Skip to main content
SpringerLink
Log in

Periodic Solutions of Non-autonomous Ordinary Differential Equations

  • Reference work entry
Mathematics of Complexity and Dynamical Systems

  • 517 Accesses

Article Outline

Glossary

Definition of the Subject

Introduction

Poincaré Operator and Linear Systems

Boundedness and Periodicity

Fixed Point Approach: Perturbation Theory

Fixed Point Approach: Large Nonlinearities

Guiding Functions

Lower and Upper Solutions

Direct Method of the Calculus of Variations

Critical Point Theory

Future Directions

Bibliography

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 600.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

Banach fixed point theorem:

If M is a complete metric space with distance d, and \( f \colon M \to M \) is contractive, i. e. \( d(f(u),f(v)) \leq \alpha d(u,v) \) for some \( \alpha \in [0,1) \) and all \( u,v \in M \), then f has a unique fixed point \( u^{\ast}\) and \( u^{\ast} = \lim_{k \to \infty} f^k(u_{0}) \) for any \( u_{0} \in M \).

Brouwer degree:

An integer \( d_{B}[f,\Omega] \) which ‘algebraically’ counts the number of zeros of any continuous mapping \( f \colon \overline \Omega \subset {\mathbb R}^n \to {\mathbb R}^n \) such that \( 0 \not \in f(\partial \Omega) \), and is invariant for sufficiently small perturbations of f. If f is of class \( C^1 \) and its zeros are non degenerate, then \( d_{B}[f,\Omega] = \sum_{x \in f^{-1}(0)} {\rm sign}\;{\rm det} f^{\prime}(x) \).

Brouwer fixed point theorem:

Any continuous mapping \( f \colon B \to B \), with B is homeomorphic to the closed unit ball in \({\mathbb R}^n \), has at least one fixed point.

Leray–Schauder degree:

The extension \( d_{LS}[I - g,\Omega] \) of the Brouwer degree, where Ω is an open bounded subset of the Banach space X, and \( g \colon \overline \Omega \to X \) is continuous, \( g(\overline \Omega) \) is relatively compact and \( 0 \not \in (I-g)(\partial \Omega) \).

Leray–Schauder–Schaefer fixed point theorem:

If X is a Banach space, \( g \colon X \to X \) is a continuous mapping taking bounded subsets into relatively compact ones, and if the set of possible fixed points of \( \varepsilon g \) \( (\varepsilon \in [0,1]) \) is bounded independently of ε, then g has at least one fixed point.

Ljusternik–Schnirelmann category:

The Ljusternik–Schnirelmann category \( \operatorname{cat}(M) \) of a metric space M into itself is the smallest integer k such that M can be covered by k sets contractible in M.

Lower and upper solutions:

A lower (resp. upper) solution of the periodic problem \( u^{\prime\prime} = f(t,u) \), \( u(0) = u(T) \), \( u^{\prime}(0) = u^{\prime}(T) \) is a function α (resp. β) of class \( C^2 \) such that \( \alpha^{\prime\prime}(t) \geq f(t,\alpha(t))\), \( \alpha(0) = \alpha(T)\), \( \alpha^{\prime}(0) \geq\alpha^{\prime}(T) \) (resp. \(\beta^{\prime\prime}(t) \leq f(t,\beta(t)), \beta(0) = \beta(T), \beta^{\prime}(0) \leq \beta^{\prime}(T)\).

Palais–Smale condition for a \( C^1 \) function \( \varphi \colon X \to \mathbb R \) :

Any sequence \( (u_{k})_{k \in \mathbb N}\) such that \( (\varphi(u_{k}))_{k \in \mathbb N}\) is bounded and \( \lim_{k \to \infty}\varphi^{\prime}(u_{k}) = 0 \) contains a convergent subsequence.

Poincaré operator :

The mapping defined in \({\mathbb R}^n \) by \( P_{T} \colon y \mapsto p(T;y) \), where \( p(t;y) \) is the unique solution of the Cauchy problem \( x^{\prime} = f(t,x),\; x(0)=y \).

Schauder fixed point theorem:

If C is a closed bounded convex subset of a Banach spaxe X, any continuous mapping \( g \colon C \to C \) such that \( g(C) \) is relatively compact has at least one fixed point.

Sobolev inequality:

For any function \( u \in L^2(0,T) \) such that \( u^{\prime} \in L^2(0,T) \) and \( \int_{0}^T u(t)\text{d} t = 0 \), one has \( \max_{t \in [0,T]} |u(t)| \leq (T^{1/2}/2\sqrt 3)[\int_{0}^T |u^{\prime}(t)|^2\text{d} t ]^{1/2}\).

