Integro-differential equations on time scales with Henstock-Kurzweil delta integrals (Q2906252)

The article is concerned with the solvability of integro-differential equations on time scales, of the form NEWLINE\[NEWLINEx^\Delta(t)=f\left(t,x(t),\int_0^t k(t,s,x(s))\Delta s\right),\;\;\;t\in[0,a]\cap{\mathbb T},\tag{1}NEWLINE\]NEWLINE where \(\mathbb T\) is a given time scale such that \(0\in\mathbb T\), and both \(f\) and \(x\) take values in a Banach space.NEWLINENEWLINEA solution is assumed to be a uniformly generalized absolutely continuous in the restricted sense (\(ACG_*\)) function for which the equality in (1) holds \(\mu_\Delta\)-almost everywhere in \([0,a]\cap{\mathbb T}\). Under these conditions, equation (1) can be written in the equivalent integral form NEWLINE\[NEWLINEx(t)=x(0)+\int_0^t f\left(z,x(z),\int_0^z k(z,s,x(s))\Delta s\right)\Delta z,\;\;\;t\in[0,a]\cap{\mathbb T},\tag{2}NEWLINE\]NEWLINE where the integral on the right-hand side is the Henstock-Lebesgue \(\Delta\)-integral (also known as the strong Henstock-Kurzweil \(\Delta\)-integral).NEWLINENEWLINEThe author uses the Mönch fixed-point theorem to obtain the local existence of a solution of (2) satisfying a~given initial condition \(x(0)=x_0\). The conditions on \(f\) and \(k\), which guarantee that the assumptions of the fixed point theorem are satisfied, are based on Kuratowski's measure of noncompactness.