixed point of the map lifts to a discrete periodic set and there exists an integer such that each point is a solution of (2.6) with the same value of .

Looking for a solution of system (2.6), an usual assumption on the map is the so-called twist condition at the boundary, which is one of the main hypotheses of the Poincaré-Birkhoff fixed point theorem. In our setting, the twist condition is expressed as follows.

Definition 2.1. We say that satisfies the twist condition if (or vice versa), for some .

If we prefer to express the twist condition directly on , we will write (or viceversa).

The celebrated Poincaré-Birkhoff “twist” theorem, in its original formulation, considers the case of a standard annulus Then we have the following.

Theorem 2.2. Assume (2.13), and suppose that is an area-preserving homeomorphism which lifts on to a homeomorphism of the form (2.3) which satisfies the following hypotheses: Boundary invariance: , , for all;Twist condition (2.11) (for some .Then, there exist at least two fixed points , for , in the interior of the annulus , with .

Usually, the hypothesis that the homeomorphism lifts to a of the form (2.3) is equivalently expressed by the assumption that is orientation preserving. One can easily modify the covering projection, for instance, to in order to have that the area-preserving homeomorphism lifts to a homeomorphism which preserves the element of area .

In order to introduce the concept of bend-twist maps we recall a (wrong) attempt of proving Theorem 2.2, as described by Wilson in a letter to Birkhoff [16]

“Won't you bother with finding out what ridiculous error there is in this simple thing that occurred to me yesterday ? The set of the points of the annulus with may be of great complexity containing ovals or ovals within ovals in the ring. But, as this set is closed and cannot be traversed by any continuous curve from the inner to the outer circles without being cut in at least one point, such set must include at least one continuous curve circling around the ring. Now, upon this curve, the shift is continuous and could not be always positive or always negative without shrinking said curve or expanding it, contrary to the supposed invariance of areas or integrals. Hence, there must be at least two points for which as well as .

We put in Italic the original words by Wilson. The notation is the original one and to make it compatible with that of the present paper we have to notice that the lifting of considered in [16] is expressed as a map . Thus our condition corresponds to in [16]. We also remark that the twist condition is assumed in [16] with (like in the original version of Poincaré-Birkhoff theorem).

The gap in this argument is not only on the fact that the set of points of the annulus where may not contain a “curve” (this perhaps is not the serious mistake), but even in the case in which there is actually a simple closed curve included in the set where , with encircling , the points of (which are supposed to exist by the area-preserving assumption) are not necessarily fixed points of . In fact, if is not star shaped, one could well have that (or ) along and, at the same time, (see Figure 1).

612041.fig.001
Figure 1: A sketch of the problem in Wilson’s argument. We depict a sector of an annular domain in which there is a portion of a non-star-shaped curve where . The points of are moved radially to the points of with preservation of the area. The points in are not fixed points for . A similar situation is described by Martins and Ureña in [6, Figures 1-2].

Of course, if we were able to prove that the radial displacement function vanishes at some points of the locus , then we would find fixed points for (making the above wrong argument meaningful). From this point of view, the study of the structure of the sets of points where may give useful information for the search of fixed points of . Such approach was considered, for instance, by Morris in [17] who proved the existence of infinitely many periodic solutions of minimal period (for each positive integer ), for the forced superlinear equation where is a smooth function with least period and mean value zero. For his proof, Morris considered the problem of existence of fixed points for the area-preserving homeomorphism of the plane where is the solution of the differential equation such that

In [17], starlike Jordan curves around the origin were constructed such that each point is mapped to on the same ray (see also [18] for a description of Morris result in comparison to other different approaches).

In [8] Ding considers the case of a topological annulus embedded in the plane having as its boundaries two simple closed curves which are starlike with respect to the origin. It is assumed that there exists an analytic function , with another starlike annulus with and, moreover, that satisfies the twist condition (2.11). It is also observed that the set of the points in where contains at least a Jordan curve which is not contractible in . The function is called a bend-twist map if there exists a Jordan curve , with noncontractible in , such that changes its sign on . Then, the following theorem holds (see [8, Theorem 7.2, page 188]).

Theorem 2.3. Let be an analytic bend-twist map. Then it has at least two distinct fixed points in .

