A Harmonic-Type Method for Nonlinear Equations in Banach Space

: In this work, we investigate the local and semi-local convergence of a harmonic mean Newton-type fourth-order technique for estimating the locally unique solutions of nonlinear systems in Banach spaces. The local analysis is established in previous works under assumptions reaching the fifth derivative of the involved operator. Therefore, the applicability of the method is restricted to solving nonlinear equations containing operators that are at least five times differentiable. However, this method may converge even if these assumptions are not satisfied. Other limitations include the lack of a priori error estimates and the isolation of the solution results. The local analysis in this work is shown using only the first derivative of the method. Moreover, a priori estimates on the error distances and uniqueness results are provided based on generated continuity assumptions on the Fréchet derivative of the operator. Furthermore, the more interesting semi-local case not studied previously is developed by means of majorizing sequences. The analysis in both cases is given not in the finite-dimensional Euclidean but in the more general setting of Banach spaces. Some numerical tests are performed to validate the theoretical results further.


Introduction
Let S 1 and S 2 be Banach spaces, and let S denote a nonempty and open subset of the space S 1 .Moreover, the notation L(S 1 , S 2 ) denotes the space of continuous operators from S 1 into S 2 , which are linear.Furthermore, H is an operator between S and S 2 , which is differentiable according to Fréchet.

H x =
The determination of a solution x * ∈ S of (1) is a very challenging undertaking in general.The analytical form of x * is most desirable.However, that is fulfilled only on rare occasions.Thus, iterative solution methods have been developed, when starting from a certain point x 0 ∈ S, a sequence is generated that approximates x * (see, for example, [4][5][6][7]).
Among those, we select the two-step Harmonic mean method (TSHM) given as  The convergence order of TSHM was established to be four in [12].This method is the generalization of one of the members of the family of higher-order multi-point methods based on the power mean for solving a single nonlinear equation by Babajee et al. [15].The benefit of using this method is that it does not require the evaluation of second-or higher-order Fréchet derivatives.Such derivatives are costlier from a computational point of view.This property of a method makes it useful to solve large-scale systems of nonlinear equations.The performance of the comparison with other methods is well explained in [12].The operator H' satisfies generalized continuity conditions, which include the Lipschitz or Hölder continuity as special cases [2].These properties of the operators H and H' are used in conditions (C 1 )-(C 3 ) and (H 1 )-(H 5 ), which are stated before the main theorems.
The following problems-limitations constitute our motivation for writing this article: The operator H must be at least five times differentiable.Notice that only H' appears on the method.(P3) There is no advanced knowledge of how many iterations should be found to achieve pre-decided accuracy.That is, computable upper and lower bounds on the norms ||x m +1 − x m || or ξ m = ||x m − x * || are not available.
(P 4 ) A computable uniqueness domain for x * is not determined.(P 5 ) The results are only local.(P 6 ) The selection of the starting guess x 0 is very challenging.According to the problem (P 2 ), there exist even simple scalar equations that cannot be solved using TSHM.As an example, let S be any open interval containing the numbers 0 and 1. Define the real function: Notice that f (1) = 0.But the function f (5) (τ) is not continuous at τ = 0. Thus, the result in [12] cannot be applied, although TSHM may converge.
Similar problems exist in the application of other high-convergence order methods [5,6,[8][9][10][11]15].The novelty of our article is that the applicability of TSHM is extended to local as well as semi-local convergence by relying only on F and F', which are on (3), and the concept of w-continuity [2].
In particular, extensions to problems (P 1 )-(P 6 ) are: (F 1 ) The convergence takes place in a Banach space.(F 2 ) Operator H is assumed to be only once differentiable.Notice also that only the first derivative of H appears on TSHM.Moreover, the w-continuity conditions are very weak and cover a wide range of problems.
(F 3 ) We know in advance the iterations needed to get pre-decided accuracy.(F 4 ) Some computable uniqueness region is specified.
The new results are local and semi-local.(F 6 ) The radius of convergence is determined, making it possible to select initial points starting from which the convergence to x * is assumed.
(F 7 ) The convergence order is recovered by the formulas: order of computational convergence (OCC) and order of approximate convergence (OAC), defined respectively by where d i = x i − x * and 1 .j j j d x x − = − Notice that no high-order derivatives are needed but only the iterates and x * .Moreover, in the case of OAC, the iterate x * is not involved.
Let k be a fixed natural number.Then, these results also extend the application of the extended TSHM (ETSHM) given as The notation zi = zi(xn) shall also be used.It was shown in [12] that the convergence order is 2k + 4, k = 1, 2, ... .Although our approach is employed to handle TSHM and ETSHM, it also applies to other methods requiring linear operators with inverses, since it is so general and does not really depend on these two methods.
The convergence analysis for TSHM and ETSHM is developed in Section 2 and Section 3, respectively.Numerical tests and conclusions appear in Section 4 and Section 5, respectively.

