Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces

Suppose K is a nonempty closed convex subset of a real Banach space E. Let S, T : K --> K be two asymptotically quasi-nonexpansive maps with sequences {u(n)},{v(n)} subset of [0,infinity) such that Sigma(infinity)(n=1) u(n) < infinity and Sigma(infinity)(n=1) v(n) < infinity, and F = F(S)boolean AND F(T) := {x is an element of K : Sx = Tx = x} not equal empty set. Suppose {x(n)} is generated iteratively by x(1) is an element of K, x(n+1) = (1 - alpha(n)) x(n) + alpha S-n(n)[(1 - beta(n)) x(n) + beta(n)T(n)x(n)], n >= 1, where {alpha(n)} and {beta(n)} are real sequences in [0,1]. It is proved that (a) {x(n)} converges strongly to some x* is an element of F if and only if liminf(n-->infinity) d(x(n), F) = 0; ( b) if X is uniformly convex and if either T or S is compact, then {x(n)} converges strongly to some x*. F. Furthermore, if X is uniformly convex, either T or S is compact and {x(n)} is generated by x(1) is an element of K, x(n+1) = alpha(n)x(n) + beta S-n(n)[alpha'(n)x(n) + beta'(n)T(n)x(n) + gamma'(n)z'(n)] + y(n)z(n), n >= 1, where {z(n)}, {z'(n)} are bounded, {alpha(n)}, {beta(n)}, {gamma(n)}, {alpha'(n)}, {beta'(n)}, {gamma'(n)} are real sequences in [0,1] such that alpha(n) + beta(n) + gamma(n) = 1 = alpha'(n) + beta'(n) + gamma'(n) and {gamma(n)}, {gamma'(n)} are summable; it is established that the sequence {x(n)} ( with error member terms) converges strongly to some x* is an element of F.