Journal of Convex Analysis 22 (2015), No. 1, 081--100
Copyright Heldermann Verlag 2015
Weak Convergence Theorems for Semigroups of Not Necessarily Continuous Mappings in Banach Spaces
Saud M. Alsulami
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
alsulami@kau.edu.sa
Nawab Hussain
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
nhusain@kau.edu.sa
Wataru Takahashi
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan
wataru@is.titech.ac.jp
We first introduce a broad semigroup of not necessarily continuous mappings in a Banach space which contains discrete semigroups generated by generalized nonspreading mappings and semigroups of φ-nonexpansive mappings. Then we establish a weak convergence theorem of Mann's type iteration for the semigroups of mappings in a Banach space. Using the result, we obtain well-known and new theorems which are connected with weak convergence results in Banach spaces.
Keywords: Attractive point, Banach space, fixed point, generalized nonspreading mapping, invariant mean, weak converegence, nonexpansive semigroup, strongly asymptotically invariant net.
MSC: 47H05, 47H09, 47H20
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