Common Fixed Points Approximation for Asymptotically Nonexpansive Semigroup in Banach Spaces

Abstract EN

Let E be a real Banach space satisfying local uniform Opial's condition, whose duality map is weakly sequentially continuous.

Let J:={T(t) t≥0} be a uniformly asymptotically regular family of asymptotically nonexpansive semigroup of E with function k:[0,∞)→[0,∞).

Let ℱ:=⋂t≥0F(T(t))≠∅ and f:E→E be weakly contractive map.

Let G:E→E be δ-strongly accretive and λ-strictly pseudocontractive map with δ+λ>1.

Let {tn} be an increasing sequence in [0,∞)and let{αn} and {βn} be sequences in [0,1] satisfying some conditions.

For some positive real number γ appropriately chosen, let {xn} be a sequence defined by x0∈E, xn+1=βnxn+(1−βn)yn, yn=(I−αnG)T(tn)xn+αnγf(xn), n≥0.

It is proved that {xn} converges strongly to a common fixed point q of the family J which is also the unique solution of the variational inequality 〈(G−γf)q,j(q−x)〉≥0, for all x∈ℱ.