Wirtinger inequality:

For any function \( u \in L^2(0,T) \) such that \( u^{\prime} \in L^2(0,T) \) and \( \int_{0}^T u(t)\text{d} t = 0 \), one has \( \int_{0}^T |u(t)|^2\text{d} t \leq \left(T^2\left/4\pi^2\right.\right)\int_{0}^T |u^{\prime}(t)|^2\text{d} t \).

Bibliography

Primary Literature

  1. Ahmad S, Lazer AC, Paul JL (1976), Elementary critical point theory and perturbations of elliptic boundary value problems. Indiana Univ Math J 25:933–944

    Article  MathSciNet  MATH  Google Scholar 

  2. Alonso JM, Ortega R (1996) Unbounded solutions of semilinear equations at resonance. Nonlinearity 9:1099–1111

    Article  MathSciNet  MATH  Google Scholar 

  3. Amann H, Ambrosetti A, Mancini G (1978) Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities. Math Z 158:179–194

    Article  MathSciNet  MATH  Google Scholar 

  4. Ambrosetti A, Prodi G (1972) On the inversion of some differentiable mappings with singularities between Banach spaces. Ann Mat Pura Appl 93(4):231–246

    Article  MathSciNet  MATH  Google Scholar 

  5. Bartsch T, Mawhin J (1991) The Leray–Schauder degree of S 1-equivariant operators associated to autonomous neutral equations in spaces of periodic functions. J Differ Equations 92:90–99

    Article  MathSciNet  MATH  Google Scholar 

  6. Campos J, Ortega R (1996) Nonexistence of periodic solutions of a complex Riccati equation. Differ Integral Equations 9:247–249

    MathSciNet  MATH  Google Scholar 

  7. Capietto A, Mawhin J, Zanolin F (1992) Continuation theorems for periodic perturbations of autonomous systems. Trans Amer Math Soc 329:41–72

    Article  MathSciNet  MATH  Google Scholar 

  8. Capietto A, Mawhin J, Zanolin F (1995) A continuation theorem for periodic boundary value problems with oscillatory nonlinearities. Nonlinear Differ Equations Appl 2:133–163

    Article  MathSciNet  MATH  Google Scholar 

  9. Chang KC (1989) On the periodic nonlinearity and the multiplicity of solutions. Nonlinear Anal 13:527–537

    Article  MathSciNet  MATH  Google Scholar 

  10. Chang KC (1993) Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Basel

    Book  MATH  Google Scholar 

  11. Conley C, Zehnder E (1984) Morse type index for flows and periodic solutions for Hamiltonian operator. Comm Pure Appl Math 37:207–253

    Article  MathSciNet  MATH  Google Scholar 

  12. Dancer EN (1982) On the use of asymptotics in nonlinear boundary value problems. Ann Mat Pura Appl 131(4):67–187

    MathSciNet  Google Scholar 

  13. Dancer EN, Ortega R (1994) The index of Lyapunov stable fixed points in two dimensions. J Dyn Differ Equations 6:631–637

    Article  MathSciNet  MATH  Google Scholar 

  14. Dolph CL (1949) Nonlinear integral equations of Hammerstein type. Trans Amer Math Soc 66:289–307

    Article  MathSciNet  MATH  Google Scholar 

  15. Duffing G (1918) Erzwungene Schwingungen bei veränderlicher Eigenfrequenz. Vieweg, Braunschweig

    MATH  Google Scholar 

  16. Fabry C, Mawhin J, Nkashama M (1986) A multiplicity result for periodic solutions of forced nonlinear second order differential equations. Bull London Math Soc 18:173–180

    Article  MathSciNet  MATH  Google Scholar 

  17. Fabry C, Mawhin J (2000) Oscillations of a forced asymmetric oscillator at resonance. Nonlinearity 13:493–505

    Article  MathSciNet  MATH  Google Scholar 

  18. Fučik S (1976) Boundary value problems with jumping nonlinearities. Časopis Pest Mat 101:69–87

    Google Scholar 

  19. Hamel G (1922) Ueber erzwungene Schwingungen bei endlichen Amplituden. Math Ann 86:1–13

    Article  MathSciNet  MATH  Google Scholar 

  20. Kazdan JL, Warner FW (1975) Remarks on some quasilinear elliptic equations. Comm Pure Appl Math 28:567–597

    Article  MathSciNet  MATH  Google Scholar 

  21. Knobloch HW (1963) Eine neue Methode zur Approximation periodischer Lösungen von Differentialgleichungen zweiter Ordnung. Math Z 82:177–197

