CUBO, A Mathematical Journal Vol. 23, no 01, pp. 97–108, April 2021 DOI: 10.4067/S0719-06462021000100097 Extended domain for fifth convergence order schemes Ioannis K. Argyros1 Santhosh George2 1 Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA. iargyros@cameron.edu 2 Department of Mathematical and Computational Sciences, NIT Karnataka, India-575 025. sgeorge@nitk.edu.in ABSTRACT We provide a local as well as a semi-local analysis of a fifth convergence order scheme involving operators valued on Ba- nach space for solving nonlinear equations. The convergence domain is extended resulting a finer convergence analysis for both types. This is achieved by locating a smaller domain included in the older domain leading this way to tighter Lip- schitz type functions. These extensions are obtained without additional hypotheses. Numerical examples are used to test the convergence criteria and also to show the superiority for our results over earlier ones. Our idea can be utilized to ex- tend other schemes using inverses in a similar way. RESUMEN Entregamos un análisis local y uno semi-local de un esquema de quinto orden de convergencia que involucra operadores con valores en un espacio de Banach para resolver ecuaciones no- lineales. El dominio de convergencia es extendido resultando en un análisis de convergencia más fino para ambos tipos. Esto se logra ubicando un dominio más pequeño incluido en el dominio antiguo, entregando funciones de tipo Lipschitz más ajustadas. Estas extensiones se obtienen sin hipótesis adicionales. Se usan ejemplos numéricos para verificar los cri- terios de convergencia y también para mostrar que nuestros resultados son superiores a otros anteriores. Nuestra idea se puede utilizar para extender otros esquemas usando inversos de manera similar. Keywords and Phrases: Fifth order convergence scheme, w-continuity, convergence analysis, Fréchet derivative, Banach space. 2020 AMS Mathematics Subject Classification: 65H10, 47H17, 49M15, 65D10, 65G99. Accepted: 25 January, 2021 Received: 02 June, 2020 ©2021 I. K. Argyros et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000100097 http://orcid.org/0000-0002-9189-9298 http://orcid.org/0000-0002-3530-5539 98 I. K. Argyros & S. George CUBO 23, 1 (2021) 1 Introduction In this article, B1,B2 are standing for Banach spaces, D ⊂ B1 is denoting a convex and open set, and F : D −→ B2 is considered differentiable according to the Fréchet notion. One of the most important tasks is the location of a solution x∗ of nonlinear equation F(x) = 0. (1.1) Solving equation F(x) = 0 is useful because using modeling (Mathematical) problems from many areas can be formulated as (1.1). The explosion of technology requires the development of higher convergence schemes. Starting from the quadratically convergent Newton’s method higher order schemes