The author is concerned with the asymptotic behaviour of solutions to differential inclusions in Banach spaces, such as \[ \begin{aligned} u_t-\Delta_p(u)+ |u|^{\gamma-1} u= f\quad &\text{on }\mathbb{R}_+\times\Omega,\\ -{\partial u\over\partial\eta}\in \beta(u)\quad &\text{on }\mathbb{R}_+\times \partial\Omega,\\ u(0,x)= u_0(x)\quad &\text{in }\Omega,\end{aligned} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), with smooth boundary \(\partial\Omega\), \(f(t,x)\) stands for an \(L^1\)-function on \(\mathbb R_+\times\Omega\), \(\gamma\geq 1\), and \(1\leq p<\infty\). \(\Delta_p\) is the \(p\)-Laplacian operator, while \(\beta\) stands for a maximal monotone graph in \(\mathbb{R}\times\mathbb{R}\), such that \(0\in\beta(0)\). The concept of an accretive operator is defined and utilized. A general criterion of accretivity is first obtained, and then applied to derive results on asymptotic behaviour for the Cauchy’s problem \(u'(t)+ A(u(t))\ni 0\), \(t\in\mathbb{R}_+\), \(u(0)= x_0\). A nonlinear boundary value problem, similar to \((P)\), is discussed in concluding the paper.