Boundary Point Method and Mann-Dotson’s Algorithm for Nonself Strict Pseudo-contractive Mappings in Uniformly Smooth Banach Spaces
Giuseppe Marino, Luigi Muglia · Numerical Functional Analysis and Optimization · 2019
Let C be a closed, convex and nonempty subset of a q-uniformly smooth Banach space X. Let T:C→X be a k−strict pseducontractive inward mapping. We consider the boundary point map hC,T:C→ℝ depending on C and T defined by hC,T(x)=max{λ∈[0,1]:[(1−λ)x+λTx]∈C}, for all x∈C. Then for a suitable step-by-step construction of the control coefficients (αn)n∈ℕ by using the function hC,T, we show the convergence of the Mann-Dotson algorithm to a fixed point of T. We obtain strong convergence if ∑nαn<∞ and weak convergence if ∑nαn=∞.