On the system of nonlinear evolution equations – existence and uniqueness via theory of M-matrices

We consider the following system of nonlinear evolution equations$ \begin{equation*} \left\{ \begin{array}{l} u_1'(t) = -A_1\big(u_1(t)\big) + F_1\big(\boldsymbol{u}(t)\big), \\ \vdots \\ u_m'(t) = -A_m\big(u_m(t)\big) + F_m\big(\boldsymbol{u}(t)\big), \end{array} \right. \end{equation*} $where $ A_1, \ldots, A_m $ are strictly monotone operators with perturbations $ F_1, \ldots, F_m $ satisfying certain counterpart of one-sided Lipschitz condition. To reach the existence and uniqueness result we propose certain remetrization method, based on M-matrices theory and utilizing the Gelfand triple setting, which enables usage of a certain type of relaxed monotonicity even in the case of not necessarily strongly monotone operators $ A_1, \ldots, A_m $. Asymptotic properties of the solution are also investigated by showing that any solution tends to the solution of a stationary problem. The results obtained are illustrated by considering a system of equations involving $ (q, p) $-Laplacians and suitable non-monotone perturbations.