A fixed point theorem in locally convex spaces

For a locally convex space[inline-graphic not available: see fulltext] 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[inline-graphic not available: see fulltext] defined on some set[inline-graphic not available: see fulltext]. We require that there exists a linear and positive operatorK, acting on functions defined on the index set Ω, such that for everyu,[inline-graphic not available: see fulltext][graphic not available: see fulltext] Under some additional assumptions, one of which is the existence of a fixed point for the operator[inline-graphic not available: see fulltext], we prove that there exists a fixed point of[inline-graphic not available: see fulltext]. For a class of elements satisfyingKn(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.