Global existence for some integro-differential equations with delay subject to non-local conditions
The authors consider the following integro-differential equation in a Banach space: \[ dx(t)/dt=Ax(t)+ F \left(t, x(\sigma_1(t)), \int_0^t g\left(t, s, x(\sigma_2(s)), \int_0^s k(\tau, s, x(\sigma_3(\tau))) d\tau \right) ds\right) \] with the non-local condition \[ x(0) + h (t_1, \dots , t_p, x(.))=x_0\in X. \] Here \(A\) is a generator of a linear strongly continuous semigroup, \(F, g, k, \sigma_j\) and \(h\) are given functions. Conditions for the global existence of solutions to that problem are derived.
scientific article; zbMATH DE number 1779547