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)
An iterative process for nonlinear Lipschitzian and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces. (English) Zbl 0801.47040

Summary: Suppose X is an s-uniformly smooth Banach space (s>1). Let T:XX be a Lipschitzian and strongly accretive map with constant k(0,1) and Lipschitz constant L. Define S: XX by Sx=f-Tx+x. For arbitrary x 0 X, the sequence {x n } n=1 is defined by

x n+1 =(1-α n )x n +α n Sy n ,y n =(1-β n )x n +β n Sx n ,n0,

where {α n } n=0 , {β n } n=0 are two real sequences satisfying:

(i) 0α n p-1 2 -1 s(k+kβ n -L 2 β n ) (w+h) -1 for each n,

(ii) 0β n p-1 min{k/L 2 ,sk/(w+h)} for each n,

(iii) n α n =,

where w=b(1+L) s and b is the constant appearing in a characteristic inequality of X, h=max{1,s(s-1)/2}, p=min{2,s}. Then {x n } n=1 converges strongly to the unique solution of Tx=f. Moreover, if p=2, α n =2 -1 s(k+kβ-L 2 β) (w+h) -1 , and β n =β for each n and some 0βmin{k/L 2 ,sk/(w+h)}, then

x n+1 -qρ n/s x 1 -q,

where q denotes the solution of Tx=f and

ρ=1 - 4 -1 s 2 (k+kβ-L 2 β) 2 (w+h) -1 (0,1)·

A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps in X. Suppose X is an m- uniformly convex Banach space (m>1) and c is the constant appearing in a characteristic inequality of X, two similar results are showed in the cases of L satisfying (1-c 2 )(1+L) m <1+c-cm(1-k) or (1-c 2 )L m <1+c-cm(1-s).

MSC:
47H06Accretive operators, dissipative operators, etc. (nonlinear)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J25Iterative procedures (nonlinear operator equations)
References:
[1]Bogin, J.: On strict pseudo-contractions and a fixed point theorem, Technion Preprint Series No. MT-219, Haifa, Israel, 1974.
[2]Browder, F. E.: Nonlinear mappings of nonexpansive and accretive type in Banach spaces,Bull. Amer. Math. Soc. 73 (1967), 875-882. · Zbl 0176.45302 · doi:10.1090/S0002-9904-1967-11823-8
[3]Browder, F. E.: Nonlinear operators and nonlinear equations of evolution in Banach spaces,Proc. Sympos. Pure Math. 18 (1976).
[4]Browder, F. E. and Petryshyn, W. V.: Construction of fixed points of nonlinear mapping in Hilbert space,J. Math. Anal. Appl. 20 (1967), 197-228. · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[5]Bruck, R. E. Jr.: The iterative solution of the equationf ?x +Tx for a monotone operator inT in Hilbert space,Bull. Amer. Math. Soc. 79 (1973), 1258-1262. · Zbl 0275.47033 · doi:10.1090/S0002-9904-1973-13404-4
[6]Chidume, C. E.: An iterative process for nonlinear Lipschitzian strongly accretive mappings inL p spaces,J. Math. Anal. Appl. 151 (1990), 453-461. · Zbl 0724.65058 · doi:10.1016/0022-247X(90)90160-H
[7]Chidume, C. E.: Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings,Proc. Amer. Math. Soc. 99(2) (1987), 283-288.
[8]Deimling, K.: Zeros of accretive operators,Manuscripta Math. 13 (1974), 365-374. · Zbl 0288.47047 · doi:10.1007/BF01171148
[9]Deng Lei: Iteration processes for nonlinear Lipschitzian strongly accretive mappings inL p spaces, (submitted).
[10]Deng Lei: On Chidume’s open questions,J. Math. Anal. Appl. 174 (1993), 741-749. · Zbl 0784.47051 · doi:10.1006/jmaa.1993.1129
[11]Dotson, W. G.: An iterative process for nonlinear monotonic nonexpansive operators in Hubert space,Math. Comp. 32(151) (1978), 223-225. · doi:10.1090/S0025-5718-1978-0470779-8
[12]Dunn, J. C.: Iterative construction of fixed points for multivalued operators of the monotone type,J. Funct. Anal. 27 (1978), 38-50. · Zbl 0422.47033 · doi:10.1016/0022-1236(78)90018-6
[13]Edelstein, M. and O’Brien, R. C.: Nonexpansive mappings, asymptotic regularity and successive approximations,J. London Math. Soc. (2)17(3) (1978), 547-554. · Zbl 0421.47031 · doi:10.1112/jlms/s2-17.3.547
[14]Gwinner, J.: On the convergence of some iteration processes in uniformly convex Banach spaces,Proc. Amer. Math. Soc. 71 (1978), 29-35. · doi:10.1090/S0002-9939-1978-0477899-4
[15]Ishikawa, S.: Fixed points by a new iteration method,Proc. Amer. Math. Soc. 44 (1974), 147-150. · doi:10.1090/S0002-9939-1974-0336469-5
[16]Istratescu, V. I.:Fixed Point Theory, D. Reidel, Dordrecht, 1981.
[17]Kato, T.: Nonlinear semigroups and evolution equations,J. Math. Soc. Japan 19 (1967), 508-520. · Zbl 0163.38303 · doi:10.2969/jmsj/01940508
[18]Lindenstrauss, J. and Tsafriri, L.:Classical Banach Spaces, II, Springer-Verlag, New York, Berlin, 1979.
[19]Mann, W. R.: Mean value methods in iteration,Proc. Amer. Math. Soc. 4 (1953), 506-510. · doi:10.1090/S0002-9939-1953-0054846-3
[20]Morales, C.: Pseudocontractive mappings and Leray Schauder boundary condition,Comment. Math. Univ. Carolin. 20(4) (1979), 745-756.
[21]Mukerjee, R. N.: Construction of fixed points of strictly pseudocontractive mappings in generalized Hubert spaces and related applications,Indian J. Pure Appl. Math. 15 (1966), 276-284.
[22]Nevalinna, O. and Reich, S.: Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces,Israel. J. Math. 32 (1979), 44-58. · Zbl 0427.47049 · doi:10.1007/BF02761184
[23]Petryshyn, W. V.: Construction of fixed points of demi-compact mappings in Hubert space,J. Math. Anal. Appl. 14 (1986), 276-284. · Zbl 0138.39802 · doi:10.1016/0022-247X(66)90027-8
[24]Reich, S.: Constructing zeros of accretive operators, II,Appl. Anal. 9 (1979), 159-163. · Zbl 0424.47034 · doi:10.1080/00036817908839264
[25]Reich, S.: Constructive techniques for accretive and monotone operators, in V. Lakshmikantham (ed.),Applied Nonlinear Analysis, Academic Press, New York, 1979, pp. 335-345.
[26]Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces,J. Math. Anal. Appl. 75 (1980), 287-292. · Zbl 0437.47047 · doi:10.1016/0022-247X(80)90323-6
[27]Rhoades, B. E.: Comments on two fixed point iteration methods,J. Math. Anal. Appl. 56 (1976), 741-750. · Zbl 0353.47029 · doi:10.1016/0022-247X(76)90038-X
[28]Xu, Hong-Kun: Fixed point theorems for uniformly Lipschitzian semigroups in uniformly convex spaces,J. Math. Anal. Appl. 152 (1990), 391-398. · Zbl 0722.47050 · doi:10.1016/0022-247X(90)90072-N
[29]Xu, Zong-Ben and Roach, G. F.: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces,J. Math. Anal. Appl. 157 (1991), 189-210. · Zbl 0757.46034 · doi:10.1016/0022-247X(91)90144-O
[30]Zarantonello, E. H.: Solving functional equations by contractive averaging, Technical Report No. 160, U.S. Army Math. Res. Centre, Madison, Wisconsin, 1960.