Homoclinic and heteroclinic solutions for a class of second-order non-autonomous ordinary differential equations: multiplicity results for stepwise potentials
© Ellero and Zanolin; licensee Springer. 2013
Received: 30 November 2012
Accepted: 11 June 2013
Published: 15 July 2013
We prove some multiplicity results for a class of one-dimensional nonlinear Schrödinger-type equations of the form
where and the weight is a positive stepwise function. Instead of the cubic term, more general nonlinearities can be considered as well.
MSC: 34C37, 34B15.
Keywordshomoclinic solutions heteroclinic solutions multiplicity nonlinear Schrödinger equation stepwise potential topological methods
Dedicated to Professor Jean Mawhin
we suppose that
(∗) is a locally Lipschitz function which is even, strictly increasing on such that and .
Examples of functions f which are suitable for our considerations are, for instance, as in , or () as in . In both these cases, the function f is even. In Section 3 we briefly describe how this restriction could be avoided.
(see ). Usually, the Planck constant and the mass are omitted in (1.3) after rescaling.
which is a particular case of (1.3) with , and . In equation (1.4) the nonlinearity is called inhomogeneous due to the presence of a non-constant weight , which in  is assumed to be positive. The sign condition on implies that nontrivial bounded solutions of (1.4) can exist only for (see [, Theorem 1]). For this reason, we prefer to set with .
which belongs to the same class of (1.1).
In spite of the apparent simplicity of equation (1.5), a throughout study of its solutions may be a rather difficult task for a general nonconstant weight function . Looking for homoclinic and heteroclinic solutions of (1.5), it will be natural to focus on the behavior of at ±∞. In similar situations, various authors have confined their study to the case in which is asymptotically constant [4–6], or eventually constant [7, 8]. These assumptions are also justified by the analysis of some physical underlying models, in which a layered structure is present. With this respect, see the introduction in , where different eventually autonomous cases are listed for related NLSEs arising in nonlinear optics. Examples in which the nonlinear term presents a piecewise constant weight function have been studied also in biological and chemical models. In particular, these situations occur in the theory of wave propagation for reaction-diffusion systems; see, for instance, [9–11]. In the context of equation (1.5), examples in which is a piecewise constant function have been considered as well (see ).
The aim of the present paper is to provide multiplicity results regarding homoclinic and heteroclinic solutions for equation (1.1) under particular assumptions on the weight function . Actually, we suppose that is an eventually constant piecewise function with only two steps.
Our approach combines phase-plane analysis with time-mapping estimates. As in [5, 7, 8, 10, 11, 13], the solutions are obtained by connecting the unstable and stable manifolds of the equilibrium points of the asymptotically autonomous equations. Such connections are performed by means of orbit paths of an intermediate equation, which represents the behavior of the system during a suitable interval of transition between the asymptotic states. Multiple connecting solutions arise when such interval length is sufficiently large. Lower estimates for will be provided in terms of time mappings, which can be expressed by Abelian integrals. This kind of approach is also reminiscent of some topological methods for the study of Sturm-Liouville boundary value problems. Indeed, a solution that satisfies the Sturm-Liouville boundary conditions can be interpreted as a trajectory in the phase-plane that connects two lines (see [14–17]). Generalized Sturm-Liouville solutions, which connect the graphs of two functions or given planar continua, have been considered as well (see [18–20]). Often these problems can be settled in the framework of the theory of ODEs with nonlinear boundary conditions (see [21, 22]).
In order to make our approach more transparent, we are going to perform our analysis for equation (1.5). This choice is motivated by the sake of avoiding unnecessary technicalities. Our arguments can be modified in a straightforward manner in the case of more general equation (1.1), with f satisfying (∗) (see Section 3). Homoclinic and heteroclinic orbits can be interpreted as solutions for some boundary value problems on unbounded intervals. In the last section we also outline possible applications of our approach to boundary value problems on a compact interval (like the Sturm-Liouville one).
Besides his manifold achievements in different areas of mathematics, Professor Jean Mawhin is one of the pioneers in the study of topological methods for nonlinear boundary value problems. It is a pleasure and an honor to have the possibility to dedicate our work to his important contributions in this area.
