Extended Newton-like Midpoint Method for Solving Equations in Banach Space

: In this study, we present a convergence analysis of a Newton-like midpoint method for solving nonlinear equations in a Banach space setting. The semilocal convergence is analyzed in two different ways. The ﬁrst one is shown by replacing the existing conditions with weaker and tighter continuity conditions, thereby enhancing its applicability. The second one uses more general ω - continuity conditions and the majorizing principle. This approach includes only the ﬁrst order Fréchet derivative and is applicable for problems that were otherwise hard to solve by using approaches seen in the literature. Moreover, the local convergence is established along with the existence and uniqueness region of the solution. The method is useful for solving Engineering and Applied Science problems. The paper ends with numerical examples that show the applicability of our convergence theorems in cases not covered in earlier studies.


Introduction
One of the most challenging problems in Engineering and Applied Sciences is to determine a locally unique solution x * of a nonlinear equation where the operator F is defined on the Banach space B 1 with values in a Banach space B 2 .
As an example, engineering problems reduce to solving differential or integral equations, which in turn are set up as (1). A solution x * of the Equation (1) is difficult to find in closed form. That forces researchers to develop iterative methods, which generate iterations convergent to x * , provided that certain initial conditions hold. A popular iterative method is defined for each m = 0, 1, 2, . . . by This is the so-called Newton's method (NM), which is only quadratically convergent [1][2][3][4]. In order to increase the order of convergence as well as the efficiency, a plethora of iterative methods have been developed (see, e.g., [5][6][7][8][9] and references therein). Among those, special attention has been given to the Newton-like midpoint method (NLMM) defined by y m =x m − F x m−1 + y m−1 2 Novelty: Due to the importance of this method, the items (1)-(5) are positively addressed. The current study includes two procedures for analyzing the semilocal convergence of NLMM. The first analysis replaces the conditions used in [10] with weaker and tighter conditions, thereby enlarging the uniqueness region. In the second semilocal convergence, the convergence conditions used in the earlier section have been replaced by more generalized ω-continuity conditions using majorizing sequences [1,4,5,[11][12][13][14][15][16][17]. The main advantage of this approach is that it uses only the first derivative, which actually appears in NLMM, for proving the convergence result instead of the second derivative used in [10], thereby enhancing its applicability. Thus, our work improves the results derived in [10] under more stringent conditions and generates finer majorizing sequences. The innovation of the study lies in the fact that extensions are achieved under weaker conditions (see also Remarks throughout the paper). The local convergence of NLMM is also established, along with the existence and uniqueness region of the solution. Moreover, the new error analysis is finer, requiring fewer iterates to achieve a predetermined error tolerance. Furthermore, more precise information is provided on the uniqueness domain of the solution. Finally, the technique can be used on other methods utilizing the inverse of an operator.
The rest of the paper is structured as follows: Section 2 includes mathematical background. In Section 3, we develop the first kind of semilocal convergence theorem based on weaker conditions. The generalized ω-continuity conditions are applied to prove the second type of semilocal convergence theorem in Section 4. The local convergence, along with the uniqueness results of NLMM, is studied in Section 5. In Section 6, numerical examples are given to illustrate the theoretical results. Concluding remarks are reported in Section 7.

Mathematical Background
The study of the behavior of a certain cubic polynomial and the corresponding scalar Newton function play a role in the semilocal convergence of NLMM. Let L > 0, M ≥ 0 and d ≥ 0 be given parameters. Define the cubic polynomial the Newton iteration function and the scalar sequences {u m }, {v m } for The proof of the following auxiliary result containing some properties of q, N q , {v m }, and {u m } can be found in [10]. Lemma 1. Suppose: Then, the following assertions hold: (i) The polynomial q given by the Formula (4) has two zeros u * , u * * with 0 < u * ≤ u * * .
(iii) q is increasing and q is convex in [0, u * ].

Semilocal Convergence I
The following conditions relating the parameters L, M, d to NLMM have been used in the semilocal convergence. Suppose: (H 1 ) There exist an initial guess x 0 ∈ D and a parameter d ≥ 0 such that The following semilocal convergence result was shown in [2,10].
and converges to the only solution x * of the equation F(x) = 0 in U[x 0 , u * ]. (11) Moreover, the following error estimates hold: Next, the preceding results are extended without additional conditions. Suppose: Define the parameter and the region Clearly, we have and It is assumed without loss of generality Otherwise, the results that follow hold with K 0 replacing K. Notice also that the computation of the parameter K 1 requires the computation of K 0 , K and K 1 = K 1 (D), K 0 = K 0 (D), but K = K(D 0 , D). Hence, no additional conditions are required to develop the results that follow.
Let us consider the cubic polynomials and It follows by (4), (12), (13), (16) and (17) that and for each t ≥ 0. Suppose that The following auxiliary result is needed.

Lemma 2.
Suppose that the condition (H 4 ) holds. Then, the conclusions of Lemma 1 with K, q 1 replace L and q, respectively.

Remark 1.
It follows by (i) of Lemma 1 and (H 4 ) that the polynomial q 1 has two zeros u * , u * * with 0 < u * ≤ u * * . Moreover, if (H 4 ) holds, then by (13) and (20), Notice also that: (a) The parameter r is the unique positive zero of the equation and but not necessarily vice versa unless if K 1 = K. Hence, we arrived at the following extension of the Theorem 1.
Proof. The assertions (1)-(6) follow with the above changes. Concerning the assertion (7), Then, we have in turn that Notice that the last inequality in (29) follows from ( Then, from (H 2 ) and the definition of y 0 and u 0 The condition (H 5 ) in Theorem 2 can be replaced by where r is given by (10). Moreover, the uniqueness ball can be enlarged from U(x 0 , u * ) given in Theorem 1 to U(x 0 , r). This can be seen using the weaker condition (H 3 ) instead of (H 3 ) used in Theorem 1 in [10] or Theorem 2 used by us. Indeed, in Theorem 1, the estimate was obtained for y * ∈ U(x 0 , u * ), and consequently, y * = x * . However, the same estimate is obtained using the tighter condition (H 3 ) with q 1 , u * replacing q, u * , respectively. (c) The Lipschitz constant K can be replaced by an at least as small.
It follows by the Banach lemma on linear operators with inverses [3,5] and (30) that showing the assertion (7). Notice that in Theorem 1, the less tight estimate than (31) is shown under the stronger and not actually needed condition (H 3 ), which is In view of the estimate (31), the rest of the proof follows as in [10].
It follows that (H 3 ), K 2 can replace (H 3 ), K, respectively, in our results and The iterates {x m } ⊂ U(y 0 , r − d).

Semilocal Convergence II
The convergence conditions of the previous section may not hold, even if the method (3) converges. As a motivational example, consider the function f defined on the interval Then, clearly the function f is unbounded on D 2 . Hence, the results of the previous section cannot guarantee the convergence of method (3) to the solution x * = 1 ∈ D 2 . That is why we drop the conditions (H 2 )-(H 5 ), (H 3 ), (H 3 ), (H 4 ) and (H 5 ) and utilize the more general ω-continuity conditions, the first derivative that actually only appears on the method (3) and majorizing sequences to present another semilocal convergence result under weaker conditions.
Let h 0 : [0, +∞) → R be a continuous and nondecreasing function. Suppose that the equation h 0 (t) − 1 = 0 has a smallest positive solution s. Moreover, suppose that there exists a function h : [0, s) → R, which is continuous and nondecreasing. Let also parameters t 0 , s 0 , t 1 be such that t 0 = 0, s 0 > 0 and s 0 < t 1 . Define the sequences {t m }, {s m } by This sequence shall be shown to be majorizing for the method (3). However, a convergence result is developed first. Then, the following items hold: and The conditions connecting the "h "functions to the operators F and F are: Proof. By subtracting the first substep of the method from the second substep, we obtain in turn that showing the estimate (34). The estimate (38) follows from the identity which is obtained by the first substep of NLMM.
Next, the semilocal convergence is developed for the method (3).
Theorem 3. Suppose that the conditions (A 1 )-(A 5 ) hold. Then, the sequence {x m } converges to a solution x * ∈ U[x 0 , α] such that Proof. Mathematical induction is applied to show and The condition (A 1 ) and the Formula (4) (for m = 0) imply that the estimate (39) holds for m = 0.
Then, we also have the iterate y 0 ∈ U(x 0 , α). Moreover, Let v ∈ U(x 0 , α) be an arbitrary point. Then, the condition (A 2 ) gives That is, F (v) −1 ∈ L(B 2 , B 1 ) and In particular, if v = x 0 + y 0 2 , then the iterate y 1 and x 1 are well defined by the method (3). Moreover, the last condition in (A 1 ) and (33) give Thus, the assertion (40) holds for m = 0. Suppose that the assertions (39) and (40) held for all integers smaller than n − 1. Then, we obtain from Then, by (37), we obtain in turn that It follows from (33), (A 3 ), (42), (43) and the induction hypotheses that and Hence, (40) holds, and the iterate x m+1 ∈ U(x 0 , α). Furthermore, by (38) and the second substep of the method (3), we have in turn that Thus, the induction for the estimates (39) and (40) is terminated. However, the sequence {t m } is Cauchy as convergent by Lemma 3. Therefore, the sequence {x m } is also Cauchy in a Banach space B 1 , and as such, it converges to some x * ∈ U[x 0 , α], since this set is closed. Furthermore, by using the continuity of F and letting n → +∞ in the calculation, we conclude that F(x * ) = 0.
Concerning the uniqueness of the solution in a neighborhood about the point x 0 , we have: Then, it follows by condition (A 2 ) and (47) that which implies v * = z * .

