Abstract
For a locally convex space
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
defined on some set
. We require that there exists a linear and positive operatorK, acting on functions defined on the index set Ω, such that for everyu,
Under some additional assumptions, one of which is the existence of a fixed point for the operator
, we prove that there exists a fixed point of
. 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.
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Kozlov, V., Thim, J. & Turesson, B.O. A fixed point theorem in locally convex spaces. Collect. Math. 61, 223–239 (2010). https://doi.org/10.1007/BF03191243
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DOI: https://doi.org/10.1007/BF03191243