2 Homoclinic and heteroclinic solutions: multiplicity results
2.1 General setting
where is a fixed coefficient and is a bounded measurable function. Solutions of (2.1) are meant in the generalized (i.e., Carathéodory) sense (see ). Actually, in our results a step function with only two jumps is considered and the solutions are piecewise smooth.
In  Belmonte-Beitia and Torres supposed that the weight function is even and T-periodic and proved the existence of a heteroclinic solution connecting two periodic stationary states. Such a result generalizes to the case of periodic coefficients the situation described above for the autonomous system.
As already observed in the introduction, equation (2.1) has been already studied by various authors for its relevance in many applicative models. The aim of the present paper is to provide multiplicity results for heteroclinic and also homoclinic solutions of equation (2.1). A possible way to obtain this goal is to assume that the weight function has a different behavior at infinity and in some intermediate time interval. In this setting, a first step consists of analyzing the situation in which is a stepwise function assuming only two values. We believe that the study of such a simplified case may lead to more general considerations, in which, for instance, is eventually periodic. In this way, multiplicity results for Belmonte-Beitia and Torres’s model can be obtained (see Section 3 for a brief overview concerning possible applications of our approach). Equation (2.1) with a periodic stepwise coefficient has recently been studied in  in the context of chaotic-like dynamics.
From the fundamental theory of differential equations it follows that the domain of Φ is an open subset of the plane and Φ is an orientation preserving homeomorphism of onto its image .
2.2 Analysis of the equation
The solutions of system (2.2) with defined as in (2.5) will be obtained by connecting suitably chosen parts of stable and unstable manifolds of (2.6) with trajectories of (2.7). More in detail, we proceed as follows.
the energy level line contains the saddle points of system (2.6).
In order to obtain homoclinic or heteroclinic solutions for system (2.2) with defined as in (2.5), we connect the unstable manifold of the point to the stable manifold of via an orbit path of for . Actually, this is the only way to get the desired solutions. If , such connection will follow a trajectory with , so it will lie in the invariant annular region ℰ bounded by and . On the contrary, for the connection lies in the set ℱ between and .
To provide more explicit details, we consider the case . A solution of (2.2), which is homoclinic to , can be produced in two ways. One consists in connecting to , using an orbit path of for . Another possibility is given by connecting a point of to a point of via an orbit path of for . For , we can obtain our solution if and only if is an integer multiple of the fundamental period of the orbit . Indeed, such a solution will be constantly equal to for , and for it coincides with the periodic solution of (2.7) such that and . This solution makes turns around the origin for .
Similar considerations can be developed with respect to heteroclinic solutions. For instance, a heteroclinic orbit from to can be obtained as follows: by connecting a point of with one of using an orbit path of for , or by a connection of two different points of via an orbit path of for . For , we can obtain our solution if and only if is an odd multiple of , where is the period of the orbit . Indeed, such a solution will be constantly equal to for and constantly equal to for . Moreover, it coincides for with the solution of (2.7) such that and . This solution makes half-turns around the origin for .
In general, the connections through the trajectories of system (2.7) will either involve only an arc of the closed curve , or they will require to perform a certain number of winds around the origin, depending on the available time. If the length of the time interval is small, only the first case is possible. However, if is large enough, more choices arise. For instance, looking at the orbit δ in Figure 4, we could make a certain number of loops from the black circle before reaching the grey square on the same orbit. Similar considerations apply to the heteroclinic trajectory described above with reference to γ.
Until now we have described the ‘external connections’, namely those which lie in the region ℱ. Further possibilities appear if we consider ‘internal connections’ by means of orbit paths contained in ℰ. For example, if we look at the orbit β in Figure 4, we can obtain trajectories homoclinic to by choosing one of the two black circles as a starting point from and one of the two grey squares as an end point on . Loops along β will be permitted if the time is sufficiently large. The construction of heteroclinic orbits from to via the line α follows a similar procedure.
2.3 Study of the Poincaré map
With reference to Figure 5, we observe that the intersection point of with (indicated by a small circle) is moved by Φ along to a point very close to . On the other hand, during the time interval , the point makes a little more than a complete turn around the origin; the final position is indicated by a cross. The arc is a spiral-like curve with one end near and the other on . For the time length , considered in Figure 5, there are precisely two intersections of with , corresponding to external homoclinic solutions. Meanwhile, we have also an external heteroclinic solution due to the intersection of with . All these three intersection points have been indicated by a grey square (following the same convention used in Figure 4).