We notice that, in Ding’s theorem, the assumptions that is area preserving and leaves the annulus invariant are not needed. This represents a strong improvement of the hypotheses required for the Poincaré-Birkhoff twist theorem. On the other hand, the assumption that a given function is a bend-twist map does not seem easy to be checked in the applications. For this purpose, the following corollary (see [8, Corollary 7.3, page 188]) provides more explicit conditions for the applicability of the abstract result.

Corollary 2.4. Let be an analytic twist map. If there are two disjoint continuous curves and in , connecting, respectively, the inner and the outer boundaries of and such that on and on , then is a bend-twist map on , and therefore it has at least two distinct fixed points.

Our aim now is to reformulate the above results in a general topological setting in order to obtain a version of Theorem 2.3 and Corollary 2.4 for general (not necessarily analytic) maps.

Let be a topological annulus (embedded in the plane) with , and let be a continuous map admitting a lifting of the form (2.3). Let us introduce the set

Lemma 2.5. Let satisfy the twist condition (2.11) for some . Then the set contains a compact connected set which is essentially embedded in and .

Proof. Our claim is an immediate consequence of Lemma 1.4 once that we have checked that . This latter property follows from the continuity of and the twist condition. Indeed, if is a path with and , then must vanish somewhere.

Our result corresponds to [8, Lemma 7.2, page 185] for a general . The Jordan curve considered in [8] in the analytic case is now replaced by our essentially embedded continuum . Following [8] we can now give the next definition.

Definition 2.6. Let be a continuous map (admitting a lifting of the form (2.3)) which satisfies the twist condition (2.11), for some . We say that is a bend-twist map in if there exists a compact connected set with essentially embedded in and such that changes its sign on .

As a consequence of this definition, the following theorem, which is a version of Theorem 2.3 for mappings which are not necessarily analytic, holds.

Theorem 2.7. Let be a bend-twist map. Then it has a fixed point in and with .

The proof is an obvious consequence of the connectedness of . If we were able to prove that is a Jordan curve, then, like in [8], the existence of at least two fixed points could be ensured.

In general, and in contrast with Theorem 2.3, we cannot hope to have more than one fixed point as shown by the following example which refers to a standard planar annulus .

Example 2.8. Let , and consider the set with , . The angular map in is defined as while, for the radial map , we set with and sufficiently small in order to have , for all . The functions and define by (2.3) a continuous map and, projecting by , a map . It is easy to check that leaves the boundaries of the annulus invariant and satisfies the twist condition (2.11) with . The set is the image of through . In accordance with Lemma 2.5 we can take . The function vanishes on the circumferences , , and , and, moreover, it is negative on the open annulus and positive on . Hence it changes its sign on . However, has a unique fixed point in which is (see Figure 2).

612041.fig.002
Figure 2: A description of the geometry in Example 2.8. The set made of the points of the annulus , where , is the union of a closed curve (contained in the part of the annulus between and ) and a small segment . The function vanishes at , and it is negative for and positive for (we have painted with a darker color the part of the annulus, where ). The point is the unique fixed point of since .

Perhaps the set in Example 2.8 is not completely satisfactory. Indeed, although it represents a compact connected set which cuts all the paths between and , it is not minimal. One could suppose that if we modify Definition 2.6by considering only minimal compact subsets of which are essentially embedded in , then we could provide the existence of at least two fixed points for , like in Ding’s theorem (see [9]). We have preferred to give a definition avoiding the concept of minimality because the existence of minimal sets will be only guaranteed by Zorn’s lemma, and, moreover, such sets could be quite pathological and thus intractable from the point of view of the applications. Further investigations should be needed in this direction.

On the other hand, we are able to recover the existence of two fixed points, as in Corollary 2.4, by the following result. In the proof we call a generalized rectangle any set which is homeomorphic to the unit square .

Theorem 2.9. Let be a continuous map (admitting a lifting of the form (2.3)) which satisfies the twist condition (2.11), for some . If there are two disjoint arcs and in , both connecting with in and such that on and on , then has at least two distinct fixed points in with .

