Convergence theorem for zeros of generalized Lipschitz generalized PHI-quasi-accretive operators

Let E be a uniformly smooth real Banach space and let A : E -> E be a mapping with N(A) not equal phi. Suppose A is a generalized Lipschitz generalized Phi-quasi-accretive mapping. Let {a(n)}, {b(n)}, and {c(n)} be real sequences in [0,1] satisfying the following conditions: (i) a(n) + b(n) + c(n) = 1; (ii) similar to(b(n) + c(n)) = 8; (iii) similar to c(n) < ;infinity (iv) lim b(n) = 0. Let {x(n)} be generated iteratively from arbitrary x(O) is an element of E by x(n+1) = a(n)x(n) + b(n)Sx(n) + c(n)u(n), n >= 0, where S : E -> E is defined by Sx := x - Ax for all x is an element of E and {u(n)} is an arbitrary bounded sequence in E. Then, there exists gamma o is an element of R such that if b(n) + c(n) <= gamma o for all n >= 0, the sequence {x(n)} converges strongly to the unique solution of the equation Au = 0. A related result deals with approximation of the unique fixed point of a generalized Lipschitz and generalized o-hemicontractive mapping.