# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Strong convergence theorems for accretive operators in Banach spaces. (English) Zbl 0712.47045
For 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) $\lim\sb{t\downarrow 0}(I+tA)\sp{-1}x$ exists, and (ii) if R(A) contains 0, then $\lim\sb{t\to +\infty}(I+tA)\sp{-1}$ exists and belongs to $A\sp{(-1)}(0).$ 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)
Full Text:
##### References:
 [1] Barbu, V.: Nonlinear semigroups and differential equations in Banach spaces. (1976) · Zbl 0328.47035 [2] Bruck, R. E.; Reich, S.: Accretive operators, Banach limits and dual ergodic theorems. Bull. acad. Polon. sci. 29, 585-589 (1981) · Zbl 0492.47030 [3] Crandall, M. G.; Pazy, A.: Semigroups of nonlinear contractions and dissipative sets. J. funct. Anal. 3, 376-418 (1969) · Zbl 0182.18903 [4] Day, M. M.: Normed linear spaces. (1973) · Zbl 0268.46013 [5] Reich, S.: Approximating zeros of accretive operators. Proc. amer. Math. soc. 51, 381-384 (1975) · Zbl 0294.47042 [6] Reich, S.: Asymptotic behavior of semigroups of nonlinear contractions in Banach spaces. J. math. Anal. appl. 53, 277-290 (1976) · Zbl 0337.47027 [7] Reich, S.: Extension problems for accretive sets in Banach spaces. J. funct. Anal. 26, 378-395 (1977) · Zbl 0378.47037 [8] Reich, S.: Constructing zeros of accretive operators II. Appl. anal. 9, 159-163 (1979) · Zbl 0424.47034 [9] Reich, S.: Product formula, nonlinear semigroups and accretive operators. J. funct. Anal. 36, 147-168 (1980) · Zbl 0437.47048 [10] Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. math. Anal. appl. 75, 287-292 (1980) · Zbl 0437.47047 [11] Reich, S.: The fixed point property for nonexpansive mappings II. Amer. math. Monthly 87, 292-294 (1980) · Zbl 0443.47057