On some measures of noncompactness in the Fréchet spaces of continuous functions
The paper is devoted to construction of new regular measures of noncompactness for Fréchet spaces of continuous functions on a bounded or an unbounded interval. For those spaces several properties of these new measures are established. The results obtained are further applied to infinite systems of functional integral equations of the form \[ x^i(t)=f_i\left( t,x^i(t),\int_{0}^{t}u_i(t,\tau,x^1(\tau),x^2(\tau),\dots)d\tau \right) \] where \( i=1,2,\dots \) and \(t\in \mathbb{R}_+\). It is proved that under some assumptions this infinite system has at least one solution \(x(t)=(x^i(t))\) such that \(x(t)\) belongs to Fréchet space \(C(\mathbb{R}_+,s)\).
scientific article; zbMATH DE number 5612944