In order to describe further these solutions, let us consider their energy at the time that we denote by , , , with . Let us define the points as for . The solution of (2.2) with or has the following behavior: for , with and convex for . Moreover, is increasing on and decreasing on . On the interval , if , the solution is concave, while if , the solution has two maxima and one (negative) minimum separated by two simple zeros. The solution of system (2.2) with is such that for with and for with ; is convex for . Moreover, the solution has one maximum and one minimum separated by one simple zero.
Even if the time interval length is small, we always have at least one intersection of with and thus a homoclinic solution. On the other hand, if grows, the spiral curve will wind more times around the origin, hence more homoclinic/heteroclinic solutions will appear.
If we look for the internal connections (made by orbit paths lying in the region ℰ), we proceed as follows.
Summarizing the above information, we conclude that, for every time interval , we have at least two solutions which are homoclinic to (one external and one internal) and one internal heteroclinic solution from to .
For the external connections, if the time interval length grows, the situation can be summarized as follows. The number of external homoclinic/heteroclinic solutions increases, depending on the number of winds around the origin of the curve . Indeed, the end point of such a curve on cannot go beyond , while the other end point is free to move on the periodic orbit . If we denote by and the number of external homoclinic and heteroclinic solutions respectively, it holds that . From the above discussion, we can conclude that if, for some nonnegative integer n, we have (where is the period of ), then . An analogous lower bound can be provided for . For a formal proof, see Theorem 2.1.
In preparation for this theorem, we introduce the following notation. Given a positive real number S and an energy level , we denote by the lower integer part of , and by the upper integer part of . By definition, during a time interval of length S, a point on the orbit makes more that half-turns around the origin, but less that half-turns. Thus the rotational gap defined above can be written as for . By definition, in the open interval , the smallest integer is , while the largest integer is . Hence, the open interval contains positive integers.
In order to make the exposition more clear, we have decided to consider separately the cases of external and internal connections.
Theorem 2.1 (External connections)
there always exist a (positive) solution homoclinic toand a (negative) solution homoclinic to;
if, there always exist a heteroclinic solution fromtoand a heteroclinic solution fromto, both with exactly one zero;
if (for some integer) then, for each integer j with, there exist at least one solution homoclinic toand one solution homoclinic to, both with exactly 2j zeros;
if (for some integer) then, for each integer j with, there exist at least one heteroclinic solution fromtoand one fromto, both with exactlyzeros.
Proof As a first step, we focus our attention on the search of solutions homoclinic to the point . The case of homoclinics to is analogous, thus it will be omitted.
Recalling that , our goal is to find points on which are moved by the Poincaré map on .
We introduce in the phase-plane for (2.7), a system of polar coordinates , with center in the origin. The initial points in the arc (which is the closure of the intersection of with the external region ℱ) are parameterized as , for , with , and strictly increasing with s. The target set is the symmetric of with respect to the x-axis. It will be denoted by and parameterized by reversing the angle .
is strictly decreasing on (this is an equivalent way to express the fact that the solutions turn around the origin in the clockwise sense).
for some nonnegative integer j. In this case, if we denote by the corresponding homoclinic solution of (2.1) such that , we have that for all . Moreover, if , for all , while has precisely 2j zeros in if .
is satisfied (for the same j).
As a consequence, and the intermediate value theorem ensures the existence of an such that . In this way we have found a positive homoclinic solution to , independently of the length T of the time interval .
As a consequence, for every integer j with , we have , and again the intermediate value theorem ensures the existence of an such that . This ends the proof for homoclinic solutions.
for some integer . Then the proof can be concluded as above, via the intermediate value theorem. We observe that the case of heteroclinic solutions from to is analogous, thus it will be omitted. □
The next theorem deals with the internal connections. For this result, it is useful to recall the rotational gap defined in (2.11).
Theorem 2.2 (Internal connections)
there always exist a solution homoclinic toand a solution homoclinic to;
there always exist a heteroclinic solution fromtoand a heteroclinic solution fromto;
if, there exist at leastsolutions homoclinic toandsolutions homoclinic to;
if, there exist at leastheteroclinic solutions fromtoandfromto.
Proof As the first step, we focus our attention on the search of solutions homoclinic to the point . The case of homoclinics to is analogous, thus it will be omitted.
