Asymptotically ɑ hemicontractive mappings in Hilbert spaces and a new algorithm for solving associated split common fixed point problem

Abstract

We introduce a novel class of asymptotically $\alpha-$hemicontractive mappings and demonstrate its relationship with the existing related families of mappings. We establish certain interesting properties of the fixed point set of the new class of mappings. Furthermore, we propose and investigate a new iterative algorithm for solving split common fixed point problem for the new class of mappings. In particular, weak and strong convergence theorems for solving split common fixed point problem for our new class of mappings in Hilbert spaces are proved. Moreover, using our method, we require no prior knowledge of norm of the transfer operator. The results presented in the paper extend and improve the results of Censor and Segal [Censor, Y.; Segal, A. The split common fixed point problem for directed operators. {\it J. Convex Anal.} {\bf 16 } (2009), no. 2, 587–600.], Moudafi [Moudafi, A. The split common fixed-point problem for demicontractive mappings. {\it Inverse Problems} {\bf 26} (2010), no. 5:055007.; Moudafi, A. A note on the split common fixed-point problem for quasi-nonexpansive operators. {\it Nonlinear Anal.} {\bf 74} (2011), no. 12, 4083–4087.], Chima and Osilike [Chima, E. E.; Osilike, M. O. Split common fixed point problem for class of asymptotically hemicontractive mappings. {\it J. Nigerian Math. Soc.} {\bf 38} (2019), no. 3, 363--390.], Fan \textit{et al} [Fan, Q.; Peng, J.; He, H. Weak and strong convergence theorems for the split common fixed point problem with demicontractive operators. {\it Optimization} {\bf 70} (2021), no. 5-6, 1409--1423.] and host of other related results in literature.