Abstract
AbstractWe discuss the following viscosity approximations with the weak contraction "Equation missing" for a non-expansive mapping sequence "Equation missing", "Equation missing", "Equation missing". We prove that Browder's and Halpern's type convergence theorems imply Moudafi's viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern's type iteration and Mouda's viscosity approximations with the weak contraction.
Highlights
The following famous theorem is referred to as the Banach Contraction Principle.Theorem 1.1 Banach 1
Let E, d be a complete metric space and let A be a contraction on X, that is, there exists β ∈ 0, 1 such that d Ax, Ay ≤ βd x, y, ∀x, y ∈ E
Let E, d be a complete metric space, and let A be a weak contraction on E, that is, d Ax, Ay ≤ d x, y − φ d x, y, ∀x, y ∈ E, 1.2
Summary
The following famous theorem is referred to as the Banach Contraction Principle. Theorem 1.1 Banach 1. In 2000, for a nonexpansive selfmapping T with Fix T / ∅ and a fixed contractive selfmapping f, Moudafi 14 introduced the following viscosity approximation method for T: xn 1 αnf xn 1 − αn T xn, 1.8 and proved that {xn} converges to a fixed point p of T in a Hilbert space. They are very important because they are applied to convex optimization, linear programming, monotone inclusions, and elliptic differential equations. We will prove that Browder’s and Halpern’s type convergence theorems imply Moudafi’s viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern’s type iteration and Moudafi’s viscosity approximations with the weak contraction
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