Strong convergence theorems for uniformly continuous pseudocontractive maps

Abstract

Let K be a nonempty closed convex subset of a real Banach space E and let T : K → K be a uniformly continuous pseudocontraction. Fix any u ∈ K . Let { x n } be defined by the iterative process: x 0 ∈ K , x n + 1 : = μ n ( α n T x n + ( 1 − α n ) x n ) + ( 1 − μ n ) u . Let δ ( ϵ ) denote the modulus of continuity of T with pseudo-inverse ϕ. If { ϕ ( t ) t : 0 < t < 1 } and { x n } are bounded then, under some mild conditions on the sequences { α n } n and { μ n } n , the strong convergence of { x n } to a fixed point of T is proved. In the special case where T is Lipschitz, it is shown that the boundedness assumptions on { ϕ ( t ) t : 0 < t < 1 } and { x n } can be dispensed with.