Solution of stochastic non-autonomous Chen – Gurtin model with multipoint initial-final condition

In this paper the authors investigate the solvability of a non-autonomous Chen – Gurtin model with a multipoint initial-final condition in the space of stochastic $\mathbf{K}$-processes. To do this, we first consider the solvability of a multipoint initial-final problem for a non-autonomous Sobolev type equation in the case when the resolving family is a strongly continuous semiflow of operators. The Chen – Gurtin model refers to non-classical models of mathematical physics. Recall that non-classical are those models of mathematical physics whose representations in the form of equations or systems of partial differential equations do not fit within one of the classical types: elliptic, parabolic or hyperbolic. For this model, multipoint initial-final conditions, which generalizing the Cauchy and Showalter-Sidorov conditions, are considered.