On the topological structure of fixed point sets for abstract Volterra operators in Fréchet spaces (Q2708290)

An operator \(F:C( [0,\infty),\mathbb{R}^{n}) \to C( [0,\infty),\mathbb{R}^{n}) \) is called an abstract Volterra operator if for each \(\varepsilon>0,\) from \(x,y\in C( [0,\infty),\mathbb{R} ^{n}) ,\) \(x(t)=y(t)\) on \([ 0,\varepsilon] \) it follows that \(F(x)(t)=F(y)(t)\) on \([ 0,\varepsilon] .\) NEWLINENEWLINENEWLINEA basic result is: For any abstract Volterra operator which is continuous, compact and satisfies \(F(x)(0)=u_{0}\) for all \(x\in C( [0,\infty),\mathbb{R}^{n}) ,\) the set \(\text{Fix}(F)\) is an \(R_{\delta}\) set (it is homeomorphic with a decreasing sequence of compact absolute retracts). Similar results are given for abstract Volterra operators on \(L_{\text{loc}}^{p}( [0,\infty),\mathbb{R}^{n}) ,\) \(1<p<\infty.\) The last section contains two applications to integral equations.