    Article  MathSciNet  MATH  Google Scholar 

  22. Krasnosel'skii MA (1968) The operator of translation along the trajectories of differential equations. Amer Math Soc, Providence

    Google Scholar 

  23. Lazer AC, Leach DE (1969) Bounded perturbations for forced harmonic oscillators at resonance. Ann Mat Pura Appl 82(4):49–68

    Article  MathSciNet  MATH  Google Scholar 

  24. Lefschetz S (1943) Existence of periodic solutions of certain differential equations. Proc Nat Acad Sci USA 29:29–32

    Article  MathSciNet  MATH  Google Scholar 

  25. Levinson N (1943) On the existence of periodic solutions for second order differential equations with a forcing term. J Math Phys 22:41–48

    MathSciNet  MATH  Google Scholar 

  26. Lichtenstein L (1915) Ueber einige Existenzprobleme der Variationsrechnung. J Reine Angew Math 145:24–85

    Google Scholar 

  27. Ljusternik L, Schnirelmann L (1934) Méthodes topologiques dans les problèmes variationnels. Hermann, Paris

    Google Scholar 

  28. Lloyd NG (1975) On a class of differential equations of Riccati type. J London Math Soc 10(2):1–10

    Article  MathSciNet  MATH  Google Scholar 

  29. Massera JL (1950) The existence of periodic solutions of systems of differential equations. Duke Math J 17:457–475

    Article  MathSciNet  MATH  Google Scholar 

  30. Mawhin J (1969) Équations intégrales et solutions périodiques de systèmes différentiels non linéaires. Bull Cl Sci Acad Roy Belgique 55(5):934–947

    MathSciNet  MATH  Google Scholar 

  31. Mawhin J (1976) Contractive mappings and periodically perturbed conservative systems. Arch Math (Brno) 12:67–73

    MathSciNet  MATH  Google Scholar 

  32. Mawhin J (1979) Topological Degree and Nonlinear Boundary Value Problems. CBMS Conf Math No 40. Amer Math Soc, Providence

    Google Scholar 

  33. Mawhin J (1983) Points fixes, points critiques et problèmes aux limites. Sémin Math Supérieures No 92, Presses Univ de Montréal, Montréal

    Google Scholar 

  34. Mawhin J (1989) Forced second order conservative systems with periodic nonlinearity. Ann Inst Henri Poincaré Anal non linéaire 6:415–434

    Google Scholar 

  35. Mawhin J (1994) Periodic solutions of some planar non-autonomous polynomial differential equations. Differ Integral Equations 7:1055–1061

    MathSciNet  MATH  Google Scholar 

  36. Mawhin J (2004) Global results for the forced pendulum equation. In: Cañada A, Drabek P, Fonda A (eds) Handbook of Differential Equations. Ordinary Differential Equations, vol 1. Elsevier, Amsterdam, pp 533–590

    Chapter  Google Scholar 

  37. Mawhin J, Ward JR (2002) Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discret Continuous Dyn Syst 8:39–54

    MathSciNet  MATH  Google Scholar 

  38. Mawhin J, Willem M (1984) Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J Differ Equations 52:264–287

    Article  MathSciNet  MATH  Google Scholar 

  39. Mawhin J, Willem M (1989) Critical point theory and Hamiltonian systems. Springer, New York

    MATH  Google Scholar 

  40. Ortega R (1995) Some applications of the topological degree to stability theory. In: Granas A, Frigon M (eds) Topological methods in differential equations and inclusions, NATO ASI C472. Kluwer, Amsterdam, pp 377–409

    Chapter  Google Scholar 

  41. Palais R (1966) Ljusternik–Schnirelmann theory on Banach manifolds. Topology 5:115–132

    Article  MathSciNet  MATH  Google Scholar 

  42. Palais R, Smale S (1964) A generalized Morse theory. Bull Amer Math Soc 70:165–171

    Article  MathSciNet  MATH  Google Scholar 

  43. Poincaré H (1892) Les méthodes nouvelles de la mécanique céleste, vol 1. Gauthier–Villars, Paris

    Google Scholar 

  44. Rabinowitz P (1978) Some minimax theorems and applications to nonlinear partial differential equations. In: Cesari L, Kannan R, Weinberger H (eds) Nonlinear Analysis, A tribute to E. Rothe. Academic Press, New York

    Google Scholar 

  45. Rabinowitz P (1988) On a class of functionals invariant under a \( \mathbb Z^n \)-action. Trans Amer Math Soc 310:303–311

