On Reich's strong convergence theorems for resolvents of accretive operators
Let E be a real Banach space with norm \(\| \|\). Then an operator \(A\subset E\times E\) with domain D(A) and range R(A) is said to be accretive if \(\| x_ 1-x_ 2\| \leq \| x_ 1-x_ 2+r(y_ 1- y_ 2)\|\) for all \(y_ i\in Ax_ i\), \(i=1,2\), and \(r>0\). An accretive operator \(A\subset E\times E\) is m-accretive if \(R(I+rA)=E\) for all \(r>0\), where I is the identity. If A is accretive, we can define, for each positive r, a single-valued mapping \(J_ r:R(I+rA)\to D(A)\) by \(J_ r=(I+rA)^{-1}\). It is called the resolvent of A. In [ibid. 75, 287-292 (1980; Zbl 0437.47047)], \textit{S. Reich} proved the following theorem: Let E be a uniformly smooth Banach space, and let \(A\subset E\times E\) be m-accretive. If \(0\in R(A)\), then for each x in E the strong \(\lim_{r\to \infty}J_ rx\) 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: \(R(I+rA)\supset cl(D(A))\) 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 \(\lim_{r\to 0}J_ rx\) 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.
scientific article; zbMATH DE number 3967228