Cyclic algorithm for common fixed points of finite family of strictly pseudocontractive mappings of Browder–Petryshyn type

Abstract

Let E be a real 2-uniformly smooth Banach space which is also uniformly convex (for example: L p or l p , 2 ≤ p < ∞ ) and K a nonempty closed convex subset of E . Let T 1 , T 2 , … , T N : K → K be strictly pseudocontractive mappings of K into K in the sense of Browder and Petryshyn with ∩ i = 1 N F ( T i ) ≠ 0̸ , where F ( T i ) = { x ∈ K : T i x = x } . Let { α n } n = 1 ∞ be a real sequence in [0,1] satisfying the condition 0 < a ≤ ( 1 − α n ) ≤ b < 2 λ c , ∀ n ≥ 1 and for some constants a , b , λ ∈ ( 0 , 1 ) and c ≥ 1 . Let { x n } be the sequence generated from an arbitrary x 1 ∈ K by x n + 1 = α n x n + ( 1 − α n ) T [ n ] x n , n ≥ 1 where T [ n ] = T i , i = n ( mod N ) , 1 ≤ i ≤ N . Weak and strong convergence theorems for the iterative approximation of common fixed points of the family { T i } i = 1 N are proved using the iterative sequence { x n } n = 1 ∞ . Furthermore, if E is an arbitrary real Banach space, it is shown that lim inf n → ∞ ‖ x n − T i x n ‖ = 0 for all i = 1 , 2 , … , N ; and a necessary and sufficient condition that guarantees the strong convergence of { x n } to a common fixed point of the family { T i } is given.