Split equality problem and multiple-sets split equality problem for quasi-nonexpansive multi-valued mappings

The multiple-sets split equality problem (MSSEP) requires finding a point x is an element of boolean AND(N)(i=1) C-i, y is an element of boolean AND j=1M Q(j) such that Ax = By, where N and M are positive integers, {C-1, C-2, ..., C-N} and {Q(1), Q(2),..., Q(M)} are closed convex subsets of Hilbert spaces H-1, H-2, respectively, and A : H-1 -> H-3, B : H-2 -> H-3 are two bounded linear operators. When N = M = 1, the MSSEP is called the split equality problem (SEP). If let B = I, then the MSSEP and SEP reduce to the well-known multiple-sets split feasibility problem (MSSFP) and split feasibility problem (SFP), respectively. Recently, some authors proposed many algorithms to solve the SEP and MSSEP. However, to implement these algorithms, one has to find the projection on the closed convex sets, which is not possible except in simple cases. One of the purposes of this paper is to study the SEP and MSSEP for a family of quasi-nonexpansive multi-valued mappings in the framework of infinite-dimensional Hilbert spaces, and propose an algorithm to solve the SEP and MSSEP without the need to compute the projection on the closed convex sets.