Fixed points of cone compression and expansion multimaps defined on Fréchet spaces: the projective limit approach
This article presents some modifications of the well-known Krasnosel'skij and Petryshyn fixed point theorems for multi-compressions and multi-expansions of a cone in a Banach space for the case of a cone in a Fréchet space \(E\). More exactly, it is assumed that \(E\) is a Fréchet space \(E = (E,\{| \cdot| _n\}_{n \in {\mathbb N}}\)) with the topology generated by a family of seminorms \(\{| \cdot| _n\}_{n \in {\mathbb N}}\) satisfying conditions \[ | x| _1 \leq | x| _2 \leq \dots \leq | x| _n \leq \ldots \quad \text{for every} \quad x \in E\tag{1} \] and for each \(n \in {\mathbb N}\), there exists a Banach space \((E_n,| \cdot| _n)\) and an isomorphism (between normed spaces) \(j_n: {\mathbf E}_n \to E_n\) (\({\mathbf E}_n\) is the completion of the quotient space \(E / \sim_n\), \(x \sim_n y\) iff \(| x - y| _n = 0\)). The assumption of contraction for \(F\) is used in the form \[ | y| _n \geq | x| _n \quad \text{for all} \quad y \in F_nx \quad \text{for small} \quad x, \] \[ | y| _n \leq | x| _n \quad \text{for all} \quad y \in F_nx \quad \text{for large} \quad x, \] where \(F_n\) are upper semicontinuous maps that approximate \(F\) in a suitable sense, and in a similar manner for the assumption of expansion. The main result describes conditions under which contraction or expansion from the viewpoint has a fixed point. As application, the nonlinear integral equation \[ y(t) = \int_0^\infty k(t,s)f(s,y(s)) \, ds \quad \text{for} \quad t \in [0,\infty) \] is considered.
scientific article; zbMATH DE number 5189227