Proof. Our argument is reminiscent of a similar one in the proof of the bifurcation result in [19]. Without loss of generality (up to a homeomorphism), we can suppose that . We also suppose (passing possibly to a sub-arc) that each intersects and exactly in one point, respectively. Let also and be the intersection points of with the circumferences and , respectively, (for ). Let be the arc of from to , and let be the arc of from to (in the counterclockwise sense). Similarly (again in the counterclockwise sense), we determine two arcs and on . The Jordan curves obtained by joining , , , and , , , bound two generalized rectangles and . We claim that in the interior of () there exists at least one fixed point for having as associated rotation number. We prove the claim for , since the proof for is exactly the same.
First of all, by the covering projection , we lift the set to the strip and observe that can be written as with a generalized rectangle contained in the strip and such that its boundary projects homeomorphically onto by . As observed above, is the compact region of the plane bounded by the Jordan curve , , , . By the Schoenflies theorem [10] we can choose a homeomorphism in such a manner that The vector field defined by is such that Thus, by the assumptions on and , we find that The above (strict) inequalities imply that we are in the setting of a two-dimensional version of the Poincaré-Miranda theorem and that where “deg” denotes the Brouwer degree. Therefore there exists at least one point such that . This, in turns, implies the existence of a fixed point such that is a fixed point of in the interior of and such that .

With the same argument of the proof of Theorem 2.9, the next result can be obtained.

Corollary 2.10. Let be a continuous map (admitting a lifting of the form (2.3)) which satisfies the twist condition (2.11), for some . Assume that there exist disjoint arcs () connecting with in . We label these arcs in a cyclic order and assume that on for odd and on for even (or viceversa). Then has at least distinct fixed points in , all the fixed points with .

Remark 2.11. Observe that Theorem 2.9, as well as Corollary 2.10, is stable with respect to small continuous perturbations of the map . This follows from the fact that (2.29) is true for any function satisfying the strict inequalities (2.28). Thus, if we perturb the function with a new continuous map with on and sufficiently small, we have that the twist condition and the conditions on on and are satisfied also for , and hence we get fixed points for as well.
On the other hand, we remark that Theorem 2.7 as well as Theorem 2.3 is not stable even in case of arbitrarily small perturbations. In order to show this fact, let us consider the following example.
Example 2.12. Let be a planar annulus with and . We consider an angular map in as , while, for the radial map, we take . The functions and define by (2.3) a continuous map and, projecting by , a map . It is easy to check that satisfies the twist condition (2.11) with . The set is the union of the circumferences and . The function vanishes on the ellipse . According to Definition 2.6, the map is a bend-twist map as changes its sign on . Indeed has exactly four fixed points which are the intersections of the ellipse with the circumference . However, for any sufficiently small, the map (where is a rotation of a small angle ) has no fixed points in . The reason is that the set disappears after an arbitrary small perturbation for , while the set is stable (in the sense that it continues into a nearby closed Jordan curve) but it is not suitable for the bend-twist map theorem since has constant sign on it.

Up to now we have presented all our results in terms of liftings of planar maps given by the standard covering projection in polar coordinates. In this manner we could make a simpler comparison with other results, like the Poincaré-Birkhoff fixed point theorem and the Ding analytic bend-twist maps theorem, which are usually expressed in the same framework. It is clear, however, that our approach works exactly the same also if different covering projections are used. For instance, in the applications to planar systems which are a perturbation of the first-order Hamiltonian system if we have an annulus filled by periodic orbits of (2.30), it could be convenient to choose as a radial coordinate the number expressing the level of the Hamiltonian and as an angular coordinate a normalized time of the corresponding orbit at level . We are going to use this remark for the application in the next section (see [19, 20] for some analogous cases).

3. An Application

It appears that the presence of bend-twist maps associated to planar differential equations is ubiquitous. This does not mean that proving their existence in concrete equations would be a simple task. It is a common belief that periodic solutions obtained for planar Hamiltonian systems via the Poincaré-Birkhoff fixed point theorem are not preserved by arbitrarily small perturbations which destroy the Hamiltonian structure of the equations. A typical example occurs when we add a small friction to a conservative system of the form passing to In general, for any continuous and each continuous function such that for all , the only possible periodic solutions of are the constant ones, corresponding to the zeros of (if any).

For (3.1) one can easily find conditions on guaranteeing the existence of an annulus in the phase plane filled by periodic orbits of the equivalent first-order Hamiltonian system To present a specific example, let us assume that there exists an open interval with such that is locally Lipschitz continuous with and The corresponding potential function is strictly decreasing on and strictly increasing on . Hence, for every constant with the energy level line defined by is a closed periodic orbit surrounding the origin. We denote by the fundamental period of . By the above assumptions it turns out that the map is continuous (see, for instance, [19, (v) page 83], where such result is proved in a more general situation).