Remark 4.
(1) If all the conditions of Theorem 3 hold, then we can set ϑ = α.
(2) The condition (A 5 ) can be replaced by (3) Suppose that d < s. Define the set Then, a tighter function h is obtained if D 5 replaces D 3 in the condition (A 2 ).

Local Convergence
We shall introduce some scalar functions and some parameters to show the local convergence analysis of NLMM. Suppose: (i) There exists a function ψ 1 : [0, +∞) → R, which is nondecreasing and continuous and a parameter M 2 ≥ 0 such that the function ψ 2 (t) − 1 has a smallest zero ρ 2 ∈ (0, +∞), where ψ 2 : [0, +∞) → R is defined by (ii) The function ψ 3 : [0, ρ 2 ) → R is such that The parameter r 3 shall be shown to be a radius of convergence for NLMM. The convergence conditions are: (l 1 ) There exists a simple solution x * ∈ Ω of the equation F(x) = 0 and a parameter Next, the local convergence is given for NLMM.
Proof. Let z ∈ U(x * , r 3 ). It follows by the conditions (l 1 ), (l 2 ) and the definition of the radius r 3 in turn that thus, F (z) −1 ∈ L(B 2 , B 1 ) and By hypothesis x 0 ∈ U(x * , r 3 ) and (49) for z = x 0 , we have that F (x 0 ) −1 ∈ L(B 2 , B 1 ) and the iterate y 0 is well defined by the first substep of NLMM. Moreover, we can write for By applying the condition (l 2 ) and (49) (for z = x 0 ) on (50), we obtain in turn that where we used that and θ ≤ 1 2 + 1 2 − θ . Thus, the iterate y 0 ∈ U(x * , r 3 ). Then, notice that r 3 ). Then, we also have so the iterates x m+1 and y m are well defined by NLMM. Then, we can write for However, the expression in the bracket can be written as In view of (49) (for z = z m ), (52), (53), we obtain where, for the numerator, we obtain where we also used y m − x * ≤ x m − x * . Similarly, the estimate we obtain as in (54) Hence, the iterates x m+1 , y m ∈ U(x * , r 3 ) for each m = 0, 1, 2, . . . . Then, from the estimates The uniqueness of the solution ball is determined in the next result.

Numerical Examples
In this section, some numerical examples are solved in order to corroborate the theoretical results obtained and the efficacy of our approach. Example 1. Let B 1 = B 2 and Ω = U(x 0 , 1 − γ) for some parameter γ ∈ (0, 1). Define the polynomial F on the interval Ω by Choose x 0 = 1. Then, if we substitute F on the "h "conditions, we see that the conditions , and M = 3 1 − γ .
where v = (ϑ 1 , ϑ 2 ) T . We shall find a solution to the equation F(v) = 0. The first and second-order Fréchet derivatives are calculated to be . Pick x 0 = (11.4, 11.4) T . Then, it follows that d = 1.9826, M = 0.47618, and K ≥ 0 can be arbitrary. By setting K = 10 −3 , we see that d < λ(K, M). Moreover, the solution x * = (9, 9) T is obtained after m = 3 iterations. Notice that it takes m = 5 iterations for NM but only three for NLMM to reach x * . Thus, this method requires fewer computations than that of Newton's method.
The numerical examples were simulated by using Mathematica 8 on Intel(R) Core(TM) i5-8250U CPU @ 1.60 GHz 1.80 GHz, with 8 GB of RAM running on Windows 10 Pro version 2017. This kind of local and semilocal convergence demonstrates that the guarantee the existence and uniqueness of the solution are especially valuable in processes where it is difficult to prove the existence of solutions.

Conclusions
The present study deals with new local and semilocal convergence results for the Newton-like midpoint method under improved initial conditions. In the first type of semilocal convergence, the previous results are extended without using any additional postulates. The estimate obtained in [10] is less tight and uses stronger conditions in comparison to our results. The second semilocal convergence utilizes more general ωcontinuity conditions and can be applied to the problems where earlier conditions fail. Notice also that the condition on F is dropped (see also the Example 1). Both semilocal convergence results are computationally verifiable and improve the previous study [10] in several directions, which are of practical importance. The local convergence theorem not given in [10] is established for the existence-uniqueness of the solution. We present varied numerical examples to show the applicability of our results. The innovation demonstrated that NLMM can also be used to extend the applicability of other methods requiring the inversion of a linear operator in an analogous way since our technique is method-free. This is the future area of research.
Author Contributions: The authors contributed equally. All authors have read and agreed to the published version of the manuscript.