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An iterative process for nonlinear lipschitzian and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces

An iterative process for nonlinear lipschitzian and strongly accretive mappings in uniformly... SupposeX is ans-uniformly smooth Banach space (s > 1). LetT: X → X be a Lipschitzian and strongly accretive map with constantk ɛ (0, 1) and Lipschitz constantL. DefineS: X → X bySx=f−Tx+x. For arbitraryx 0 ɛ X, the sequence {xn} n=1 ∞ is defined byx n+1=(1−α n)xn+α nSyn,y n=(1−Β n)xn+Β nSxn,n⩾0, where {αn} n=0 ∞ , {Βn} n=0 ∞ are two real sequences satisfying: (i) 0⩽α n p−1 ⩽ 2−1s(k+kΒ n−L 2Βn)(w+h)−1 for eachn, (ii) 0⩽Β n p−1 ⩽ min{k/L2, sk/(Ω+h)} for eachn, (iii) ⌆n αn=∞, wherew=b(1+L)s andb is the constant appearing in a characteristic inequality ofX, h=max{1, s(s-l)/2},p=min {2, s}. Then {xn} n=1 ∞ converges strongly to the unique solution ofTx=f. Moreover, ifp=2, α n=2−1s(k +kΒ−L2Β)(w+h)−1, andΒ n=Β for eachn and some 0 ⩽Β ⩽ min {k/L2, sk/(w + h)}, then ∥xn + 1−q∥ ⩽ρ n/s∥x1-q∥, whereq denotes the solution ofTx=f andρ=(1 − 4−1s2(k +kΒ − L 2Β)2(w + h)−1 ɛ (0, 1). A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps inX. SupposeX ism-uniformly convex Banach spaces (m > 1) andc is the constant appearing in a characteristic inequality ofX, two similar results are showed in the cases of L satisfying (1 − c2)(1 + L)m < 1 + c − cm(l − k) or (1 − c2)Lm < 1 + c − cm(1 − s). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

An iterative process for nonlinear lipschitzian and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces

Deng, Lei
Acta Applicandae Mathematicae , Volume 32 (2) – Jan 3, 2005
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References (31)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/BF00998152
Publisher site
See Article on Publisher Site

Abstract

SupposeX is ans-uniformly smooth Banach space (s > 1). LetT: X → X be a Lipschitzian and strongly accretive map with constantk ɛ (0, 1) and Lipschitz constantL. DefineS: X → X bySx=f−Tx+x. For arbitraryx 0 ɛ X, the sequence {xn} n=1 ∞ is defined byx n+1=(1−α n)xn+α nSyn,y n=(1−Β n)xn+Β nSxn,n⩾0, where {αn} n=0 ∞ , {Βn} n=0 ∞ are two real sequences satisfying: (i) 0⩽α n p−1 ⩽ 2−1s(k+kΒ n−L 2Βn)(w+h)−1 for eachn, (ii) 0⩽Β n p−1 ⩽ min{k/L2, sk/(Ω+h)} for eachn, (iii) ⌆n αn=∞, wherew=b(1+L)s andb is the constant appearing in a characteristic inequality ofX, h=max{1, s(s-l)/2},p=min {2, s}. Then {xn} n=1 ∞ converges strongly to the unique solution ofTx=f. Moreover, ifp=2, α n=2−1s(k +kΒ−L2Β)(w+h)−1, andΒ n=Β for eachn and some 0 ⩽Β ⩽ min {k/L2, sk/(w + h)}, then ∥xn + 1−q∥ ⩽ρ n/s∥x1-q∥, whereq denotes the solution ofTx=f andρ=(1 − 4−1s2(k +kΒ − L 2Β)2(w + h)−1 ɛ (0, 1). A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps inX. SupposeX ism-uniformly convex Banach spaces (m > 1) andc is the constant appearing in a characteristic inequality ofX, two similar results are showed in the cases of L satisfying (1 − c2)(1 + L)m < 1 + c − cm(l − k) or (1 − c2)Lm < 1 + c − cm(1 − s).

Journal

Acta Applicandae Mathematicae – Springer Journals

Published: Jan 3, 2005

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