Convergence for TSHM
In what follows, the local and semi-local convergence is studied.The former relies on some real functions and the latter on the concept of a majoring sequence [16].
The functions φ0, φ, and the radius R are connected to the operators on TSHM if x * ∈ S is a simple solution for the equation H(x) = 0 as follows. Suppose: We are equipped to show the local convergence of TSHM.

( )
, where the radius R is defined by the formula (8) and the functions h 1 and h 2 are as previously given.

Semi-local convergence
Let ψ 0 and ψ be continuous and nondecreasing real functions defined on the intervals D and D 1 , respectively.Moreover, define the scaler sequences {am}, {βm} for t 0 = 0, some s 0 ≥ 0 and each m = 0, 1, 2, ... by ) Proof.It follows from the formula (28) and the conditions (29) that the assertion (30) holds.Then, the assertion (31) follows (30).Remark 2.5 (i) The limit t * is the unique least upper bound of the sequences {α m }.
Next, the uniqueness of the solution region is specified.Proposition 2.7 Suppose: There exists a solution _ x ∈ U(x 0 , r 0 ) of the equation H(x) = 0 for some r 0 > 0; the condition (H 2 ) holds on the ball U(x 0 , r) and there exists r 1 > r 0 , such that ( ) Thus, we conclude that _ y = _ x.Remark 2.8 (i) The point t * may be switched with δ or t ** in the condition (H 5 ).(ii) Under Theorem 2.6, we can take _ x = x * and r 0 = t * in Proposition 2.7.

Convergence for ETSHM
As in the local convergence of TSHM, some real functions are defined.Define the functions on interval D 1 by By Theorem 2.1, we have ( ) Hence, by the rest of the substeps, we get in turn that ( )

Experiments
We take into account some numerical examples to estimate the real parameters in order to validate the theoretical deductions.The parameters R 1 and R 2 are obtained as solutions of the nonlinear equations h 1 (t) − 1 = 0, h 2 (t) − 1 = 0, defined earlier, respectively, for specialized functions φ 0 and φ.Then, the radius R is given by the formula (8).Numerical computations are performed in Mathematica software, with multi-precision arithmetic that uses a floatingpoint representation of 50 decimal digits of mantissa.

Conclusion
This paper thoroughly discusses the convergence (local and semi-local) of a fourth-order technique in Banach space.We have simply taken into account the first derivative in our process, unlike other techniques that rely on higher derivatives and the Taylor series.In this way, the method's uses are broadened because it may be used for a wider class of functions.The production of an error estimate and convergence ball, within which the iterates lie, is another benefit of convergence analysis.The theoretical outcomes of this approach are verified through numerical testing on a few problems.The idea is applicable to other methods cited in the introduction and can be extended to handle more advanced equations [19].
the only solution of the equation H(x) = 0 in the region S 2 is x * .Proof.Let 2 y S ∈ be a solution of the equation H(x) = 0. Consider the operator 1 0 (

r 1 ]
. Then, the point _ x is the only solution of the equation H(x) = 0 in the region S 4 .Proof.Let _ y ∈ S 4 be a solution of the equation H(x) = 0. Consider the operator

3 . 1
equation d(τ) − 1 = 0 has the smallest solution in the interval (0, R].Denote such a solution by R * .The condition (C 3 ) is replaced by have the local convergence analysis for ETSHM.Theorem Under the conditions (C 1 ), (C 2 ), and (C 3 )', the sequence {x m } is convergent to x * .Proof.As in Theorem 2.1, we have in turn the estimates ( )