Composite Halpern Iteration for Approximating Common Zero of Countably Infinite Family of m-accretive Mappings
Ronghua Zhang · Journal of Civil Aviation University of China · 2009
Let K be a closed convex nonempty subset of a strictly convex and real reflexive Banach space E which has a uniformly Gateaux differentiable norm.Let Ai ∶ K→E(i∈N) be a countably infinite family of m-accretive mappings such that ∩∞i=1 N(Ai)≠φ.For arbitrary u,x1∈K,{αn}n∞=1 and {βn}n∞=1 are real sequences in the [0,1] satisfying the conditions:(i)limn→∞αn= 0,∑n∞=1 αn= ∞,∑n∞=1 |αn+1-αn| ∞;(ii)limn→∞βn= 0,∑n∞=1 |βn+1-βn| ∞。 {xn}n∞=1be composite Halpern iteration definited by: yn = βnxn +(1-βn)Sxn,n≥1 xn + 1 = αnu +(1-αn)yn≥ Where S = ∑∞i=1 ξi JAi,JAi=(I + Ai)-1(i∈N),then,{xn}n∞=1 converges strongly to a common zero of {Ai}i∈N.The results presented in this paper improve and extend the correspoinding ones of Hegeye and Shahzad and Ofoedu and others.