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

Let E be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm, D a nonempty closed convex subset of E , and T : D → K ( E ) a nonself multimap such that F ( T ) ≠ 0 and P T is nonexpansive, where F ( T ) is the fixed point set of T , K ( E ) is the family of nonempty compact subsets of E and P T ( x ) = { u x ∈ T x : ‖ x − u x ‖ = d ( x , T x ) } . Suppose that D is a nonexpansive retract of E and that for each v ∈ D and t ∈ ( 0 , 1 ) , the contraction S t defined by S t x = t P T x + ( 1 − t ) v has a fixed point x t ∈ D . Let { α n } , { β n } and { γ n } be three real sequences in ( 0 , 1 ) satisfying approximate conditions. Then for fixed u ∈ D and arbitrary x 0 ∈ D , the sequence { x n } generated by x n ∈ α n u + β n x n − 1 + γ n P T ( x n ) , ∀ n ≥ 0 , converges strongly to a fixed point of T .