  We propose a forward-backward splitting dynamical system for solving
inclusion problems of the form $0\in A(x)+B(x)$ in Hilbert spaces, where $A$ is
a maximal operator and $B$ is a single-valued operator. Involved operators are
assumed to satisfy a generalized monotonicity condition, which is weaker than
the standard monotone assumptions. Under mild conditions on parameters, we
establish the fixed-time stability of the proposed dynamical system. In
addition, we consider an explicit forward Euler discretization of the dynamical
system leading to a new forward backward algorithm for which we present the
convergence analysis. Applications to other optimization problems such as
Constrained Optimization Problems (COPs), Mixed Variational Inequalities
(MVIs), and Variational Inequalities (VIs) are presented and some numerical
examples are given to illustrate the theoretical results.
