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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 . Then an operator AE×E with domain D(A) and range R(A) is said to be accretive if x 1 -x 2 x 1 -x 2 +r(y 1 -y 2 ) for all y i Ax i , i=1,2, and r>0. An accretive operator AE×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)D(A) by J r =(I+rA) -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 AE×E be m-accretive. If 0R(A), then for each x in E the strong lim r 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(I+rA)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 r0 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.

47H06Accretive operators, dissipative operators, etc. (nonlinear)
47J10Nonlinear spectral theory, nonlinear eigenvalue problems