Strong convergence of Picard and Mann iterations for strongly demicontractive multi-valued mappings

Abstract

A class of demicontractive mappings was first introduced in [Hicks, T. L. and Kubicek, J. D.,On the Mann ite-ration process in a Hilbert space, J. Math. Anal. Appl.,59(1977) 498–504 and M ̆arus ̧ter, S ̧ .,The solution by iterationof nonlinear equations in Hilbert spaces, Proc. Amer. Math. Soc.,63(1977), 69–73] and was first mentioned in thecase of multi-valued mappings in [Chidume, C. E., Bello, A. U. and Ndambomve, P.,Strong and∆-convergencetheorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces, Abstr.Appl. Anal.,2014(2014), https://doi.org/10.1155/2014/805168 and Isiogugu, F. O. and Osilike, M. O.,Conver-gence theorems for new classes of multivalued hemicontractive-type mappings, Fixed Point Theory Appl.,2014(2014),https://doi.org/10.1186/1687-1812-2014-93]. The demicontractivity with some weak smoothness conditionsensures only weak convergence of Mann iteration. In 2015, M ̆arus ̧ter and Rus [Kannan contractions and stronglydemicontractive mappings, Creat. Math. Inform.,24(2015), No. 2, 173–182], introduced a class of strongly de-micontractive mappings, and also discussed some relationships between strongly demicontractive mappingsand Kannan contractions. In this paper, we introduce a new class of strongly demicontractive multi-valuedmappings in Hilbert spaces. Strong convergence theorems of Picard and Mann iterative methods for stronglydemicontractive multi-valued mappings are established under some suitable coefficients and control sequences.