# zbMATH — the first resource for mathematics

An iterative process for nonlinear Lipschitzian strongly accretive mappings in $$L_ p$$ spaces. (English) Zbl 0724.65058
A very accurate description of the content of the paper under review is given in the author’s abstract:
“Suppose $$X=L_ p$$ (or $$l_ p)$$, $$p\geq 2$$. Let $$T: X\to X$$ be a Lipschitzian and strongly accretive map with constant $$k\in (0,1)$$ and Lipschitz constant L. Define $$S: X\to X$$ by $$Sx=f-Tx-x$$. Let $$\{C_ n\}^{\infty}_{n=1}$$ be a real sequence satisfying:
(i) $$0<C_ n\leq k[(p-1)L^ 2+2k-1]^{-1}$$ for each n,
(ii) $$\sum^{\infty}_{n=0}C_ n=\infty.$$
Then, for arbitrary $$x_ 0\in X$$, the sequence $x_{n+1}=(1-C_ n)x_ n+C_ nSx_ n,\;n\geq 0,$ converges strongly to the unique solution of $$Tx=f$$. Moreover, if $$C_ n=k[(p-1)L^ 2+2k-1]^{-1}$$ for each n, $$then$$
$$\| x_{n+1}-q\| \leq \rho^{n/2}\| x_ 1-q\|$$, where q denotes the solution of $$Tx=f$$ and $\rho =(1-k[(p-1)L^ 2+2k-1]^{- 1})\in (0,1).$ A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps in X.”
Reviewer: I.Marek (Praha)

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 47H06 Nonlinear accretive operators, dissipative operators, etc. 47J05 Equations involving nonlinear operators (general) 47J25 Iterative procedures involving nonlinear operators
Full Text:
##### References:
  Bogin, J, On strict pseudo-contractions and a fixed point theorem, Technion preprint series no. MT-219, (1974), Haifa, Israel  Browder, F.E, Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. amer. math. soc., 73, 875-882, (1967) · Zbl 0176.45302  Browder, F.E, Nonlinear operators and nonlinear equations of evolution in Banach spaces, () · Zbl 0176.45301  Browder, F.E; Petryshyn, W.V, Construction of fixed points of nonlinear mappings in Hilbert space, J. math. anal. appl., 20, 197-228, (1967) · Zbl 0153.45701  Bynum, W.L, Weak parallelogram laws for Banach spaces, Canad. math. bull., 19, No. 3, (1976) · Zbl 0347.46015  Bruck, R.E, The iterative solution of the equation fϵx + tx for a monotone operator in T in Hilbert space, Bull. amer. math. soc., 79, 1258-1262, (1973)  Chidume, C.E, Iterative approximation of fixed points of Lipschitzian strictly pseudo-contractive mappings, (), 283-288, No. 2 · Zbl 0646.47037  Deimling, K, Zeros of accretive operators, Manuscripta math., 13, 365-374, (1974) · Zbl 0288.47047  Dotson, W.G, An iterative process for nonlinear monotonic nonexpansive operators in Hilbert space, Math. comp., 32, No. 151, 223-225, (1978) · Zbl 0374.47028  Dunn, J.C, Iterative construction of fixed points for multivalued operators of the monotone type, J. funct. anal., 27, 38-50, (1978) · Zbl 0422.47033  Edelstein, M; O’Brian, R.C, Nonexpansive mappings, asymptotic regularity and successive approximations, J. London math. soc. (2), 17, No. 3, 547-554, (1978) · Zbl 0421.47031  Gwinner, J, On the convergence of some iteration processes in uniformly convex Banach spaces, (), 29-35 · Zbl 0393.47040  Ishikawa, S, Fixed points by a new iteration method, (), 147-150 · Zbl 0286.47036  Ishikawa, S, Fixed points and iteration of a nonexpansive mapping in a Banach space, (), 65-71 · Zbl 0352.47024  Kato, T, Nonlinear semigroups and evolution equations, J. math. soc. Japan, 18/19, 508-520, (1967) · Zbl 0163.38303  Kirk, W.A; Morales, C, On the approximation of fixed points of locally non-expansive mappings, Cand. math. bull., 24, No. 4, 441-445, (1981) · Zbl 0475.47041  Mann, W.R, Mean value methods in iteration, (), 506-510 · Zbl 0050.11603  Martin, R.H, A global existence theorem for autonomous differential equations in Banach spaces, (), 307-314 · Zbl 0202.10103  Morales, C, Pseudocontractive mappings and Leray Schauder boundary condition, Comment. math. univ. carolin., 20, No. 4, 745-756, (1979) · Zbl 0429.47021  Mukerjee, R.N, Construction of fixed points of strictly pseudocontractive mappings in generalized Hilbert spaces and related applications, Indian J. pure appl. math., 15, 276-284, (1966)  Nevalinna, O; Reich, S, Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel J. math., 32, 44-58, (1979) · Zbl 0427.47049  Petryshyn, W.V, Construction of fixed points of demi-compact mappings in Hilbert space, J. math. anal. appl., 14, 276-284, (1986) · Zbl 0138.39802  Reich, S, Constructing zeros of accretive operators, II, Appl. anal., 9, 159-163, (1979) · Zbl 0424.47034  Reich, S, Constructive techniques for accretive and monotone operators, (), 335-345  Reich, S, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. anal. appl., 85, 287-292, (1980) · Zbl 0437.47047  Rhoades, B.E, Comments on two fixed point iteration methods, J. math. anal. appl., 56, 741-750, (1976) · Zbl 0353.47029  Zarantonello, E.H, Solving functional equations by contractive averaging, ()
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.