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The aim of this paper is to introduce a viscosity iterative algorithm for the implicit midpoint rule of nonexpansive mappings in uniformly smooth spaces. Under some appropriate conditions on the parameters, we prove some strong convergence theorems. As applications, we apply our main results to solving fixed point problems of strict pseudocontractive mappings, variational inequality problems in Banach spaces and equilibrium problems in Hilbert spaces. Finally, we give some numerical examples for supporting our main results.

In this paper, our aim is to introduce a viscosity type algorithm for solving proximal split feasibility problems and prove the strong convergence of the sequences generated by our iterative schemes in Hilbert spaces. First, we prove strong convergence result for a problem of finding a point which minimizes a convex function f such that its image under a bounded linear operator A minimizes another convex function g. Secondly, we prove another strong convergence result for the case where one of the two involved functions is prox-regular. In all our results in this work, our iterative schemes are proposed by way of selecting the step sizes such that their implementation does not need any prior information about the operator norm because the calculation or at least an estimate of the operator norm A is not an easy task. Finally, we give a numerical example to study the efficiency and implementation of our iterative schemes. Our results complement the recent results of Moudafi and Thakur (

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