Our goal is to find points on which are moved by the Poincaré map on .
As a preliminary remark, we note that is a simple arc contained in ℰ, connecting its end points and on through the ‘intermediate’ point on . Observe also that and are antipodal. In fact, and are antipodal and is an odd map (indeed, is a solution of (2.7) if and only if is a solution of the same equation). Now, if the trivial situation occurs, we have . Otherwise, we find that and belong to the two different components of . Since and are the end points of the arc , by an elementary connectivity argument, we conclude that . This proves the first assertion of the theorem.
For the search of heteroclinic solutions from to , we adopt a similar procedure, choosing as a target the set ℒ itself. The fact that, for every , there is always at least one intersection point between and ℒ follows by the same argument developed for homoclinic solutions. Namely, if the trivial situation occurs, we have . Otherwise, we find that and belong to the two different components of . Therefore, it follows that .
We study now the problem of multiplicity of solutions. As before, we consider at first the case of solutions homoclinic to .
We introduce in the inner region ℰ of a phase-plane for (2.7) a system of modified polar coordinates with center in the origin. In this system, every point is determined by its angular coordinate θ and its energy E defined in (2.8). Since the energy level lines in ℰ are strictly star-shaped with respect to the origin, we obtain a coordinate system equivalent to the polar one.
Observe that the set intersects every half-line from the origin exactly in one point. Accordingly, we can parameterize those points using the angular coordinate. With this convention, the initial points in the arc ℒ are parameterized as for (s is the angle) with , and . As a consequence, is strictly decreasing for and strictly increasing in . The target set (the symmetric of ℒ with respect to the x-axis) is parameterized with the angle .
Hence, the natural parametrization of ℒ in the -plane is given by for .
(recall that for ).
(this follows from the fact that the orbit has a period and is symmetric with respect to the x-axis).
Then, the open interval contains at least one odd integer.
By definition, for any nonnegative integer i, the set is a simple arc connecting the points and in the rectangle and, similarly, is a simple arc connecting the points and in the rectangle . On the other hand, the set is a simple arc connecting the points and in and, similarly, is a simple arc connecting the points and in the same strip.
for an even integer. Hence the points and are separated by the arc , while the points and are separated by and . As a consequence, contains at least three points, precisely the nonempty intersections of , and .
for an even integer. Hence the points and are separated by the arcs and . The points and are separated by , and . As a consequence, contains at least five points: two points coming from the intersections of with the target set Ξ and three from the intersections of .
Proceeding by induction, with the same argument, the following result is obtained.
Claim 1 has at least solutions, where is the number of integers less than or equal to which are contained in the open interval .
A completely symmetric argument leads to the same multiplicity result for solutions which are homoclinic to .
where is the function defined in (2.15).
As before, we assume the validity of condition (2.18). Then the open interval contains at least one even integer.
For any positive integer j with , we have that (2.20) holds too. At this point, we have simply to repeat (with obvious changes due to the fact that the target set is which is a shift of Ξ by ) the argument presented above for the case of homoclinic connections and obtain the following conclusion.
Claim 2 has at least solutions, where is the number of integers less than or equal to which are contained in the open interval .
A completely symmetric argument leads to a multiplicity result for heteroclinic solutions from to .
This concludes the proof of the theorem. □
3 Remarks and related results
A problem which naturally arises from the analysis that we have performed concerns what happens if, for a stepwise weight function satisfying (2.5), we suppose that . Repeating the preliminary phase-plane analysis of Section 2.2, one can easily check that for any gap , there always exist a solution homoclinic to and another homoclinic to , as well as a heteroclinic from to and another one from to . However, in general, if , one cannot obtain multiplicity results like those achieved in Section 2 without some extra assumptions on . This is the reason for which, in the study of equation (2.1), we have considered only the case . For different examples on related equations in which a weight coefficient can be above or below its limits at infinity, see, for instance, [4–6].
- 2.We notice that the same argument of the proofs applies to an equation of the form
- 3.Due to the special form of the weight coefficient, it is standard to verify (via a simple rescaling procedure) that (1.1) is equivalent to an equation of the form(3.1)
The approach used in the proofs, based on the properties of the Poincaré map, guarantees that our results are stable with respect to small perturbations. More precisely, fixed a suitable length for the time interval, Theorems 2.1 and 2.2 provide a lower bound for the number of solutions. We can state that the same lower bound persists for a small perturbation of the coefficient in the -norm on . Therefore, the assumption that the weight in (1.1) or in (3.1) are stepwise functions can be slightly relaxed, so we can ‘smooth’ them.
