Strong convergence of an iterative algorithm for nonself multimaps in Banach spaces

Let EE be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm, DD a nonempty closed convex subset of EE, and T:D→K(E)T:D→K(E) a nonself multimap such that F(T)≠0̸F(T)≠0̸ and PTPT is nonexpansive, where F(T)F(T) is the fixed point set of TT, K(E)K(E) is the family of nonempty compact subsets of EE and PT(x)={ux∈Tx:‖x−ux‖=d(x,Tx)}PT(x)={ux∈Tx:‖x−ux‖=d(x,Tx)}. Suppose that DD is a nonexpansive retract of EE and that for each v∈Dv∈D and t∈(0,1)t∈(0,1), the contraction StSt defined by Stx=tPTx+(1−t)vStx=tPTx+(1−t)v has a fixed point xt∈Dxt∈D. Let {αn},{βn}{αn},{βn} and {γn}{γn} be three real sequences in (0,1)(0,1) satisfying approximate conditions. Then for fixed u∈Du∈D and arbitrary x0∈Dx0∈D, the sequence {xn}{xn} generated by xn∈αnu+βnxn−1+γnPT(xn),∀n≥0, converges strongly to a fixed point of TT.