×

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) \(\lim_{t\downarrow 0}(I+tA)^{-1}x\) exists, and (ii) if R(A) contains 0, then \(\lim_{t\to +\infty}(I+tA)^{-1}\) exists and belongs to \(A^{(-1)}(0).\)
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 Nonlinear accretive operators, dissipative operators, etc.
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, (1976), Editura Academiei R. S. R Bucharest
[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), Springer-Verlag Berlin/New York · Zbl 0268.46013
[5] Reich, S., Approximating zeros of accretive operators, (), 381-384 · 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.