The variational approach to nonlinear evolution equations

In this paper, we present a few recent existence results via variational approach for the Cauchy problem
$$ \frac{dy}{dt}(t)+A(t)y(t)\ni f(t),\quad y(0)=y_0,\qquad t\in[0,T], $$
where $A(t)\colon V\to V'$ is a nonlinear maximal monotone operator of subgradient type in a dual pair $(V,V')$ of reflexive Banach spaces. In this case, the above Cauchy problem reduces to a convex optimization problem via Brezis–Ekeland device and this fact has some relevant implications in existence theory of infinite-dimensional stochastic differential equations.