Split feasibility problem for quasi-nonexpansive multi-valued mappings and total asymptotically strict pseudo-contractive mapping

The purpose of this article is to study the split feasibility problem for a family of quasi-nonexpansive multi-valued mappings and a total asymptotically strict pseudo-contractive mapping in infinitely dimensional Hilbert spaces. The main results presented in the paper improve and extend some recent results of Censor et al. [Numer. Algorithms 8 (1994) 221–239; Inverse Problem 21 (2005) 2071–2084; J. Math. Anal. Appl. 327 (2007) 1244–1256], Byrne [Inverse Problem 18 (2002) 441–453], Yang [Inverse Problem 20 (2004) 1261–1266], Moudafi [Inverse Problem 26 (2010) 055007], Xu [Inverse Problem 26 (2010) 105018], Censor and Segal [J. Convex Anal. 16 (2009) 587–600], Masad and Reich [J. Nonlinear Convex Anal. 8 (2007) 367–371] and others.