In this setting we propose an application of the Poincaré-Birkhoff twist theorem and the bend-twist maps theorem to equations which are small perturbations of (3.1).

Just to start, we suppose that there exist and such that for . For convenience in the next exposition, we also suppose that The case in which can be treated analogously. The planar annulus is filled by periodic trajectories whose period varies continuously with the parameter . In particular the inner boundary and the outer boundary of the annulus are the energy level lines and , respectively.

Consider the level line with . By (3.5) it follows that is strictly star shaped around the origin. Hence, for every angle , the line intersects exactly in one point. From this fact, we immediately obtain another covering projection map onto the annulus which is equivalent to the projection in polar coordinates . In this manner, we can describe the points of by means of pairs , where, for each point , we have that is the usual angle in polar coordinates and .

The continuity of the map implies that, for every with equation (3.1) has at least one -periodic solution , where, for we have that . Actually, due to the autonomous nature of the system, there is at least a continuum of periodic solutions given by the shifts in time of , that is, the functions , with . From the point of view of the Poincaré map, which is the map where is the solution of (3.4) with , we have that has a continuum of fixed points which are all the points of the closed curve . The uniqueness of the periodic trajectory is not guaranteed (unless we assume some further conditions, like the strict monotonicity of the period with respect to ). In this autonomous case, as we have observed above, an arbitrarily small perturbation destroying the Hamiltonian structure of the equation may have the effect that the nontrivial -periodic solutions disappear.

As a next step, we consider a perturbation of (3.1) in the form of where is a -periodic function. For our purposes, only weak regularity assumptions on are needed. For instance, we can suppose that and consider the solutions of (3.16) in the generalized (Carathéodory) sense (see [21]). In this case, by the theorem of continuous dependence of the solutions in the Carathéodory setting, the Poincaré map associated to the planar system is well defined on if is sufficiently small in the -norm on . Then the following theorem holds.

Theorem 3.1. Assume (3.13). Then there exists such that for each with (3.16) has at least two -periodic solutions with initial value in , for defined in (3.11).

Theorem 3.1 is substantially a variant of a result of Buttazzoni and Fonda [22]. The proof follows a version of the Poincaré-Birkhoff fixed point theorem due to Ding [23] which applies to an area-preserving twist homeomorphism of a planar annulus with star-shaped boundaries. To be more precise, it should be remarked that recently the counterexamples in [6, 7] have shown that the theorem fails for annular domains with non-star-shaped boundaries. Here we use a result by Rebelo [14, Corollary 2] which holds for an area-preserving homeomorphism of the plane such that and with satisfying a twist condition on the boundary of a starlike annulus surrounding the origin.

We give a sketch of the proof of Theorem 3.1 for the reader’s convenience.

Proof. If we denote by the solution of (3.17) with and by the corresponding Poincaré map we have that is defined on (if is sufficiently small) as an area-preserving homeomorphism of onto with and , for all and . Passing to the polar coordinates we can determine an angular function so that It turns out that, in terms of the lifting associated to (compare to (2.3)), we have that with for (see [24] for the details). Assumption (3.13) for system (3.4) which now is viewed as a comparison system for (3.17) implies that if the perturbation is sufficiently small, then on and on and thus the twist condition (2.11) holds for .
Finally, using the fact that and are strictly star shaped with respect to the origin with , we can apply Ding’s version of the Poincaré-Birkhoff theorem [14, 23] and the existence of at least two distinct fixed points for in the interior of is ensured.

A natural question that now can arise is whether such (nontrivial) -periodic solutions would persist if a sufficiently small perturbation which destroys the Hamiltonian structure of the equation is performed. In the abstract setting of the Poincaré-Birkhoff theorem an answer can be found in the papers by Neumann [25] and Franks [26, 27] according to which if we have a finite number of fixed points then there are also fixed points with nonzero index. Actually, in [25, Theorem 2.1], the more general situation that the set of fixed points does not separate the boundaries is considered as well. In such cases, a standard application of the fixed point index theory (or the topological degree theory for maps of the plane) guarantees the persistence of fixed points for maps which are close to the Poincaré map, and hence the existence of nontrivial -periodic solutions also for sufficiently small perturbations of (3.16) holds. From this point of view, we could say that the bend-twist map theorem, in the form of Theorem 2.9 provides an effective criterion to prove the persistence of periodic solutions under small perturbations. In order to show an example of (3.16) to which our result can be applied, we consider a special form of the -periodic weight . For simplicity in the exposition we confine ourselves to the case of a continuous and -periodic function such that there is an interval such that By the continuity of we can get the following corollary of Theorem 3.1 where the smallness of in the -norm is expressed in terms of .

