Mediterr. J. Math. DOI 10.1007/s00009-015-0541-3 c  Springer Basel 2015 Split Common Fixed Point Problem of Nonexpansive Semigroup Mohammad Eslamian and Javad Vahidi Dedicated to Professor Ghasem Alizadeh Afrouzi on the occasion of his 55th birthday. Abstract. In this paper, we first introduce a new algorithm with a vis- cosity iteration method for solving the split common fixed point prob- lem (SCFP) for a finite family of nonexpansive semigroups. We also present a new algorithm for solving the SCFP for an infinite family of quasi-nonexpansive mappings. We establish strong convergence of these algorithms in an infinite-dimensional Hilbert spaces. As application, we obtain strong convergence theorems for split variational inequality prob- lems and split common null point problems. Our results improve and extend the related results in the literature. Mathematics Subject Classification. 47J25, 47N10, 65J15, 90C25. Keywords. Nonexpansive semigroup, quasi-nonexpansive mapping, split common fixed point problem, viscosity iteration method. 1. Introduction Let C and Q be nonempty closed convex subsets of real Hilbert spaces H and K, respectively. The split feasibility problem (SFP) is formulated as: to finding x ∗ ∈ C and Ax ∗ ∈ Q, where A : H→K is a bounded linear operator. In 1994, Censor and Elfving [1] first introduced the SFP in finite-dimensional Hilbert spaces for model- ing inverse problems which arise from phase retrievals and in medical image reconstruction [2]. It has been found that the SFP can also be used in vari- ous disciplines such as image restoration, computer tomograph and radiation therapy treatment planning [3–5]. The problems have been investigated by many researchers (see [6–14] and references therein). It is known that find- ing a solution to the split feasibility problem is equivalent to finding the minimum-norm fixed point of the mapping x → P C (I − γ A ∗ (I − P Q )A)x, where P C and P Q are the nearest point projections from H onto C and from K onto Q, respectively, γ> 0 is a positive constant and A ∗ is the adjoint of Downloaded from http://www.elearnica.ir