On the Gauss-Newton method for solving equations

Abstract

We use a combination of the center-Lipschitz condition with the Lipschitz condition condition on the Fréchet-derivative of the operator involved to provide a semilocal convergence analysis of the GaussNewton method to a solution of an equation. Using more precise estimates on the distances involved, under weaker hypotheses, and under the same computational cost, we provide an analysis of the GaussNewton method with the following advantages over the corresponding results in [8]: larger convergence domain; finer error estimates on the distances involved, and an at least as precise information on the location of the solution AMS Subject Classification. 65F20, 65G99, 65H10, 49M15.

Weak convergence of an iterative algorithm for accretive operators

“…In this paper, we will focus our attention on problem (1.1), which is very general in the sense that it includes, as special cases, convexly constrained linear inverse problems, split feasibility problem, convexly constrained minimization problems, fixed point problems, variational inequalities, Nash equilibrium problem in noncooperative games and others; see, for instance, [2,3,9,10,19] and the references therein. Because of their importance, forward-backward splitting methods, which were proposed by Passty [18], and, in a dual form for convex programming, by Han and Lou [13], for solving (1.1) have been studied extensively recently; see, for instance, [4,15,20,21,25] and the references therein.…”

A general iterative algorithm for common solutions of quasi variational inclusion and fixed point problems

“…As a result of the interaction between different branches of mathematical and engineering sciences, we now have a variety of techniques to suggest and analyze various iterative algorithms for solving these problems and related convex optimization problems. Variational inclusions involving two operators are useful and important extension and generalizations of the variational inequalities with a wide range of applications in economics, decision sciences, network, mathematical, and engineering sciences, see [1,2,6,18,21,[25][26][27] and the references therein. It is well known that the projection method and its variant forms including the Wiener-Hopf equations cannot be extended and modified for solving the variational inclusions, which motivate us to use new techniques and methods.…”

Demiclosed principals and convergence theorems for asymptotically pseudocontractive nonself-mappings in intermediate sense

“…Fixed point theory of nonlinear operators provides us with a general and unified framework in which to study a wide class of problems arising in pure and applied sciences; see, for example, [1,10,18,20,24] and the references therein. In addition to the existence results, many authors have extensively investigated the approximation of fixed points of nonlinear operators via various kind of iterative processes, in particular, the mean valued iterative process; see, [2,3,6,7,19], and the references therein.…”