Collectanea mathematica

, Volume 61, Issue 2, pp 223–239 | Cite as

A fixed point theorem in locally convex spaces



For a locally convex space Open image in new window with the topology given by a family {p(┬; α)} α ∈ ω of seminorms, we study the existence and uniqueness of fixed points for a mapping Open image in new window defined on some set Open image in new window . We require that there exists a linear and positive operatorK, acting on functions defined on the index set Ω, such that for everyu, Open image in new window
Under some additional assumptions, one of which is the existence of a fixed point for the operator Open image in new window , we prove that there exists a fixed point of Open image in new window . For a class of elements satisfyingK n (p)u;┬))(α) → 0 asn → ∞, we show that fixed points are unique. This class includes, in particular, the class for which we prove the existence of fixed points.

We consider several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms.


Fixed point theorem Locally convex spaces Ordinary differential equations Pseudodifferential operators 


47H10 46N20 47G30 34A34 


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  1. 1.
    J. Dugundji and A. Granas,Fixed Point Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.MATHGoogle Scholar
  2. 2.
    M.S.P. Eastham,The Asymptotic Solution of Linear Differential Systems, London Mathematical Society Monographs4, The Clarendon Press, Oxford University Press, New York, 1989.MATHGoogle Scholar
  3. 3.
    E. Hille and R. Phillips,Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications31, Providence, R.I., 1957.Google Scholar
  4. 4.
    V. Kozlov and V. Maz’ya,Theory of a Higher-Order Sturm-Liouville Equation, Lecture Notes in Mathematics1659, Springer-Verlag, Berlin, 1997.MATHGoogle Scholar
  5. 5.
    V. Kozlov and V. Maz’ya,Differential Operators and Spectral Theory, Amer. Math. Soc. Transl.2, Providence, R.I., 1999.Google Scholar
  6. 6.
    V. Kozlov and V. Maz’ya, An asymptotic theory of higher-order operator differential equations with nonsmooth nonlinearites,J. Funct. Anal. 217 (2004), 448–488.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    V. Kozlov, J. Thim, and B.O. Turesson, Riesz potential equations in localL p-spaces,Complex Var. Elliptic Equ. 54 (2009), 125–151.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J.L. Lions and E. Magenes,Non-homogeneous Boundary Value Problems and Applications II, Springer-Verlag, New York-Heidelberg, 1972.MATHGoogle Scholar
  9. 9.
    E.M. Stein,Harmonic Analysis: RealVariable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical43, Princeton University Press, Princeton, N.J., 1993.Google Scholar
  10. 10.
    W. Wasow,Asymptotic Expansions for Ordinary Differential Equations, Pure and Applied Mathematics14, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965.MATHGoogle Scholar
  11. 11.
    E. Zeidler,Applied Functional Analysis, Applied Mathematical Sciences108, Springer-Verlag, New York, 1995.Google Scholar

Copyright information

© Universitat de Barcelona 2010

Authors and Affiliations

  • Vladimir Kozlov
    • 1
  • Johan Thim
    • 1
  • Bengt Ove Turesson
    • 1
  1. 1.Department of MathematicsLinköping UniversityLinköpingSweden

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