Local Convergence for Some Third-Order Iterative Methods Under Weak Conditions

Abstract

Abstract. The solutions of equations are usually found using iterative methods whose convergence order is determined by Taylor expansions. In particular, the local convergence of the method we study in this paper is shown under hypotheses reaching the third derivative of the operator involved. These hypotheses limit the applicability of the method. In our study we show convergence of the method using only the first derivative. This way we expand the applicability of the method. Numerical examples show the applicability of our results in cases earlier results cannot.

Increasing the order of convergence for iterative methods in Banach space under weak conditions

“…Our analysis includes computable radius of convergence. error bounds and a uniqueness result not given in [19] or earlier similar works [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] using higher than one order Fréchet derivatives although these derivatives do not appear in the methods. Hence, we extend the applicability of these methods under weaker conditions.…”

“…For example consider the following; 1], Ω =B(x * , 1). Consider the nonlinear integral equation of the mixed Hammerstein-type [1,2,[6][7][8][9]12] defined by…”

“…Our goal is to weaken the assumptions in [19] and apply the method for solving equation (1.1) in Banach spaces, so that the applicability of the method (1.2) can be extended. This approach can be applied on other iterative methods [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Notice that earlier studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] also use hypotheses on higher order than two derivatives of H although these derivatives do not appear in the method.…”

“…This approach can be applied on other iterative methods [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Notice that earlier studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] also use hypotheses on higher order than two derivatives of H although these derivatives do not appear in the method. We also provide computable radius of convergence error bounds on the distances x n − x * and a uniqueness result based on Lipschitz-type conditions not given in [19] or related methods [1][2][3][4][5][6][7][8][9][10]…”

A Third Order Newton-Like Method and Its Applications

“…for each n ≥ 0, where F ′ x denotes the Fréchet derivative of F at point x ∈ D. The Newton method and the Newton-like methods are attractive because it converges rapidly from any sufficient initial guess. A number of researchers [7,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] have generalized and established local as well as semilocal convergence analysis of the Newton method (1.2) under the following conditions:…”