- Open Access
Constructive analysis of periodic solutions with interval halving
© Rontó et al.; licensee Springer 2013
- Received: 27 December 2012
- Accepted: 28 February 2013
- Published: 22 March 2013
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.
- periodic solution
- periodic boundary value problem
- periodic successive approximations
- numerical-analytic method
- interval halving
- Mawhin’s theorem
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 . 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 depends on the period T. The improvement of condition (0.1) consists in maximising the value of the constant .
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 , which are stated in the course of exposition when appropriate.
The passage to problem (1.3), (1.4) is justified by assumption (1.2).
for all and .
Here and below, the obvious notation 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., for any , where , , is defined as the column vector with the components , .
is the unit matrix of dimension n.
is the maximal, in modulus, eigenvalue of a matrix K.
- 3.Given a closed interval , we define the vector by setting(2.1)
, : see (10.5).
∂ Ω is the boundary of a domain Ω.
: see Definition 10.1.
The notion of a set 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 and is arbitrary but fixed.
for and , where the vector is regarded as a parameter, the value of which is to be determined later.
Proposition 3.1 ([, Theorem 3.17])
- 1.Sequence (3.4) converges to a limit function(3.7)
- 2.The limit function (3.7) satisfies the p-periodic boundary conditions
- 4.Given an arbitrarily small positive ε, one can choose a number such that the estimate
Recall that, according to (2.2), condition (3.6) means the non-emptiness of the inner -neighbourhood of the set D, where 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 and , . We provide the formulation here for a clearer understanding of the constants appearing in the estimates.
Lemma 3.2 ([, Lemma 3])
for all and .
It should be noted that estimate (3.13) is optimal in the sense that ε can never be put equal to zero.
then in Lemma 3.2 (here, of course, we think of 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).
- 1.Given a , the function 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(3.15)
For any solution of problem (3.2), (3.3) with , there exists a such that .
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 . This topic is discussed in detail, in particular, in , 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 , , and , , , respectively. This is impossible, however, because the boundary conditions on the half-intervals, with trivial exceptions, are never -periodic.
Proposition 4.1 ()
Conversely, if a certain function is a solution of problem (1.3), (1.4), then its restrictions and 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 at is allowed. Note that function (4.6) is always continuous at .
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 ()
Conversely, if a certain is a solution of problem (1.3), (1.4), then the functions and 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.
for all η and λ from .
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., ). One can also derive these formulae directly from Proposition 3.1 by carrying out, respectively, the substitutions , , and , , after which one arrives at parametrised -periodic boundary value problems on the corresponding half-intervals.
It is important to note that all the members of the sequences , , and , , satisfy, respectively, conditions (4.2) and (4.4).
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 and , whence the assertion is obvious for . Similarly, if , then, according to (5.2) and (5.4), we have and and, consequently, relation (5.7) is equivalent to (5.8) for any m. □
Recall that we use notation (2.3). In other words, a couple of vectors belongs to if and only if every convex combination of ξ and η lies in D together with its r-neighbourhood. The inclusion implies, in particular, that and , i.e., the vectors ξ and η both belong to the set defined by formula (2.2). It is also obvious from (6.3) that 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 which are admissible in the sense that with r sufficiently large.
- 1.The uniform, in , limit(6.6)
for all and , where is given by (3.11).
Recall that the constant involved in condition (6.4) is given by equality (2.4), while the vector arising in (6.5) is defined according to (2.1).
is given by formula (3.14). Consequently, inequality (6.11) with holds for an arbitrary value of .
By analogy with Theorem 6.1, under similar conditions, we can establish the uniform convergence of sequence (6.2). Namely, the following statement holds.
- 1.The uniform, in , limit(6.13)
for all and , where 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 provided that with given by formula (3.14).
Theorems 6.1 and 6.3 are improved versions of Theorems 1 and 2 from , 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 , and we omit it.
we arrive immediately at the following statement summarising the last two theorems.
