Strong convergence theorems for accretive operators in Banach spaces.

*(English)*Zbl 0712.47045For a nonlinear m-accretive operator $A\subset E\times E$ with domain D(A) and range R(A) in a real Hilbert space E, it is known that for each $x\in E$, (i) ${lim}_{t\downarrow 0}{(I+tA)}^{-1}x$ exists, and (ii) if R(A) contains 0, then ${lim}_{t\to +\infty}{(I+tA)}^{-1}$ exists and belongs to ${A}^{(-1)}\left(0\right)\xb7$

The paper under review extends the assertions (i) and (ii) to the case where E is a real reflexive Banach space with a uniformly Gâteaux differentiable norm, and $I+tA$ is replaced by $S+tA$ with S a bounded, strongly accretive and continuous operator of the closure of D(A) into E, assuming existence of a nonexpansive retraction of E onto the closure of D(A).

Reviewer: T.Ichinose

##### MSC:

47H06 | Accretive operators, dissipative operators, etc. (nonlinear) |