Strong Convergence Theorems for Asymptotically WeakG-Pseudo-Ψ-Contractive Non-Self-Mappings with the Generalized Projection in Banach Spaces

A new concept of the asymptotically weakG-pseudo-Ψ-contractive non-self-mappingT:G↦Bis introduced and some strong convergence theorems for the mapping are proved by using the generalized projection method combined with the modified successive approximation method or with the modified Mann iterative sequence method in a uniformly and smooth Banach space. The proof methods are also different from some past common methods.

Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings

Suppose thatEis a real normed linear space,Cis a nonempty convex subset ofE,T:C→Cis a Lipschitzian mapping, andx*∈Cis a fixed point ofT. For givenx0∈C, suppose that the sequence{xn}⊂Cis the Mann iterative sequence defined byxn+1=(1-αn)xn+αnTxn,n≥0, where{αn}is a sequence in [0, 1],∑n=0∞αn2<∞,∑n=0∞αn=∞. We prove that the sequence{xn}strongly converges tox*if and only if there exists a strictly increasing functionΦ:[0,∞)→[0,∞)withΦ(0)=0such thatlimsup n→∞inf j(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}≤0.