Then, for any , the assertions of Theorems 6.1 and 6.3 hold.
and, consequently, z is a convex combination of and . By virtue of (2.2), (6.24) and (6.27), both vectors and 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 , 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 . The linear mapping , which obviously transforms the space 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., ). 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).
- 1.For ,(7.2)
- 2.For ,(7.3)
Recall that and are functions (4.12), (4.13), and the vectors , are defined according to (2.1). The proof of Lemma 7.1 is almost a literal repetition of that of [, Lemma 7] and uses the estimate obtained in [, Lemma 3].
for . This means that, at every point t from , the value of is a convex combination of ξ and η. Recalling definition (6.3) of the set and using assumption (7.6), we conclude that all the values of the function lie in D, i.e., (7.4) holds with .
for all . It follows from (7.11) that, at every point , the value lies in the -neighbourhood of a convex combination of the vectors ξ and η. Since ξ and η satisfy (7.6) and, by (6.19), , it follows from definition (6.3) of the set that all the values of the function belong to D, i.e., (7.9) holds. Thus, inclusion (7.4) is true for all . Recalling notation (6.1), we arrive immediately to (7.4).
for all . Since , inequality (7.16) implies that all the values of the function belong to the -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 .
The proof of Lemma 7.3 consists in passing to the limit in (7.4) and (7.5) as , 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 and defined according to equalities (6.10) and (6.17) that provide such a characterisation.
we obtain a function , which is well defined for the same values of . This function is obviously continuous.
The following theorem, which is a modified version of [, 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.
- 1.The function defined by (8.1) is a solution of the periodic boundary value problem (1.3), (1.4) if and only if the pair satisfies the system of 2n equations(8.2)
For every solution of problem (1.3), (1.4) with , there exists a pair such that .
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 . 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.
Note that, unlike system (8.2), the m th approximate determining system (9.3) contains only terms involving the functions and and, thus, known explicitly.
which is an ‘approximate’ version of (8.1) well defined for all and .
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 determined by equality (9.4) is continuously differentiable on .
and, therefore, is continuous at . The continuous differentiability of the function 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 and , , defined by (9.1) and (9.2).
hold for any values of and .
which estimate coincides with (9.8). Note that the invertibility of the matrix is guaranteed by condition (9.7).
Consequently, by virtue of relations (9.12) and (9.20), inequality (9.18) leads us directly to the required estimate (9.9). □
where m is fixed. In the formulation of the theorem given below, the following notion is used.
Definition 10.1 ()
holds for all .
and stands for the usual inner product in . The binary relation introduced by Definition 10.1 is a kind of strict inequality for vector functions and its properties are similar to those of the usual strict inequality sign. For example, and imply that . The last named property will be used below in the proof of Theorem 10.2.
We are now able to formulate a statement guaranteeing the solvability of the original periodic problem (1.3), (1.4) based on the information obtained in the course of computation of iterations. In contrast to the unmodified scheme of periodic successive approximations (Proposition 3.1, ), here the iterations are proved to be convergent under the assumption that is twice as weak as in the former case (Theorem 6.5, ). A similar observation can be made concerning the assumption on the domain D (see Corollary 6.7 and the remarks related to conditions (6.30) and (6.31)).
When stating the existence theorem, we restrict our consideration to a slightly weaker version of condition (6.4), where the value is replaced by 0.3, and thus neglect the gap () for ε in estimates (6.11) and (6.18).
Then there exist certain values such that the function is a solution of the periodic boundary value problem (1.3), (1.4).
Recall that the symbol in (10.7) is understood in the sense of Definition 10.1. It should be noted that condition (10.7) involves the values of functions on the boundary of Ω only.
and therefore, in view of (10.6), we conclude that . Consequently, there exist vectors and possessing the properties indicated, and it only remains to refer to Theorem 8.1. The theorem is proved. □
for , which, as is easy to see from (9.4), follow directly from Theorem 6.5. A uniform inequality, not given here, can be obtained by estimating the mapping for any fixed .
It is worth to emphasise the role of the unknown parameters whose values appearing in (11.1) are determined from equations (9.3): is an approximation of the initial value of the periodic solution and is that of its value at .
As regards the practical application of Theorem 10.2, it should be noted that, according to (10.2), the mapping is known in an analytic form because it is determined solely by the m th iteration, which is already constructed at the moment. Of course, the degree in (10.6) is the Brouwer degree because all the vector fields are finite-dimensional. Likewise, all the terms in the right-hand side of inequality (10.7) are computed explicitly (e.g., by using computer algebra systems).
is a solution of problem (12.1), (12.2). This solution has values in the domain , where, as one can verify, the convergence condition (3.5) is not satisfied. However, the corresponding condition with the doubled constant (3.18) does hold, and therefore, the interval halving technique can be used.
Approximate values of parameters at several steps of iteration for problem ( 12.1 ), ( 12.2 ). The last row corresponds to the exact solution ( 12.3 )
Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.
The work supported in part by RVO: 67985840 (A. Rontó). This research was carried out as part of the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project with support from the European Union, co-financed by the European Social Fund (M. Rontó).
- Rontó A, Rontó M: Periodic successive approximations and interval halving. Miskolc Math. Notes 2012, 13(2):459-482.MATHMathSciNetGoogle Scholar
- Nirenberg L: Topics in Nonlinear Functional Analysis. Courant Institute of Mathematical Sciences New York University, New York; 1974. (With a chapter by E. Zehnder, Notes by R. A. Artino, Lecture Notes, 1973-1974)MATHGoogle Scholar
- Gaines RE, Mawhin JL Lecture Notes in Mathematics 568. In Coincidence Degree, and Nonlinear Differential Equations. Springer, Berlin; 1977.Google Scholar
- Cesari L Ergebnisse der Mathematik und Ihrer Grenzgebiete, N. F. 16. In Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. 2nd edition. Academic Press, New York; 1963.View ArticleGoogle Scholar
- Hale JK: Oscillations in Nonlinear Systems. McGraw-Hill, New York; 1963.MATHGoogle Scholar
- Samoilenko AM: A numerical-analytic method for investigation of periodic systems of ordinary differential equations. I. Ukr. Math. J. 1965, 17(4):82-93. 10.1007/BF02526569View ArticleGoogle Scholar
- Samoilenko AM: A numerical-analytic method for investigation of periodic systems of ordinary differential equations. II. Ukr. Math. J. 1966, 18(2):50-59. 10.1007/BF02537778View ArticleGoogle Scholar
- Samoilenko AM: On a sequence of polynomials and the radius of convergence of its Abel-Poisson sum. Ukr. Math. J. 2003, 55(7):1119-1130. doi:10.1023/B:UKMA.0000010610.69570.13View ArticleGoogle Scholar
- Samoilenko AM, Ronto NI: Numerical-Analytic Methods of Investigating Periodic Solutions. Mir, Moscow; 1979. (With a foreword by Yu. A. Mitropolskii)Google Scholar
- Samoilenko AM, Ronto NI: Numerical-Analytic Methods of Investigation of Boundary-Value Problems. Naukova Dumka, Kiev; 1986. (In Russian, with an English summary, edited and with a preface by Yu. A. Mitropolskii)Google Scholar
- Samoilenko AM, Ronto NI: Numerical-Analytic Methods in the Theory of Boundary-Value Problems for Ordinary Differential Equations. Naukova Dumka, Kiev; 1992. (In Russian, edited and with a preface by Yu. A. Mitropolskii)Google Scholar
- Rontó A, Rontó M: Successive approximation techniques in non-linear boundary value problems for ordinary differential equations. Handb. Differ. Equ. In Handbook of Differential Equations: Ordinary Differential Equations. Vol. IV. Elsevier/North-Holland, Amsterdam; 2008:441-592.Google Scholar
- Rontó A, Rontó M: Successive approximation method for some linear boundary value problems for differential equations with a special type of argument deviation. Miskolc Math. Notes 2009, 10: 69-95.MATHMathSciNetGoogle Scholar
- Rontó A, Rontó M: On a Cauchy-Nicoletti type three-point boundary value problem for linear differential equations with argument deviations. Miskolc Math. Notes 2009, 10(2):173-205.MATHMathSciNetGoogle Scholar
- Rontó A, Rontó M: On nonseparated three-point boundary value problems for linear functional differential equations. Abstr. Appl. Anal. 2011., 2011: Article ID 326052. doi:10.1155/2011/326052Google Scholar
- Ronto A, Rontó M: A note on the numerical-analytic method for nonlinear two-point boundary-value problems. Nonlinear Oscil. 2001, 4: 112-128.MATHMathSciNetGoogle Scholar
- Rontó A, Rontó M: On some symmetric properties of periodic solutions. Nonlinear Oscil. 2003, 6: 82-107. doi:10.1023/A:1024827821289 10.1023/A:1024827821289MathSciNetView ArticleGoogle Scholar
- Rontó M, Shchobak N: On the numerical-analytic investigation of parametrized problems with nonlinear boundary conditions. Nonlinear Oscil. 2003, 6(4):469-496. doi:10.1023/B:NONO.0000028586.11256.d7MathSciNetView ArticleGoogle Scholar
- Rontó M, Shchobak N: On parametrization for a non-linear boundary value problem with separated conditions. Electron. J. Qual. Theory Differ. Equ. 2007, 18: 1-16.View ArticleGoogle Scholar
- Ronto AN, Ronto M, Shchobak NM: On the parametrization of three-point nonlinear boundary value problems. Nonlinear Oscil. 2004, 7(3):384-402. 10.1007/s11072-005-0019-5MathSciNetView ArticleGoogle Scholar
- Ronto AN, Rontó M, Samoilenko AM, Trofimchuk SI: On periodic solutions of autonomous difference equations. Georgian Math. J. 2001, 8: 135-164.MATHMathSciNetGoogle Scholar
- Rontó M, Mészáros J: Some remarks on the convergence of the numerical-analytical method of successive approximations. Ukr. Math. J. 1996, 48: 101-107. doi:10.1007/BF02390987 10.1007/BF02390987MATHView ArticleGoogle Scholar
- Rontó M, Samoilenko AM: Numerical-Analytic Methods in the Theory of Boundary-Value Problems. World Scientific, River Edge; 2000. (With a preface by Yu. A. Mitropolsky and an appendix by the authors and S. I. Trofimchuk)MATHView ArticleGoogle Scholar
- Rontó A, Rontó M: Existence results for three-point boundary value problems for systems of linear functional differential equations. Carpath. J. Math. 2012, 28: 163-182.MATHGoogle Scholar
- Kwapisz M: On modifications of the integral equation of Samoilenko’s numerical-analytic method of solving boundary value problems. Math. Nachr. 1992, 157: 125-135.MATHMathSciNetGoogle Scholar
- Kwapisz M: On modification of Samoilenko’s numerical-analytic method of solving boundary value problems for difference equations. Appl. Math. 1993, 38(2):133-144.MATHMathSciNetGoogle Scholar
- Rontó A, Rontó M, Holubová G, Nečesal P: Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions. Bound. Value Probl. 2011., 2011: Article ID 58. doi:10.1186/1687-2770-2011-58Google Scholar
- Mawhin J CBMS Regional Conference Series in Mathematics 40. In Topological Degree Methods in Nonlinear Boundary Value Problems. Am. Math. Soc., Providence; 1979. (Expository lectures from the CBMS Regional Conference held at Harvey Mudd College, Claremont, Calif., June 9-15, 1977)Google Scholar
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