Continuous-modified-Newton’s-type-method-for-nonlinear-operator-equations.pdfVIP

Continuous modified Newton’s-type method for nonlinear operator equations Alexander G. Ramm 1 Alexandra B. Smirnova E-mail: ramm@math.ksu.edu E-mail: smirn@cs.gsu.edu Department of Mathematics Department of Math and Stat Kansas State University Georgia State University Manhattan, KS 66506, USA Atlanta, GA 30303, USA Angelo Favini E-mail: favini@dm.unibo.it Dipartimento di Matematica Universita di Bologna 540127 Bologna, Italy Abstract A nonlinear operator equation F (x) = 0, F : H → H, in a Hilbert space is considered. Continuous Newton’s-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point x0 and becoming asymptotically close to a solution of F (x) = 0 as t → +∞ are discussed. Well-posed and ill-posed problems are investigated. Key words: nonlinear problem, integral inequality, Fr´echet derivative, Newton method. AMS subject classification: 65J15, 58C15, 47H17 1. Introduction The theme of this paper is solving nonlinear operator equations of the form: (1.1) F (x) = 0, F : H → H, in a real Hilbert space H . We consider a real Hilbert space for the sake of simplicity: numerical algorithms for solving (1.1) in a complex Hilbert space can be treated similarly. In order to approximate a solution to equation (1.1) we use the idea developed in [2]-[4], wh