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Collectanea mathematica

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

A fixed point theorem in locally convex spaces

  • Vladimir Kozlov
  • Johan Thim
  • Bengt Ove Turesson
Article

Abstract

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.

Keywords

Fixed point theorem Locally convex spaces Ordinary differential equations Pseudodifferential operators 

MSC2000

47H10 46N20 47G30 34A34 

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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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