On the Solvability of Some Abstract Differential Equations

This paper is concerned with the solvability of some abstract differential equation of type $$\dot{u}(t)+Au(t)+Bu(t)\ni f(t),t\in (0,T],u(0)=0,$$ where $A$ is a linear selfadjoint operator and $B$ is a nonlinear(possibly multi-valued)maximal monotone operator in a (real) Hilbert space $\mathbb{H}$ with the normalization $0\in B(0)$. We use the concept of variational sum introduced by H. Attouch, J.-B. Baillon, and M. Th'era, to investigate solutions to the given abstract differential equation. Several applications will be discussed, among them the case where $B=\text{subdifferential}\varphi$, the subdifferential of a convex semicontinuous proper function $\varphi$.