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Convergence theorem for zeros of generalized Lipschitz generalized Φ-quasi-accretive operators. (English) Zbl 1072.47062

Summary: Let E be a uniformly smooth real Banach space and let A:EE be a mapping with N(A). Suppose that A is a generalized Lipschitz generalized Φ-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) (b n +c n )=; (iii) c n <; (iv) limb n =0· Let {x n } be generated iteratively from arbitrary x 0 E by

x n+1 =a n x n +b n Sx n +c n u n ,n0,

where S:EE is defined by Sx:=x-Ax forall xE and {u n } is an arbitrary bounded sequence in E. Then there exists γ 0 such that if b n +c n γ 0 n0, 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 ϕ-hemi-contractive mapping.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces