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Strong convergence theorems for accretive operators in Banach spaces. (English) Zbl 0712.47045

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

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)