Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.


Introduction
Let B 1 and B 2 be Banach spaces and Ω be a convex subset of B 1 .Further, suppose that L(B 1 , B 2 ) is the set of bounded linear operators from B 1 into B 2 .In applied mathematics many problems can be modeled in the form wherein F: Ω ⊂ B 1 → B 2 is a Fréchet differentiable operator.Most of the methods for finding a solution x * of Equation ( 1) are iterative, since closed form solutions can be found only in some special cases (see [1,2]).In this paper, we study the local convergence of the method defined by where Q(x n ) = F (x n ) −1 F (y n ).Method (2) was studied in [3], when , where m is a positive integer.The method was compared favorably to existing higher-order methods.
The eighth-order convergence of Equation ( 2) was shown in [3] using Taylor expansions and assumptions on F (i) , i = 1, 2, ...., 8.Such assumptions restrict the applicability of this method, especially since only the first derivative is used in the method.As a motivational example, define function F on Ω = [− 5  2 , 2] and We have that and Notice that F (x) is not bounded on Ω, so earlier results cannot be applied.In this work, our goal is to weaken the assumptions considered in [3].Consequently, we study the local convergence of Method (2) using hypotheses on the first Fréchet-derivative only by taking advantage of the Lipschitz continuity of the first Fréchet-derivative.There exist many studies which deal with the local and semilocal convergence of iterative methods (see, for example, [2,[4][5][6][7][8][9][10][11][12][13][14][15][16]).In particular, relevant work can be found in [17] for the special case The rest of the paper is structured as follows.In Section 2, the local convergence analysis is studied.In the analysis, we also provide a radius of convergence, error bounds, and a uniqueness result of Method (2).Some numerical examples are presented in Section 3. Concluding remarks are given in Section 4.
In the sequel, we present local convergence analysis of Method (2) using the preceding notation and the hypotheses (H).
Theorem 1. Suppose that the hypotheses (H) hold and we choose x 0 ∈ U(x * , r * ) − {x * }.Then, sequence {x n } starting at x 0 and generated by Method (2) is well defined, remains in U(x * , r) for every n = 0, 1, 2......, and converges to x * .Additionally, the following error bounds hold and where functions ψ i are given previously and r is defined in Equation (6).Furthermore, the limit point x * is only a solution of the equation F(x) = 0 in Ω 1 given in (h 5 ).

Conclusions
In the forgoing study, we have studied the local convergence of an efficient eighth-order method by assuming conditions only on the first derivative of the operator.The iterative scheme does not use second or higher-order derivative of the considered function.However, in an earlier study of convergence, the hypotheses used were based on Taylor series expansions reaching up to the eighthor higher-order derivatives of function, although the iterative scheme uses first-order derivative.These conditions restrict the usage of the iterative scheme.We have extended the suitability of the method by considering suppositions only on the first-order derivative.The local convergence we have studied is also important in the sense that it provides estimates on the radius of convergence and the error bounds of the solution.Such estimates are not provided in the procedures that use Taylor expansions of higher derivatives, which may not exist or may be very expensive to compute.We have also verified the theoretical results so derived on some numerical problems.

( 10 )
Denote by U(µ, a) the open ball in B 1 with center µ ∈ B 1 and of radius a > 0.Moreover, denote by Ū(µ, a) the closure of U(µ, a).