Abstract
We show that the convergence of Picard iteration is equivalent to the convergence of Mann iteration schemes for various Zamfirescu operators. Our result extends of Soltuz (2005).
Highlights
Let E be a real normed space, D a nonempty convex subset of E, and T a self-map of D, let p0, u0, x0 ∈ D
The Mann iteration is defined by un+1 = 1 − an un + anTun, n ≥ 0
The Ishikawa iteration is defined by yn = 1 − bn xn + bnTxn, n ≥ 0, xn+1 = 1 − an xn + anT yn, n ≥ 0, (1.3)
Summary
Let E be a real normed space, D a nonempty convex subset of E, and T a self-map of D, let p0, u0, x0 ∈ D. 2 Fixed Point Theory and Applications if, for each pair x, y in D, T satisfies at least one of the following conditions given in (1)–(3): (1) Tx − T y ≤ a x − y ; (2) Tx − T y ≤ b( x − Tx + y − T y ); (3) Tx − T y ≤ c( x − T y + y − Tx ). Soltuz [1] had studied that the equivalence of convergence for Picard, Mann, and Ishikawa iterations, and proved the following results.
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