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

SupposeX is ans-uniformly smooth Banach space (s > 1). LetT: X X be a Lipschitzian and strongly accretive map with constantk (0, 1) and Lipschitz constantL. DefineS: X X bySx=f–Tx+x. For arbitraryx ₀ X, the sequence {x_n} _n=1 is defined byx _n+1=(1– _n)x_n+ _nSy_n,y _n=(1– _n)x_n+ _nSx_n,n0, where {_n} _n=0 , {_n} _n=0 are two real sequences satisfying: (i) 0 _n ^p–1 2^–1s(k+k _n–L ²_n)(w+h)^–1 for eachn, (ii) 0 _n ^p–1 min{k/L², sk/(+h)} for eachn, (iii) _{n n}=, 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 {x_n} _n=1 converges strongly to the unique solution ofTx=f. Moreover, ifp=2, _n=2^–1s(k +k–L²)(w+h)^–1, and _n= for eachn and some 0 min {k/L², sk/(w + h)}, then x_{n + 1}–q ^n/sx₁-q, whereq denotes the solution ofTx=f and=(1 – 4^–1s²(k +k – L ²)²(w + h)^–1 (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)L^m < 1 + c – cm(1 – s).