On the Monotone Convergence of Implicit Newton-like Methods

. In this study, we examine the convergence of implicit Newton-like methods to a solution of a nonlinear equation on a partially ordered topological space. We generate two sequences, one increasing and one decreasing that enclose a solution of the nonlinear equation. Finally, we provide an example that shows that our method converges whereas the standard explicit Newton-like method does not. 1991 A.M.S. (MOS) Subject Classification Codes. 65J15, 65B05, 47H17, 49D15. Key Words and Phrases. Newton-like method, monotone convergence, partially ordered topological space. I. INTRODUCTION In this study, we are concerned with the problem of approximating a solution z of the nonlinear operator equation (1) F (x) = 0 where F is a continuous nonlinear operator defined on a subset D of a partially ordered topological space (POTL-space) E 1 with values in a POTL-space E 2 [2], [3], [5]. We introduce implicit Newton-like methods of the form (2) F (xn+1 ) + A 1 (xn\Gamma1 ; xn )(x n+1 \Gamma ..