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In this paper, we introduce an iterative scheme using the gradient projection method with a new step size, which is not depend on the related matrix inverses and the largest eigenvalue (or the spectral radius of the self-adjoint operator) of the related matrix, based on Moudafi’s viscosity approximation method for solving the  split feasibility problem  (SFP), which is to find a point in a given closed convex subset of a real Hilbert space such that its image under a bounded linear operator belongs to a given closed convex subset of another real Hilbert space. We suggest and analyze this iterative scheme under some appropriate conditions imposed on the parameters such that another strong convergence theorems for the SFP are obtained. The results presented in this paper improve and extend the main results of Tian and Zhang (J. Inequal. Appl. 2017:Article ID 13,  2017 ), and Tang et al. (Acta Math. Sci. 36B(2):602–613,  2016 ) (in a single-step regularized method) with a new step size, and many others. The examples of the proposed SFP are also shown through numerical results.

Gradient projection method with a new step size for the split feasibility problem