Strong convergence of iteration algorithms for a countable family of nonexpansive mappings Original Research Article

For a countable family View the MathML source{Tn}n=1+∞ of nonexpansive mappings, a strong convergence of Halpern type iteration is shown in order to find a common fixed point of View the MathML source{Tn}n=1+∞ in a reflexive Banach space when αn∈[0,1]αn∈[0,1] satisfies the conditions (C1) limn→∞αn=0limn→∞αn=0 and (C2) View the MathML source∑n=1∞αn=∞, and several examples satisfying the condition (B) is given also. We also research the strong convergence of the proximal point algorithm. Finally, we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings.