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

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 arbitraryx0ɛ X, the sequence {xn}n=1∞ is defined byxn+1=(1−αn)xn+αnSyn,yn=(1−Βn)xn+ΒnSxn,n⩾0, where {αn}n=0∞, {Βn}n=0∞ are two real sequences satisfying: (i) 0⩽αnp−1 ⩽ 2−1s(k+kΒn−L2Βn)(w+h)−1 for eachn, (ii) 0⩽Βnp−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, αn=2−1s(k +kΒ−L2Β)(w+h)−1, andΒn=Β for eachn and some 0 ⩽Β ⩽ min {k/L2, sk/(w + h)}, then ∥xn + 1−q∥ ⩽ρn/s∥x1-q∥, whereq denotes the solution ofTx=f andρ=(1 − 4−1s2(k +kΒ − L2Β)2(w + h)−1ɛ (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).