develop all the time [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Recently, Singh et al. [13] provided a semi-local convergence for efficient fifth order scheme under Lipschitz continuity on F ′′ defined as follows yn = xn −F ′(xn)−1F(xn) zn = yn −F ′(xn)−1F(yn) (1.2) xn+1 = zn −F ′(yn)−1F(zn). Later in [14] the applicability of scheme (1.2) was extended using w- continuity conditions. In general, the convergence domain is small. That is why we develop a technique where a tighter domain than before is obtained containing the iterates. This way the new w-functions are tighter leading to a finer semi-local convergence analysis. It is worth noticing that these extensions do not involve new hypotheses because the new w-functions are specializations of the old one. Hence, we extend the applicability of the method. It turns out that the local convergence analysis can be extended too. For example: Let B1 = B2 = R, Ω = [−12, 3 2 ]. Define G on Ω by G(x) =   x 3 log x2 + x5 −x4, x 6= 0 0, x = 0. Then, we get x∗ = 1, and G′(x) = 3x2 log x2 + 5x4 − 4x3 + 2x2, G′′(x) = 6x log x2 + 20x3 − 12x2 + 10x, G′′′(x) = 6 log x2 + 60x2 − 24x + 22. Obviously G′′′(x) is not bounded on Ω. So, the convergence of scheme (1.2) is not guaranteed by the analysis in [13, 14]. In this study we use only assumptions on the first derivative to prove our results. Relevant studies can be found in [6, 19]. The structure of the rest of the article involves local and semi-local convergence analysis in Section 2 and Section 3, respectively. The numerical experiments appear in Section 4. CUBO 23, 1 (2021) Extended domain for fifth convergence order schemes 99 2 Local convergence It is easier for the local convergence of method (1.2), if we develop some real functions. We start with a function ω0 defined on the interval I = [0,∞) with values in I satisfying ω0(0) = 0. Assume equation ω0(t) = 1 (2.1) has a least positive solution called ρ0. Assume the existence of function ω, continuous increasing defined on I0 = [0,ρ0) with values in I satisfying ω(0) = 0. Define functions λ1 and µ1 on I0 as follows λ1(t) = ∫ 1 0 ω((1 −θ)t)dθ 1 −ω0(t) and µ1(t) = λ1(t) − 1. These definitions lead to µ1(0) = −1 and µ1(t) −→ ∞ as t −→ ρ−0 . Then, the theorem on intermediate value assure the existence of solutions for the equation µ1(t) = 0 in (0,ρ0). Set R1 to be the least such solution. Assume equation ω0(λ1(t)t) = 1 (2.2) has a least positive solution called ρ1. Set I1 = [0,ρ2), ρ2 = min{ρ0,ρ1}. Define functions λ2 and µ2 on I1 as follows λ2(t) = ∫ 1 0 ω((1 −θ)λ1(t)t)dθλ1(t) 1 −ω0(λ1(t)t) and µ2(t) = λ2(t) − 1. This time we also have λ2(0) = −1 and λ2(t) −→∞ as t −→ ρ−2 . Call R2 the smallest solution of equation λ2(t) = 0 in (0,ρ2). Assume equation ω0(λ2(t)t) = 1 (2.3) has a least positive solution called ρ3. Set I2 = [0,ρ4), ρ4 = min{ρ2,ρ3}. Consider functions λ3 and µ3 on I2 as follows λ3(t) = [∫ 1 0 ω((1 −θ)λ2(t)t)dθ 1 −ω0(λ2(t)t) + (ω0(λ2(t)t) + ω0(λ1(t)t) ∫ 1 0 v(θλ2(t)t)dθ (1 −ω0(λ2(t)t))(1 −ω0(λ1(t)t)) ] λ2(t) and µ3(t) = λ3(t) − 1, 100 I. K. Argyros & S. George CUBO 23, 1 (2021) where v : I2 −→ I is an increasing and continuous function. By these functions, we obtain µ3(0) = −1 and µ3(t) −→ ∞ as t −→ ρ−4 . Let R3 stand for the smallest solution of equation µ3(t) = 0 in (0,ρ4). A radius of convergence can be given as follows R = min{Ri}, i = 1, 2, 3. (2.4) Then, for all t ∈ [0,R). 0 ≤ ω0(t) < 1 (2.5) 0 ≤ ω0(λ1(t)t) < 1 (2.6) 0 ≤ ω0(λ1(t)t) < 1 (2.7) 0 ≤ ω0(λ2(t)t) < 1 (2.8) and 0 ≤ λi(t) < 1. (2.9) Denote by U(x∗,γ) a ball of center x∗ and with a radius γ > 0. Then, Ū(x∗,γ) stands for the closure of U(x∗,γ). We base the local convergence on this notation and the conditions (C). (c1) F : D −→ B2 is differentiable according to Fréchet, and x∗ ∈ D with F(x∗) = 0 is a simple solution. (c2) There exists an increasing and continuous real function ω0 on I satisfying ω0(0) = 0 and such that for all x ∈ D ‖F ′(x∗)−1(F ′(x) −F ′(x∗))‖≤ ω0(‖x−x∗‖). Set U0 = D ∩U(x∗,ρ0). (c3) There exists a function ω on I0 continuous and increasing satisfying ω(0) = 0 such that for all x,y ∈ U0 ‖F ′(x∗)−1(F ′(y) −F ′(x))‖≤ ω(‖y −x‖). Set U1 = D ∩U(x∗,ρ4). (c4) There exists a function v on I2 continuous and increasing, such that for all x ∈ U1 ‖F ′(x∗)−1F ′(x)‖≤ v(‖x−x∗‖). (c5) Ū(x∗,R) ⊆ D. (c6) There exists R1 ≥ R such that ∫ 1 0 ω0(θR1)dθ < 1. Set U2 = D ∩ Ū(x∗,R1). CUBO 23, 1 (2021) Extended domain for fifth convergence order schemes 101 Theorem 2.1. Assume hypotheses (C) hold and starting point x0 ∈ U(x∗,R) −{x∗}. Then the following assertions are valid, sequence {xn} belongs in U(x∗,R) −{x∗} and converges to x∗ ∈ U(x∗,R) so that this limit point uniquely solves equation F(x) = 0 in the set U2. Proof. Let z ∈ U(x∗,R) −{x∗} and utilize (c2), (2.4) and (2.5) to obtain ‖F ′(x∗)−1(F ′(z) −F ′(x∗))‖≤ ω0(‖z −x∗‖) ≤ ω0(R) < 1, which together with a result by Banach [12] for linear operators whose inverse exists imply ‖F ′(z)−1F ′(x∗)‖≤ 1 1 −ω0(‖z −x∗‖) . (2.10) In particular, by scheme (1.2) y0,z0 are well defined since if we set z = x0 ∈ U(x∗,R) −{x∗}, and F ′(x0) is invertible. Then, by (2.4), (2.8) (for k = 1), (c1), (c3) and (2.10) (for z = x0), we have ‖y0 −x∗‖ = ‖x0 −x∗ −F ′(x0)−1F(x0)‖ ≤ ‖F ′(x0)−1F ′(x∗)‖ [∫ 1 0 ‖F ′(x∗)−1[F ′(x0 + θ(x0 −x∗)) −F ′(x0)](x0 −x∗)dθ‖ ] ≤ ∫ 1 0 ω((1 −θ)‖x0 −x∗‖)dθ 1 −ω0(‖x0 −x∗‖) ‖x0 −x∗‖ ≤ λ1(‖x0 −x∗‖)‖x0 −x∗‖≤‖x0 −x∗‖ < R. (2.11) Hence, y0 ∈ U(x∗,R). Using the second substep of method (1.2) and replacing x0,y0, by y0,z0, respectively as in (2.10) and (2.11), we get ‖z0 −x∗‖ ≤ ∫ 1 0 ω((1 −θ)‖y0 −x∗‖)dθ 1 −ω0(‖y0 −x∗‖) ‖y0 −x∗‖ ≤ ∫ 1 0 ω((1 −θ)λ1(‖x0 −x∗‖)‖x0 −x∗‖)dθλ1(‖x0 −x∗‖)‖x0 −x∗‖ 1 −ω0(λ1(‖x0 −x∗‖)‖x0 −x∗‖ ≤ λ2(‖x0 −x∗‖)‖x0 −x∗‖≤‖x0 −x∗‖. (2.12) That is z0 ∈ U(x∗,R) and also x1 exists (for y0 = z, in (2.10)). Notice that (c1), (c4), (2.12) and F(z0) = F(z0) −F(x∗) = ∫ 1 0 F ′(x∗ + θ(z0 −x∗))dθ(z0 −x∗), we obtain that ‖F ′(x∗)−1F ′(z0)‖ ≤ ∫ 1 0 v(θ‖z0 −x∗‖)dθ‖z0 −x∗‖ ≤ ∫ 1 0 v(θλ2(‖x0 −x∗‖)‖x0 −x∗‖dθλ2(‖x0 −x∗‖)‖x0 −x∗‖. (2.13) Moreover, by the last substep of method (1.2), (2.4), (2.5), (2.8) (for k = 3), (2.10), (2.13) (for 102 I. K. Argyros & S. George CUBO 23, 1 (2021) z = x0,y0), (2.11) and (2.12), we have in turn that ‖x1 −x∗‖ ≤ ‖z0 −x∗ −F ′(z0)−1F(z0)‖ (2.14) +‖F ′(z0)−1[(F ′(y0) −F ′(x∗)) + (F ′(x∗) −F ′(z0))]F ′(y0)−1F(z0)‖ ≤ [∫ 1 0 ω((1 −θ)‖z0 −x∗‖)dθ 1 −ω0(‖z0 −x∗‖) + (ω0(‖z0 −x∗‖) + ω0(‖y0 −x∗‖)) ∫ 1 0 v(θ‖z0 −x∗‖)dθ (1 −ω0(‖z0 −x∗‖))(1 −ω0(‖y0 −x∗‖)) ] ‖z0 −x∗‖ ≤ λ3(‖x0 −x∗‖)‖x0 −x∗‖≤‖x0 −x∗‖, (2.15) so x1 ∈ U(x∗,R). Replacing x0,y0,z0,x1 by xk,yk,zk,xk+1, in the previous computations we obtain ‖yk −x∗‖≤ λ1(‖xk −x∗‖)‖xk −x∗‖≤‖xk −x∗‖ < R, (2.16) ‖zk −x∗‖≤ λ2(‖xk −x∗‖)‖xk −x∗‖≤‖xk −x∗‖ (2.17) and ‖xk+1 −x∗‖≤ λ3(‖xk −x∗‖)‖xk −x∗‖≤‖xk −x∗‖, (2.18) so yk,zk,xk+1 stay in U(x∗,R) and limk−→∞xk = x∗. Furthermore, let x 1 ∗ ∈ U2 with F(x1∗) = 0. In view of (c2) and (c6) we obtain∣∣∣∣ ∣∣∣∣F ′(x∗)−1 (∫ 1 0 F ′(x∗ + θ(x 1 ∗ −x∗))dθ −F ′(x∗) )∣∣∣∣ ∣∣∣∣ ≤ ∫ 1 0 ω0(θ‖x1∗ −x∗‖)dθ ≤ ∫ 1 0 ω0(θR1)dθ < 1, so x1∗ = x∗, since T = ∫ 1 0 F ′(x∗ + θ(x 1 ∗ −x∗))dθ is invertible and 0 = F(x1∗) −F(x∗) = T(x 1 ∗ −x∗). Remark 2.2. 1. In view of (2.10) and the estimate ‖F ′(x∗)−1F ′(x)‖ = ‖F ′(x∗)−1(F ′(x) −F ′(x∗)) + I‖ ≤ 1 + ‖F ′(x∗)−1(F ′(x) −F ′(x∗))‖≤ 1 + L0‖x−x∗‖ condition (2.13) can be dropped and M can be replaced by M(t) = 1 + L0t or M(t) = M = 2, since t ∈ [0, 1 L0 ). CUBO 23, 1 (2021) Extended domain for fifth convergence order schemes 103 2. The results obtained here can be used for operators F satisfying autonomous differential equations [2] of the form F ′(x) = P(F(x)) where P is a continuous operator. Then, since F ′(x∗) = P(F(x∗)) = P(0), we can apply the results without actually knowing x∗. For example, let F(x) = ex − 1. Then, we can choose: P(x) = x + 1. 3. Let ω0(t) = L0t, and ω(t) = Lt. In [2, 3] we showed that rA = 2 2L0+L is the convergence radius of Newton’s method: xn+1 = xn −F ′(xn)−1F(xn) for each n = 0, 1, 2, · · · (2.19) under the conditions (2.11) and (2.12). It follows from the definition of R in (2.4) that the convergence radius R of the method (1.2) cannot be larger than the convergence radius rA of the second order Newton’s method (2.19). As already noted in [2, 3] rA is at least as large as the convergence radius given by Rheinboldt [12] rR = 2 3L , (2.20) where L1 is the Lipschitz constant on D. The same value for rR was given by Traub [15]. In particular, for L0 < L1 we have that rR < rA and rR rA → 1 3 as L0 L1 → 0. That is the radius of convergence rA is at most three times larger than Rheinboldt’s. 4. It is worth noticing that method (1.2) is not changing when we use the conditions of Theo- rem 2.1 instead of the stronger conditions used in [13, 14]. Moreover, we can compute the computational order of convergence (COC) defined by ξ = ln ( ‖xn+1 −x∗‖ ‖xn −x∗‖ ) / ln ( ‖xn −x∗‖ ‖xn−1 −x∗‖ ) or the approximate computational order of convergence ξ1 = ln ( ‖xn+1 −xn‖ ‖xn −xn−1‖ ) / ln ( ‖xn −xn−1‖ ‖xn−1 −xn−2‖ ) . This way we obtain in practice the order of convergence in a way that avoids the bounds involving estimates using estimates higher than the first Fréchet derivative of operator F. Note also that the computation of ξ1 does not require the usage of the solution x ∗. 104 I. K. Argyros & S. George CUBO 23, 1 (2021) 3 Semi-local convergence analysis Let Γ0 = F ′(x0) −1 ∈ L(B2,B1) exists at x0 ∈ D, where L(B2,B1) denotes the set of bounded linear operators from B2,B1 and the following conditions hold. (1) ‖Γ0‖≤ β0. (2) ‖Γ0F(x0)‖≤ η0. (3)’ ‖F ′(x) −F ′(x0)‖≤ M0‖x−x0‖ for all x ∈ D. Set D0 = D ∩U ( x0, 1 β0M0 ) . (3) ‖F ′′(x)‖≤ M for all x ∈ D0. (4) ‖F ′′(x) − F ′′(y)‖ ≤ ω(‖x − y‖) for all x,y ∈ D0 for a continuous nondecreasing function ω, ω(0) ≥ 0 such that ω(tx) ≤ tpω(x) for t ∈ [0, 1],x ∈ (0,∞) and p ∈ [0, 1]. Then, as in [13, 14], let r0 = Mβ0η0, s0 = β0η0ω(η0) and define sequences {rk},{sk} and {ηk} for k = 0, 1, 2, . . . , by rk+1 = rkϕ(rk) 2ψ(rk,sk), (3.1) sk+1 = skϕ(rk) 2+pψ(rk,sk) 1+p, (3.2) ηk+1 = ηkϕ(rk)ψ(rk,sk), (3.3) where ϕ(t) = 1 1 − tg(t) (3.4) g(t) = ( 1 + t 2 + t2 2(1 − t) ( 1 + t 4 )) (3.5) and ψ(t,s) = t2 2(1 − t) (1 + t 4 ) [ s 1 + p ( t1+p 21+p + 1 2 + p ( t2 2(1 − t) ( 1 + t 4 ))1+p) + t 2 ( t + t2 2(1 − t) ( 1 + t 4 ))] . (3.6) Remark 3.1. In [14] the following conditions were used instead of (3), (4), respectively (3)’ ‖F ′′(x)‖≤ M1 for all x ∈ D (4)’ ‖F ′′(x) −F ′′(y)‖≤ ω1(‖x−y‖) for all x,y ∈ D and ω1 as ω. But, we have D0 ⊆ D, so M0 ≤ M1 CUBO 23, 1 (2021) Extended domain for fifth convergence order schemes 105 M ≤ M1 and ω(θ) ≤ ω1(θ). Examples where the preceding items are strict can be found in [1, 2, 3, 4, 5, 6]. Notice that (3)’ is used to determine D0 leading to M = M(D0,x)). Hence, the results in [13, 14] can be rewritten with M replacing M1. So, if M < M1 the new semi-local convergence analysis is finer. This is also done under the same computational effort because in practice finding ω1,M1 requires finding ω,M0,M as special cases. This technique can be used to extend the applicability of other schemes involving inverses in an analogous fashion. Hence, the proof of the following semi-local convergence result for scheme (1.2) is omitted. Theorem 3.2. Let r0 = Mβ0η0 < ν,s0 = β0η0ω(η0) and assumptions (1)-(4) hold. Then, for Ū(x0,Rη0) ⊆ D, where R = g(r0) 1−δγ , the sequence {xk} generated by (1.2) converges to the solution x∗ of F(x) = 0. Moreover, yk,zk,xk+1,x∗ ∈ Ū(x0,Rη0) and x∗ is the unique solution in U ( x0, 2 M0β0 −Rη0 ) ∩D. Furthermore, we have ‖xk −x∗‖≤ g(r0)δk γ (4+q)k−1 3+q 1 − δγ(4+q)k η0. 