Accessibility of solutions of operator equations by Newton-like methods

Abstract

The concept of a majorizing sequence introduced and applied by Rheinboldt in 1968 is taken up to develop a convergence theory of the Picard iteration xn+1=G(xn) for each n≥0 for fixed points of an iteration mapping G: D0⊂X→X in a complete metric space X satisfying iterated contraction-like condition: d(G(y), G(x)) ≤ ψ(d(y, x), d(y, x0), d(x, x0))d(y, x) for all x ∈ D0 with y=G(x) ∈ D0, where x0 and ψ ∈ Φ(3). Here 3 is a suitable set of (ℝ+)3 to be defined in Section 2. We study the region of accessibility of fixed points of G by the Picard iteration un+1=G(un), where the starting point u0 ∈ D0 is not necessarily x0. Our convergence theory is applied to the Newton-like iterations in Banach spaces under the center Lipschitz condition ∥ Fx′-Fx0′∥≤ ω(x-x0) for a given point x0 ∈ D0. Our results extend and improve the previous ones in the sense of the center Lipschitz condition and the region of accessibility of solutions. We apply our results to solve the nonlinear Fredholm operator equations of second kind. © 2015 Elsevier Inc.