On shrinking projection method for cutter type mappings with nonsummable errors

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276-286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303-311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301-310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105-120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301-310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105-120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space.