Integrodifferential equations on time scales with Henstock-Kurzweil-Pettis delta integrals
Summary: We prove existence theorems for integro-differential equations \[ x^\Delta(t)=f(t,x(t),\int^t_0k(t,s,x(s))\,\Delta s),\quad x(0)=x_0,\;t\in I_a=[0,a]\cap T,\;a\in\mathbb R_+, \] where \(T\) denotes a time scale (nonempty closed subset of real numbers \(\mathbb R\)), and \(I_a\) is a time scale interval. The functions \(f\), \(k\) are weakly-weakly sequentially continuous with values in a Banach space \(E\), and the integral is taken in the sense of Henstock-Kurzweil-Pettis delta integral. This integral generalizes the Henstock-Kurzweil delta integral and the Pettis integral. Additionally, the functions \(f\) and \(k\) satisfy some boundary conditions and conditions expressed in terms of measures of weak noncompactness. Moreover, we prove Ambrosetti's lemma.
scientific article; zbMATH DE number 5791637
scientific article