The paper is concerned with solving iteratively nonlinear operator equations involving set valued accreative operators in Banach spaces (in Hilbert spaces they are called monotone operators). The authors are focused on three types of iterations: the continuous Picard type iteration, the approximate Picard type iteration and the Halpern type iteration [see B. Halpern, Bull. Am. Math. Soc. 73, 957–961 (1967; Zbl 0177.19101)]. All these three iterative methods are depending on the resolvent of the considered accreative operator and on a real sequence from the interval satisfying certain conditions.
The results proved by the authors are refering to the strong convergence of these iterative methods under the assumptions that the Banach space is a reflexive one, with weakly continuous duality mapping and the accretive operator satisfies the range condition. The obtained results are then applied to a viscosity approximation with weak contraction for equilibrium problems and for solving variational inequalities.