- 5.With reference to equations (1.1) or (3.1) with stepwise coefficients, we observe that our approach can be adapted to boundary value problems on a compact interval like, e.g., the Dirichlet (two-point) or the Neumann problem. In these cases, we have to find solutions connecting given lines which depart from the origin. For the sake of conciseness, we cannot describe the most general situation, but we just outline a possible application for the Neumann problem(3.2)
As a final remark, we mention the fact that combining our technique with Ważewski’s method , following the approach developed by Conley in , one can deal with some more general classes of weight functions. For example, one could tackle with these techniques the cases of asymptotically constant or asymptotically periodic coefficients. These extensions, however, need a more delicate analysis and they are beyond the goals of the present paper.
We hope that the abundance of multiplicity results found in the present work (in the special case of stepwise coefficients) may suggest possible directions for extending Theorem 2.1 and Theorem 2.2 to more general weight functions. This will be our goal for a future investigation of the problem.
The authors are deeply indebted with the referee for the careful checking of the manuscript and for his/her remarks, including a correction to an erroneous argument in the previous version of the proof. This research was partially supported by the project PRIN-2009 Equazioni Differenziali Ordinarie e Applicazioni.
- Belmonte-Beitia J, Torres PJ: Existence of dark soliton solutions of the cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity. J. Nonlinear Math. Phys. 2008, 15(3):65–72.MathSciNetView ArticleGoogle Scholar
- Torres PJ, Konotop VV: On the existence of dark solitons in a cubic-quintic nonlinear Schrödinger equation with a periodic potential. Commun. Math. Phys. 2008, 282: 1–9. 10.1007/s00220-008-0527-0MathSciNetView ArticleMATHGoogle Scholar
- Ambrosetti A, Badiale M, Cingolani S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 1997, 140: 285–300. 10.1007/s002050050067MathSciNetView ArticleMATHGoogle Scholar
- Gavioli A: On the existence of heteroclinic trajectories for asymptotically autonomous equations. Topol. Methods Nonlinear Anal. 2009, 34: 251–266.MathSciNetMATHGoogle Scholar
- Gavioli A: Monotone heteroclinic solutions to non-autonomous equations via phase plane analysis. Nonlinear Differ. Equ. Appl. 2011, 18: 79–100. 10.1007/s00030-010-0085-yMathSciNetView ArticleMATHGoogle Scholar
- Gavioli A, Sanchez L: Heteroclinics for non-autonomous second-order differential equations. Differ. Integral Equ. 2009, 22: 999–1018.MathSciNetMATHGoogle Scholar
- Holmes PJ, Stuart CA: Homoclinic orbits for eventually autonomous planar flows. Z. Angew. Math. Phys. 1992, 43: 598–625. 10.1007/BF00946253MathSciNetView ArticleMATHGoogle Scholar
- Jones CKRT, Küpper T, Schaffner K: Bifurcation of asymmetric solutions in nonlinear optical media. Z. Angew. Math. Phys. 2001, 52: 859–880. 10.1007/PL00001578MathSciNetView ArticleMATHGoogle Scholar
- Aronson DG, Mantzaris NV, Othmer HG: Wave propagation and blocking in inhomogeneous media. Discrete Contin. Dyn. Syst. 2005, 13: 843–876.MathSciNetView ArticleMATHGoogle Scholar
- Lewis TJ, Keener JP: Wave-block in excitable media due to regions of depressed excitability. SIAM J. Appl. Math. 2000, 61: 293–316. 10.1137/S0036139998349298MathSciNetView ArticleMATHGoogle Scholar
- Yang J, Kalliadasis S, Merkin JH, Scott SK: Wave propagation in spatially distributed excitable media. SIAM J. Appl. Math. 2002, 63: 485–509.MathSciNetView ArticleMATHGoogle Scholar
- Rodrigues AS, Kevrekidis PG, Porter MA, Frantzeskakis DJ, Schmelcher P, Bishop AR: Matter-wave solitons with a periodic, piecewise-constant scattering length. Phys. Rev. A 2008., 78: Article ID 013611Google Scholar
- Margheri A, Rebelo C, Zanolin F: Connected branches of initial points for asymptotic BVPs, with application to heteroclinic and homoclinic solutions. Adv. Nonlinear Stud. 2009, 9: 95–135.MathSciNetMATHGoogle Scholar
- Boscaggin A, Zanolin F: Pairs of nodal solutions for a class of nonlinear problems with one-sided growth conditions. Adv. Nonlinear Stud. 2013, 13: 13–55.MathSciNetMATHGoogle Scholar
- Capietto A, Henrard M, Mawhin M, Zanolin F: A continuation approach to some forced superlinear Sturm-Liouville boundary value problems. Topol. Methods Nonlinear Anal. 1994, 3: 81–100.MathSciNetMATHGoogle Scholar
- Fonda, A, Garrione, M: Generalized Sturm-Liouville boundary conditions for first order differential systems in the plane. Topol. Methods Nonlinear Anal. (to appear)
- Rebelo C, Zanolin F: On the existence and multiplicity of branches of nodal solutions for a class of parameter-dependent Sturm-Liouville problems via the shooting map. Differ. Integral Equ. 2000, 13: 1473–1502.MathSciNetMATHGoogle Scholar
- Conley C: An application of Wazewski method to a non-linear boundary value problem which arises in population genetics. J. Math. Biol. 1975, 2: 241–249. 10.1007/BF00277153MathSciNetView ArticleMATHGoogle Scholar
- Rebelo C, Sanchez L: Existence and multiplicity for an O.D.E. with nonlinear conditions. Differ. Equ. Dyn. Syst. 1995, 3: 383–396.MathSciNetMATHGoogle Scholar
- Struwe M: Multiple solutions of anticoercive boundary value problems for a class of ordinary differential equations of second order. J. Differ. Equ. 1980, 37: 285–295. 10.1016/0022-0396(80)90099-6MathSciNetView ArticleMATHGoogle Scholar
- Gaines RE, Mawhin J: Ordinary differential equations with nonlinear boundary conditions. J. Differ. Equ. 1977, 26: 200–222. 10.1016/0022-0396(77)90191-7MathSciNetView ArticleMATHGoogle Scholar
- Gaines RE, Mawhin J Lecture Notes in Mathematics 568. In Coincidence Degree, and Nonlinear Differential Equations. Springer, Berlin; 1977.Google Scholar
- Hale JK: Ordinary Differential Equations. 2nd edition. Krieger, Huntington; 1980.MATHGoogle Scholar
- Zanini C, Zanolin F: An example of chaos for a cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity. Adv. Nonlinear Stud. 2012, 12: 481–499.MathSciNetMATHGoogle Scholar
- Gradshteyn IS, Ryzhik IM: Table of Integrals, Series, and Products. 7th edition. Academic Press, Boston; 2007. Translation Edited and with a Preface by Alan JeffreyMATHGoogle Scholar
- Meyer KR: Jacobi elliptic functions from a dynamical systems point of view. Am. Math. Mon. 2001, 108: 729–737. 10.2307/2695616View ArticleMathSciNetMATHGoogle Scholar
- Belmonte-Beitia J, Cuevas J: Existence of dark solitons in a class of stationary nonlinear Schrödinger equations with periodically modulated nonlinearity and periodic asymptotics. J. Math. Phys. 2011., 52: Article ID 032702Google Scholar
- Chicone C: The monotonicity of the period function for planar Hamiltonian vector fields. J. Differ. Equ. 1987, 69: 310–321. 10.1016/0022-0396(87)90122-7MathSciNetView ArticleMATHGoogle Scholar
- Frauenkron H, Grassberger P: Interaction of nonlinear Schrödinger solitons with an external potential. J. Phys. A 1995, 28: 4987–4996. 10.1088/0305-4470/28/17/027MathSciNetView ArticleMATHGoogle Scholar
- Sakaguchi H, Tamura M: Scattering and trapping of nonlinear Schrödinger solitons in external potentials. J. Phys. Soc. Jpn. 2004, 73: 503–506. 10.1143/JPSJ.73.503MathSciNetView ArticleGoogle Scholar
- Ważewski T: Sur un principe topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles ordinaires. Ann. Soc. Polon. Math. 1947, 20: 279–313.MathSciNetMATHGoogle Scholar