4 Numerical Examples Example 4.1. Let us consider a system of differential equations governing the motion of an object and given by F ′1(x) = e x, F ′2(y) = (e− 1)y + 1, F ′ 3(z) = 1 with initial conditions F1(0) = F2(0) = F3(0) = 0. Let F = (F1,F2,F3). Let B1 = B2 = R3,D = Ū(0, 1),p = (0, 0, 0)T . Define function F on D for w = (x,y,z)T by F(w) = ( ex − 1, e− 1 2 y2 + y,z )T . The Fréchet-derivative is defined by F ′(v) =   ex 0 0 0 (e− 1)y + 1 0 0 0 1   . Notice that using the (A) conditions, we get for α = 1, w0(t) = (e−1)t,w(t) = e 1 e−1 t,v(t) = e 1 e−1 . The radii are R1 = 0.38269191223238574472986783803208,R2 = 0.33841523581069998805048726353562, R3 = 0.32249343047238987480795913143083 and R = R3. 106 I. K. Argyros & S. George CUBO 23, 1 (2021) Example 4.2. Let B1 = B2 = C[0, 1], the space of continuous functions defined on [0, 1] be equipped with the max norm. Let D = U(0, 1). Define function F on D by F(ϕ)(x) = ϕ(x) − 5 ∫ 1 0 xθϕ(θ)3dθ. (4.1) We have that F ′(ϕ(ξ))(x) = ξ(x) − 15 ∫ 1 0 xθϕ(θ)2ξ(θ)dθ, for each ξ ∈ D. Then, we get that x∗ = 0, so w0(t) = 7.5t,w(t) = 15t and v(t) = 2. Then the radii are R1 = 0.066666666666666666666666666666667,R2 = 0.059338915721683857529278327547217, R3 = 0.047722035514509826559237382070933 and R = R3. Example 4.3. Returning back to the motivational example at the introduction of this study, we have w0(t) = w(t) = 96.6629073t and v1(t) = 2. The parameters for method (1.2) are R1 = 0.0068968199414654552878434223828208,R2 = 0.0061008926455964288676492301988219, R3 = 0.004463243021326804456372361329386 and R = R3. 5 Conclusion In general, the convergence domain of iterative schemes is small limiting their applications. Hence, any attempt to increase it is very important. This is achieved here by finding smaller ω− functions than before which are also specialization of the previous ones. Hence, the extensions are obtained under the same computational cost. Our idea can be used to extend the usage of other schemes in a similar way. Numerical experiments further demonstrate the superiority of our findings. CUBO 23, 1 (2021) Extended domain for fifth convergence order schemes 107 References [1] I. K. Argyros, “A new convergence theorem for the Jarratt method in Banach space”, Comput. Math. Appl., vol. 36, pp. 13–18, 1998. [2] I. K. Argyros, Convergence and Application of Newton-Type Iterations, Springer, New York, 2008. [3] I. K. Argyros, D. Chen, and Q. Qian, “The Jarratt method in Banach space setting”, J. Comput. Appl. Math, vol. 51, pp. 103–106, 1994. [4] I. K. Argyros, and A. A. Magreñañ, Iterative Methods and their dynamics with applications: A Contemporary Study, CRC Press, 2017. [5] I. K. Argyros, and S. George, Mathematical modeling for the solution of equations and systems of equations with applications, Volume-IV, Nova Publishes, New York, 2020. [6] M. Chen, Y. Khan, Q. Wu, and A. Yildirim, “Newton–Kantorovich Convergence Theorem of a Modified Newton’s Method Under the Gamma-Condition in a Banach Space”, Journal of Optimization Theory and Applications, vol. 157, no. 3, pp. 651–662. [7] J. L. Hueso, and E. Mart́ınez, “Semi-local convergence of a family of iterative methods in Banach spaces”, Numer. Algorithms, vol. 67, pp. 365–384, 2014. [8] A. Kumar, D. K. Gupta, E. Mart́ınez, and S. Singh, “Semi-local convergence of a Steffensen type method under weak Lipschitz conditions in Banach spaces”, J. Comput. Appl. Math., vol. 330, pp. 732–741, 2018. [9] A. A. Magreńãn, “Different anomalies in a Jarratt family of iterative root finding methods”, Appl. Math. Comput., vol. 233, pp. 29–38, 2014. [10] A. A. Magreńãn, “A new tool to study real dynamics: The convergence plane”, Appl. Math. Comput., vol. 248, pp. 29–38, 2014. [11] E. Mart́ınez, S. Singh, J. L. Hueso, and D. K. Gupta, “Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces”, Appl. Math. Comput., vol. 281, pp. 252–265, 2016. [12] W. C. Rheinboldt, “An adaptive continuation process for solving systems of nonlinear equa- tions”, In: Mathematical models and numerical methods (A.N.Tikhonov et al. eds.) pub.3, pp. 129–142, 1977, Banach Center, Warsaw, Poland. [13] S. Singh, D. K. Gupta, E. Mart́ınez, and J. L. Hueso, “Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces”, Mediterr. J. Math., vol. 13, pp. 4219–4235, 2016. 108 I. K. Argyros & S. George CUBO 23, 1 (2021) [14] S. Singh, E. Mart́ınez, A. Kumar, and D. K. Gupta, “Domain of existence and uniqueness for nonlinear Hammerstein integral equations”, Mathematics, vol. 8, no. 3, 2020. [15] J. F. Traub, Iterative methods for the solution of equations, AMS Chelsea Publishing, 1982. [16] X. Wang, J. Kou, and C. Gu, “Semi-local convergence of a class of Modified super Halley method in Banach space”, J. Optim. Theory. Appl., vol. 153, pp. 779–793, 2012. [17] Q. Wu, and Y. Zhao, “Newton-Kantorovich type convergence theorem for a family of new deformed Chebyshev method”, Appl. Math. Comput., vol. 192, pp. 405–412, 2008. [18] Y. Zhao, and Q. Wu, “Newton-Kantorovich theorem for a family of modified Halley’s method under Hölder continuity conditions in Banach space”, Appl. Math. Comput., vol. 202, pp. 243–251, 2008. [19] A. Emad, M. O. Al-Amr, A. Yıldırım, W. A. AlZoubi, “Revised reduced differential transform method using Adomian’s polynomials with convergence analysis”, Mathematics in Engineer- ing, Science & Aerospace (MESA), vol. 11, no. 4, pp. 827–840, 2020. Introduction Local convergence Semi-local convergence analysis Numerical Examples Conclusion