Strong convergence of iteration algorithms for a countable family of nonexpansive mappings

For a countable family {Tn}n=1+∞ of nonexpansive mappings, a strong convergence of Halpern type iteration is shown in order to find a common fixed point of {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) ∑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.