On semigroups generated by $m$-accretive operators in a strict sense

Abstract:

Let $\left \{ {S(t):t \geqslant 0} \right \}$ be a nonlinear semigroup generated by an $m$-accretive operator $A$ in a real Banach space $\left ( {X,\left | \cdot \right |} \right )$. It is shown that (1) for any $x \in D(A)$ belongs to ${L^p}(0,T;V)(T < 0)$ if $A$ satisfies \[ {\left | {{u_1} - u} \right |^p} + C\lambda {\left \| {{u_1} - {u_2}} \right \|^p} \leqslant {\left | {{u_1} + \lambda A{u_1} - {u_2} - \lambda A{u_2}} \right |^p}\] $(p{\text { > 1,}}C{\text { > }}0)$ for $\lambda {\text { > }}0$ and ${u_i} \in D(A)(i = 1,2)$, where $\left ( {V,\left \| \cdot \right \|} \right )$ is a Banach space including $D(A)$ and included continuously in $X$; and that (2) $A + B$ has similar properties to those of the above $A$ if $B$ is a Lipschitz continuous operator from $V$ to $X$.