Corollary 3.2. Assume (3.13), and let be a continuous and -periodic function satisfying (3.23). Then there exists such that, if equation (3.16) has at least two -periodic solutions with initial value in .

In comparison to this result obtained via the Poincaré-Birkhoff fixed point theorem, using Corollary 2.10 we can obtain the following.

Theorem 3.3. Assume (3.13), and let be a continuous and -periodic function satisfying (3.23). Then there exists such that, if equation (3.16) has at least four -periodic solutions with initial value in . Moreover, the result is robust with respect to small perturbations. In particular, for there are at least four -periodic solutions with initial value in the annulus if is sufficiently small.

Proof. Without loss of generality (via a time shift leading to an equivalent equation), we can suppose that where we have set
To begin with the proof, we consider the Poincaré map on the annulus , with defined as in (3.18). Passing to the polar coordinates and following the same argument as in the proof of Theorem 3.1, we find a constant such that if then on and on and hence the twist condition (2.11) holds for .
In order to check the validity of the condition on the map , it is for us convenient to enter in the setting of the modified polar coordinates , instead of the standard polar coordinates . In this case, we can express the function as
We split now the map as with and defined as follows: where is the solution of the autonomous system (3.4) with and where is the solution of system (3.17) which departs from the point at the time . In this splitting we have also used the fact that system (3.17) coincides with the autonomous system (3.4) on . Hence we have
Let us consider now a solution of (3.17) and evaluate the energy along such solution. We obtain For we have that , and therefore the energy evaluated on a solution for the time interval is decreasing as long as the solution remains in the first or in the third quadrant and is increasing as long as the solution remains in the second or in the fourth quadrant.
Let be a fixed angle (the smaller we take, the larger will be allowed). Recalling the definition of in (3.12), let be the intersection of the line with the annulus . We are interested in the motion of the points of under the action of . Since , the points of move in the clockwise sense, and therefore they remain in the first quadrant if is sufficiently small. Hence is negative for when . This proves that Arguing in the same manner, we have that All these relations hold provided that is chosen suitably small (say ) so that the solutions of (3.17) which depart at the time from , remain in the same quadrant of for all . In order to make such argument more precise we can evaluate the angular displacements and choose such that holds for all ().
Finally, recalling (3.33) and the definition of in the -coordinates given in (3.29) and setting we conclude that on for odd and on for even. The thesis is thus achieved using Corollary 2.10.
An analysis of the proof and of inequality (3.37) shows that our argument is still valid if we take , where is a fixed positive function in .
Clearly, the same result holds also for (3.26) which can be viewed as a perturbation of (3.16). Of course, for such an application we exploit also the fact that in Corollary 2.10 no area-preserving-type hypothesis is required. The smallness of will depend on the smallness of .

We have achieved our result for a very special form of the weight function. A natural question concerns which kind of shape for a -periodic coefficient may be required in order to obtain a similar result for Generally speaking our argument may work (modulo technical difficulties) whenever we can split the behavior of the solutions of the equivalent system in the phase plane into two regimes, depending on a different shape of in two subintervals of its domain. In at least one of these regimes, we need to have a control of the trajectories and prove that they do not go too far from an annular region described by the level lines of an associated autonomous system. In the other regime, we need to show that there are at least some trajectories which are, in some sense, transverse to the annulus (and move into opposite directions). A different application of our technique is contained in [9] where we have considered a model of fluid mixing which is reminiscent to the case in which changes its sign.

A theorem about the existence of four solutions in this setting appears rather unusual (with respect to Corollary 3.2 and other analogous results following from the Poincaré-Birkhoff twist theorem). For previous multiplicity results in a completely different setting (namely, the Floquet problem for a superlinear equation), see [28].

Acknowledgments

The authors are grateful to Professor Tongren Ding for providing them with a copy of his interesting book [8] on its publication. They also thank the referees for the careful checking of their paper and for some useful remarks.

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