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On Reich’s strong convergence theorems for resolvents of accretive operators. (English) Zbl 0599.47084

Let E be a real Banach space with norm $\parallel \parallel$. Then an operator $A\subset E×E$ with domain D(A) and range R(A) is said to be accretive if $\parallel {x}_{1}-{x}_{2}\parallel \le \parallel {x}_{1}-{x}_{2}+r\left({y}_{1}-{y}_{2}\right)\parallel$ for all ${y}_{i}\in A{x}_{i}$, $i=1,2$, and $r>0$. An accretive operator $A\subset E×E$ is m-accretive if $R\left(I+rA\right)=E$ for all $r>0$, where I is the identity. If A is accretive, we can define, for each positive r, a single-valued mapping ${J}_{r}:R\left(I+rA\right)\to D\left(A\right)$ by ${J}_{r}={\left(I+rA\right)}^{-1}$. It is called the resolvent of A. In [ibid. 75, 287-292 (1980; Zbl 0437.47047)], S. Reich proved the following theorem: Let E be a uniformly smooth Banach space, and let $A\subset E×E$ be m-accretive. If $0\in R\left(A\right)$, then for each x in E the strong ${lim}_{r\to \infty }{J}_{r}x$ exists and belongs to ${A}^{-1}0$. He remarked also that the assumption that A is m-accretive can be replaced with the assumption that cl(D(A)), the closure of D(A), is convex, and that A satisfies the range condition: $R\left(I+rA\right)\supset cl\left(D\left(A\right)\right)$ for all $r>0·$

In this paper we first prove a theorem that generalizes simultaneously the above results. Furthermore, we generalize another Reich strong convergence theorem asserting that the strong ${lim}_{r\to 0}{J}_{r}x$ exists in uniformly convex and uniformly smooth Banach spaces. Though our proofs are similar to those in the cited paper they are slightly simple on account of using Banach limits.

##### MSC:
 47H06 Accretive operators, dissipative operators, etc. (nonlinear) 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems