Weakly singular functional-integral equation in infinite dimensional Banach spaces.
The author is investigating functional-integral equations of the form \[ y(t)= \phi\Biggl(t, \lambda\int_\Omega k(t,s) g(s,y(s))\,ds\Biggr),\tag{1} \] with \(t\) a.e. in \(\mathbb{R}^n\), where \(\Omega\) stands for a compact set of \(\mathbb{R}^n\). The values of \(y\) belong to a Banach space \(E\). The kernel \(k(t,s)\) has the form \(k(t,s)= h(t,s)| t-s|^{-\alpha}\), \(0< \alpha< n\), \(t\neq s\). The solution is sought in the space \(L^p(\Omega, E)\), \(p> 1\). Under suitable conditions on data, among them the boundedness and strong measurability of \(h\) on \(\Omega\), one proves the existence of at least one \(L^p\)-solution to the equation (1). Kuratowski's measure of noncompactness and a fixed point theorem due to Mönch are the main tools in carrying out the proof.
scientific article; zbMATH DE number 1861375