A fixed point theorem in locally convex spaces

Abstract

For a locally convex space $\X$ with the topology given by a family $\{\pa{\cdot}\}_{\alpha \in \I}$ of seminorms, we study the existence and uniqueness of fixed points for a mapping $\K \colon \DK \rightarrow \DK$ defined on some set $\DK \subset \X$. We require that there exists a linear and positive operator $\KK$, acting on functions defined on the index set $\I$, such that for every $u,v \in \DK$ \[ \pa{\K(u) - \K(v)} \leq \KK \apa{u-v}(\alpha) , \quad \alpha \in \I. \] Under some additional assumptions, one of which is the existence of a fixed point for the operator $\KK + \pa[\cdot]{\K(0)}$, we prove that there exists a fixed point of $\K$. For a class of elements satisfying $\KK^{n} \apa{u}(\alpha) \rightarrow 0$ as $n\rightarrow \infty$, 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 pseudo-differential equations with nonlinear terms.