A fixed point theorem in locally convex spaces
In this interesting paper, the authors study the existence and uniqueness of fixed points for a mapping \(\mathcal K : \mathcal D_{\mathcal K} \rightarrow \mathcal D_{\mathcal K}\) defined on some set \(\mathcal D_{\mathcal K} \subset \mathcal X\), where \(\mathcal X\) is a locally convex space with the topology given by a family \(\{p(\cdot;\alpha)\}_{\alpha \in \Omega}\) of seminorms. By assuming that there exists a linear and positive operator \(K\), acting on functions defined on the index set \(\Omega\), such that for every \(u,v \in \mathcal D_{\mathcal K}\), \[ p( {\mathcal K}(u) - {\mathcal K }(v);\alpha ) \leq K (p(u-v;\cdot))(\alpha) , \quad \alpha \in \Omega, \] and some additional assumptions including the existence of a fixed point for the operator \(K + p({\mathcal K}(0);\cdot)\), they prove that there exists a fixed point of \(\mathcal K\). For a class of elements satisfying \(K^{n} (p(u;\cdot))(\alpha) \rightarrow 0\) as \(n\rightarrow \infty\), including, in particular, the class for which they prove the existence of fixed points, the authors show that fixed points are unique. In addition, the authors present several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. They also consider pseudodifferential equations with nonlinear terms.
scientific article; zbMATH DE number 5711740
scientific article