    MathSciNet  MATH  Google Scholar 

  46. Reissig R (1964) Ein funktionenanalytischer Existenzbeweis für periodische Lösungen. Monatsber Deutsche Akad Wiss Berlin 6:407–413

    MathSciNet  MATH  Google Scholar 

  47. Scorza Dragoni G (1931) Il problema dei valori ai limite studiate in grande per le equazione differenziale del secondo ordine. Math Ann 105:133–143

    Article  MathSciNet  Google Scholar 

  48. Srzednicki R (1994) On periodic planar solutions of planar polynomial differential equations with periodic coefficients. J Differ Equations 114:77–100

    Article  MathSciNet  MATH  Google Scholar 

  49. Stoppelli F (1952) Su un'equazione differenziale della meccanica dei fili. Rend Accad Sci Fis Mat Napoli 19(4):109–114

    MathSciNet  Google Scholar 

  50. Villari G (1965) Contributi allo studio dell'esistenzadi soluzioni periodiche per i sistemi di equazioni differenziali ordinarie. Ann Mat Pura Appl 69(4):171–190

    Article  MathSciNet  MATH  Google Scholar 

  51. Willem M (1981) Oscillations forcées de systèmes hamiltoniens. Publications du Séminaire d'Analyse non linéaire, Univ Besançon

    Google Scholar 

Books and Reviews

  1. Bobylev NA, Burman YM, Korovin SK (1994) Approximation procedures in nonlinear oscillation theory. de Gruyter, Berlin

    Book  MATH  Google Scholar 

  2. Bogoliubov NN, Mitropolsky YA (1961) Asymptotic methods in the theory of nonlinear oscillations. Gordon and Breach, New York

    Google Scholar 

  3. Cesari L (1971) Asymptotic behavior and stability problems for ordinary differential equations, 3rd edn. Springer, Berlin

    Book  Google Scholar 

  4. Coddington EA, Levinson N (1955) Ordinary differential equations. McGraw-Hill, New York

    MATH  Google Scholar 

  5. Cronin J (1964) Fixed points and topological degree in nonlinear analysis. Amer Math Soc, Providence

    MATH  Google Scholar 

  6. Ekeland I (1990) Convexity methods in hamiltonian mechanics. Springer, Berlin

    Book  MATH  Google Scholar 

  7. Farkas M (1994) Periodic motions. Springer, New York

    Book  MATH  Google Scholar 

  8. Gaines RE, Mawhin J (1977) Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics, vol 568. Springer, Berlin

    Google Scholar 

  9. Hale JK (1969) Ordinary differential equations. Wiley Interscience, New York

    MATH  Google Scholar 

  10. Mawhin J (1993) Topological degree and boundary value problems for nonlinear differential equations. Lecture Notes in Mathematics, vol 1537. Springer, Berlin, pp 74–142

    Google Scholar 

  11. Mawhin J (1995) Continuation theorems and periodic solutions of ordinary differential equations. In: Granas A, Frigon M (eds) Topological methods in differential equations and inclusions, NATO ASI C472. Kluwer, Amsterdam, pp 291–375

    Chapter  Google Scholar 

  12. Pliss VA (1966) Nonlocal problems of the theory of oscillations. Academic Press, New York

    MATH  Google Scholar 

  13. Rabinowitz P (1986) Minimax methods in critical point theory with applications to differential equations. CBMS Reg Conf Ser Math No 65. Amer Math Soc, Providence

    Google Scholar 

  14. Reissig R, Sansone G, Conti R (1963) Qualitative Theorie nichtlinearer Differentialgleichungen. Cremonese, Roma

    Google Scholar 

  15. Reissig R, Sansone G, Conti R (1974) Nonlinear differential equations of higher order. Noordhoff, Leiden

    Google Scholar 

  16. Rouche N, Mawhin J (1980) Ordinary differential equations. Stability and periodic solutions. Pitman, Boston

    MATH  Google Scholar 

  17. Sansone G, Conti R (1964) Non-linear differential equations. Pergamon, Oxford

    MATH  Google Scholar 

  18. Verhulst F (1996) Nonlinear differential equations and dynamical systems, 2nd edn. Springer, Berlin

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématique, Université Catholique de Louvain, Maryland, USA

    Jean Mawhin

Authors
  1. Jean Mawhin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Mawhin, J. (2012). Periodic Solutions of Non-autonomous Ordinary Differential Equations. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_75

Download citation

Publish with us

Policies and ethics

Search

Navigation