Equivalent theorems of the convergence between proximal type algorithms

The main aim of this work is to give the equivalence of the convergence between proximal type algorithms. Our convergence analysis covers Rockafellar type proximal point algorithm (with a weak contraction) for finding zeros of accretive operators as well as Browder and Halpern type iterations and viscosity approximation methods for finding fixed points of nonexpansive type mappings in Banach spaces. We also provide simple proofs of some results of Wong, Sahu and Yao [N.C. Wong, D.R. Sahu, J.C. Yao, Solving variational inequalities involving nonexpansive type mappings, Nonlinear Anal. 69 (12) (2008) 4732–4753].