**2. Convergence Analysis**

Let *b* > 0, *α* > 0, *γ* > 0, *β* ∈ S, *k* ∈ N and *M* ≥ 1 be given constants. Furthermore, we consider that *H* : S × S → S, *h* : [0, ∞) → [0, ∞) are continuous functions such that:

$$|H(\upsilon,\,\eta)| \le |H(|\upsilon|,\,|\eta|)| \le h(\upsilon),\tag{4}$$

for each *υ*, *η* ∈ S with |*η*| ≤ *υ*, and that |*H*| and *h* are nondecreasing functions on the interval 0, 1*γ* 2, 0, 1*γ* 2, respectively. For the local convergence analysis of (2), we need to introduce a few functions and parameters. Let us define the parameters *R*0 and *R*1 given by:

$$R\_0 = \frac{1}{(1+a)\gamma},\ R\_1 = \frac{1}{(1+a)\gamma + \gamma a (b|\beta|M+a)},\tag{5}$$

and function *g*1 on the interval [0, *<sup>R</sup>*1) by:

$$\log\_1(\upsilon) = \frac{\gamma \alpha (b|\beta|M + \alpha)\upsilon}{1 - (1 + \mathfrak{a})\gamma \upsilon}. \tag{6}$$

From the above functions, it is easy to see that *R*1 < *R*0 < 1*γ* , *g*1(*<sup>R</sup>*1) = 1 and 0 ≤ *g*1(*υ*) < 1, for *υ* ∈ [0, *<sup>R</sup>*1). Moreover, we consider the functions *q* and *q*¯ on [0, *<sup>R</sup>*1) as:

$$
\eta(\upsilon) = \gamma(\mathfrak{a} + \mathfrak{g}\_1(\upsilon))\upsilon \quad \text{and} \quad \mathfrak{q}(\upsilon) = \mathfrak{q}(\upsilon) - 1.
$$

It is straightforward to find that *q*¯(0) = −1 < 0 and that *q*¯(*υ*) → +∞ as *υ* → *r*−1 . By the intermediate value theorem, we know that *q*¯ has zeros in the interval (0, *<sup>R</sup>*1). Let us assume that *Rq* is the smallest zero of function *q*¯ on (0, *<sup>R</sup>*1), and set:

$$
\vec{r} = \min\{\mathcal{R}\_1, \mathcal{R}\_q\}.\tag{7}
$$

Furthermore, let us define functions *g*2 and *g*¯2 on [0, *r*¯) such that:

$$\mathcal{g}\_2(\upsilon) = \left(1 + \frac{Mh(\upsilon)}{1 - q(\upsilon)}\right) \mathcal{g}\_1(\upsilon) \tag{8}$$

and:

$$\mathfrak{g}\_2(\upsilon) = \mathfrak{g}\_2(\upsilon) - 1. \tag{9}$$

Suppose that:

$$\mathfrak{g}\_2(\upsilon) \to \text{ a positive number or } +\infty, \text{ as } \upsilon \to \mathbb{r}^-. \tag{10}$$

From (8), we have that *g*¯2(0) < 0 and from (10) that *g*¯2(*υ*) > 0 as *υ* → *r*¯−1. Further, we assume that *R* is the smallest zero of function *g*¯2 on (0, *r*¯). Therefore, we have that for each *υ* ∈ [0, *r*):

$$0 \le \mathcal{g}\_1(\upsilon) < 1,\tag{11}$$

$$0 \le \varrho\_2(v) < 1,\tag{12}$$

$$0 \le q(v) < 1.\tag{13}$$

Let us denote by *<sup>U</sup>*(*μ*, *r*) and *U*¯ (*μ*, *r*) the open and closed balls in S with center *μ* ∈ *S* and of radius *r* > 0, respectively.

**Theorem 1.** *Let us assume that* Ω : P ⊂ S → S *is a differentiable function and* [·, · ; Ω] : P × P → *<sup>L</sup>*(S, S) *is a divided difference of first order of* Ω*. Furthermore, we consider that h and H are functions satisfying* (4)*,* (9)*, p*∗ ∈ P, *b* > 0, *α* > 0, *γ* > 0, *M* ≥ 1, *k* ∈ N*, β* ∈ *S and that for each x*, *y* ∈ P*, we have:*

$$
\Omega(p\_\*) = 0, \quad \Omega'(p\_\*) \neq 0,\\

$$<|\Omega'(p\_\*)^{-1}([\mathbf{x}, y, \Omega] - \Omega'(p\_\*) | \le \gamma (|\mathbf{x} - p\_\*| + |y - p\_\*|),\tag{15}$$

$$h(\upsilon) = H\left(\frac{M\gamma(|\beta|Mb + a)\upsilon}{(1 - \gamma av)(1 - \gamma(1 + a)\upsilon)}, \frac{Mg\_1(\upsilon)}{1 - \gamma v}\right) \tag{16}$$

$$|I + \beta[\mathbf{x}, \ p\_\*; \Omega]^k (\mathbf{x} - p\_\*)^{k-1}| \le a,\tag{17}$$

$$|\Omega'(p\_\*)^{-1}[\mathbf{x},\ p\_{\*\prime},\ \Omega]| \le M,\tag{18}$$

$$
\mathcal{U}(p\_\*, \mathfrak{a}r) \subseteq \mathbb{P}.\tag{19}
$$

*Then, the sequence* {*<sup>δ</sup>s*} *obtained for λ*0 ∈ *<sup>U</sup>*(*p*<sup>∗</sup>, *R*) − {*x*∗} *by* (2) *is well defined, remains in <sup>U</sup>*(*p*<sup>∗</sup>, *R*) *for each n* = 0, 1, 2, . . . , *and converges to p*<sup>∗</sup>*, so that:*

$$|\lambda\_{\mathfrak{s}} - p\_{\mathfrak{s}}| \le \mathfrak{a} |\delta\_{\mathfrak{s}} - p\_{\mathfrak{s}}| \prec \mathcal{R}\_{\prime} \tag{20}$$

$$|\mu\_s - p\_\*| \le g\_1(|\delta\_s - p\_\*|) |\delta\_s - p\_\*| \le |\delta\_s - p\_\*| < R,\tag{21}$$

$$|\delta\_{s+1} - p\_\*| \le g\_2(|\delta\_s - p\_\*|) |\delta\_s - p\_\*| < |\delta\_s - p\_\*|,\tag{22}$$

*and G* ∈ [*<sup>R</sup>*, 1*γ* )*. Moreover, the limit point p*∗ *is the unique root of equation* <sup>Ω</sup>(*x*) = 0 *in U*¯ (*p*<sup>∗</sup>, *G*) ∩ P.

**Proof.** By hypotheses *λ*0 ∈ *<sup>U</sup>*(*p*<sup>∗</sup>,*<sup>r</sup>*) − {*x*∗}, (14), (17) and (19), we further obtain:

$$\begin{aligned} \delta\_0 - p\_\* &= \lambda\_0 - p\_\* + \beta \left( \Omega(\lambda\_0) - \Omega(p\_\*) \right)^k \\ &= \left( I + \beta [\lambda\_0, p\_\*; \Omega]^k (\lambda\_0 - p\_\*)^{k-1} \right) (\lambda\_0 - p\_\*), \end{aligned}$$

so that:

$$\begin{aligned} |\delta\_0 - p\_\*| &= \left| I + \beta[\lambda\_{0\prime} \ p\_\*; \, \Omega]^k (\lambda\_0 - p\_\*)^{k-1} \right| |\lambda\_0 - p\_\*| \\ &\le ar, \end{aligned} \tag{23}$$

which leads to (20) for *s* = 0 and *δ*0 ∈ *<sup>U</sup>*(*p*<sup>∗</sup>, *<sup>α</sup>r*). We need to show that [*<sup>λ</sup>*0, *δ*0; Ω] = 0. Using (15) and the definition of *R*, we obtain:

$$\begin{split} \left| \Omega'(p\_\*)^{-1} ([\lambda\_0, \delta\_0; \, \Omega] - \Omega'(p\_\*) \, \right| &\leq \gamma \left( |\lambda\_0 - p\_\*| + |\delta\_0 - p\_\*| \right) \\ &\leq \gamma \left( |\lambda\_0 - p\_\*| + a |\lambda\_0 - p\_\*| \right) \\ &\leq \gamma (1 + a) |\lambda\_0 - p\_\*| < \gamma (1 + a)R < 1. \end{split} \tag{24}$$

From the Banach lemma on invertible functions [7,14], it follows that [*<sup>λ</sup>*0, *δ*0; Ω] = 0 and:

$$\left| [\lambda\_0, \delta\_0; \,\Omega]^{-1} \Omega'(p\_\*) \right| \le \frac{1}{1 - \gamma(1 + a)|\lambda\_0 - p\_\*|}. \tag{25}$$

In view of (14) and (18), we have:

$$\begin{aligned} \left| \Omega'(p\_\*)^{-1} \Omega(\lambda\_0) \right| &= \left| \Omega'(p\_\*)^{-1} \left( \Omega(\lambda\_0) - \Omega(p\_\*) \right) \right| \\ &= \left| \Omega'(p\_\*)^{-1} [\lambda\_{0'}, p\_\*, \Omega](\lambda\_0 - p\_\*) \right| \\ &\leq M |\lambda\_0 - p\_\*| \end{aligned} \tag{26}$$

and similarly:

$$\left|\Omega'(p\_\*)^{-1}\Omega(\delta\_0)\right| \le M|\delta\_0 - p\_\*|\,. \tag{27}$$

since *δ*0 ∈ P. Then, using the second substep of Methods (2), (11), (14), (16), (25) and (27), we obtain:

$$\begin{split} |p\_0 - p\_\*| &= \left| \delta\_0 - p\_\* - [\lambda\_0, \delta\_0, \Omega]^{-1} \Omega(\delta\_0) \right| \\ &\le \left| [\lambda\_0, \delta\_0, \Omega]^{-1} \Omega'(p\_\*) \right| \left| \Omega'(p\_\*)^{-1} \left( [\lambda\_0, \delta\_0, \Omega][\delta\_0 - p\_\*) - (\Omega(\delta\_0) - \Omega(p\_\*)) \right) \right| \\ &\le \left| [\lambda\_0, \delta\_0, \Omega]^{-1} \Omega'(p\_\*) \right| \left| \Omega'(p\_\*)^{-1} \left( [\lambda\_0, \delta\_0, \Omega] - [\delta\_0, p\_{\*\*}, \Omega] \right) (\delta\_0 - p\_\*) \right| \\ &\le \frac{\gamma \left( [\lambda\_0 - \delta\_0] + [\delta\_0 - p\_\*) \right) [\delta\_0 - p\_\*]}{1 - \gamma (1 + a) \lambda\_0 - p\_\* |} \\ &\le \frac{\gamma \left( [\beta | bM | \lambda\_0 - p\_\* | + a | \lambda\_0 - p\_\* | \right) a | \lambda\_0 - p\_\* \right)}{1 - \gamma (1 + a) | \lambda\_0 - p\_\*|} \\ &\le \frac{\gamma a \left( [\beta | bM + a \right) \left| \lambda\_0 - p\_\* \right|^2 \\ &= \frac{\gamma a \left( [\beta | bM + a \right) \left| \lambda\_0 - p\_\* \right|^2 \right)}{1 - \gamma (1 + a) \left| \lambda\_0 - p\_\* \right|} \\ &= g\_1 \left( |\lambda\_0 - p\_\*| \right) |\lambda\_0 - p\_\*| < |\lambda\_0 - p\_\*| < R, \end{split} \tag{28}$$

and so, (21) is true for *s* = 0 and *μ*0 ∈ *<sup>U</sup>*(*p*<sup>∗</sup>, *<sup>R</sup>*). Next, we need to show that <sup>Ω</sup>(*<sup>λ</sup>*0) = 0 and <sup>Ω</sup>(*<sup>δ</sup>*0) = 0, for *δ*0 = *p*<sup>∗</sup>. Using (14) and (15), and the definition of *R*, we obtain:

$$\begin{split} & \left| \left( (\lambda\_0 - p\_\*) \Omega'(p\_\*) \right)^{-1} \left[ \Omega(\lambda\_0) - \Omega(p\_\*) - \Omega'(p\_\*) (\lambda\_0 - p\_\*) \right] \right| \\ & \leq |\lambda\_0 - p\_\*|^{-1} \left| \Omega'(p\_\*)^{-1} \left( [\lambda\_{0\prime} \ p\_\*; \Omega] - \Omega'(p\_\*) (\lambda\_0 - p\_\*) \right) \right| \\ & \leq \gamma |\lambda\_0 - p\_\*|^{-1} |\lambda\_0 - p\_\*|^2 = \gamma |\lambda\_0 - p\_\*| < \gamma R < 1. \end{split} \tag{29}$$

Hence, <sup>Ω</sup>(*<sup>λ</sup>*0) = 0 and:

$$|\Omega'(\lambda\_0)^{-1}\Omega'(p\_\*)| \le \frac{1}{|\lambda\_0 - p\_\*|(1 - \gamma|\lambda\_0 - p\_\*|)}.\tag{30}$$

Similarly, we have that:

$$|\Omega'(\delta\_0)^{-1}\Omega'(p\_\*)| \le \frac{1}{|\delta\_0 - p\_\*|(1 - \gamma|\delta\_0 - p\_\*|)} \le \frac{1}{|\delta\_0 - p\_\*|(1 - a\gamma|\lambda\_0 - p\_\*|)}.\tag{31}$$

Then, by using (4) and (12) (for *δ*0 = *μ*0), (16), (27), (28), (30) and (31), we have:


Adopting (13), we get:

$$\begin{split} \left| \Omega'(p\_\*)^{-1} \left( [\delta\_0, \,\mu\_0; \,\Omega] - \Omega'(p\_\*) \right) \right| &\leq \gamma \left( |\delta\_0 - p\_\*| + |\mu\_0 - p\_\*| \right) \\ &\leq \gamma \left( a|\lambda\_0 - p\_\*| + g\_1(|\lambda\_0 - p\_\*|) |\lambda\_0 - p\_\*| \right) \\ &= q(|\lambda\_0 - p\_\*|) < q(R) < 1. \end{split} \tag{33}$$

Hence, we have:

$$\left| [\delta\_0, \,\mu\_0; \,\Omega]^{-1} \Omega'(p\_\*) \right| \le \frac{1}{1 - q(|\lambda\_0 - p\_\*|)}. \tag{34}$$

Furthermore, *λ*1 is well defined by (24), (32) and (34). Using the third substep of (2), (12), (27) (for *δ*0 = *μ*0), (28), (32) and (34), we get:

$$\begin{split} |\lambda\_1 - p\_\*| &\le |\mu\_0 - p\_\*| + |H(v\_{0\*}, \eta\_0)| \left| [\lambda\_{0\*} \,\delta\_0; \,\Omega]^{-1} \Omega'(p\_\*) \right| \left| \Omega'(p\_\*)^{-1} \Omega(\mu\_0) \right| \\ &\le \left[ 1 + \frac{Mh(|\lambda\_0 - p\_\*|)}{1 - q(|\lambda\_0 - p\_\*|)} \right] |\mu\_0 - p\_\*| \\ &\le \left[ 1 + \frac{Mh(|\lambda\_0 - p\_\*|)}{1 - q(|\lambda\_0 - p\_\*|)} \right] g\_1(|\lambda\_0 - p\_\*|) |\lambda\_0 - p\_\*| \\ &\le g\_2(|\lambda\_0 - p\_\*|) |\lambda\_0 - p\_\*| < |\lambda\_0 - p\_\*| < R, \end{split} \tag{35}$$

showing that (22) is true for *s* = 0 and *λ*1 ∈ *<sup>U</sup>*(*p*<sup>∗</sup>, *<sup>R</sup>*). Replacing *λ*0, *δ*0, and *μ*0 by *λ<sup>s</sup>*, *δ<sup>s</sup>*, and *μ<sup>s</sup>*, respectively, in the preceding estimates, we arrive at (20)–(22). From the estimates *<sup>δ</sup>s*+<sup>1</sup> − *p*∗ < *<sup>δ</sup>s* − *p*∗ < *r*, we conclude that lim*s*→∞ *δs* = *p*∗ and *xs*+1 ∈ *<sup>U</sup>*(*p*<sup>∗</sup>, *<sup>R</sup>*). Finally, to illustrate the uniqueness, let *p*∗∗ ∈ *U* ¯ (*p*<sup>∗</sup>, *T*) such that <sup>Ω</sup>(*p*∗∗) = 0. We assume *Q* = [*p*<sup>∗</sup>, *p*∗∗; <sup>Ω</sup>]. Adopting (15), we get:

$$\begin{split} \left| \Omega'(p\_\*)^{-1} (Q - \Omega'(p\_\*)) \right| &\leq \gamma \left( |p\_\* - p\_\*| + |p\_{\*\*} - p\_\*| \right) \\ &= \gamma T < 1. \end{split} \tag{36}$$

Therefore, *Q* = 0, and in view of the identity <sup>Ω</sup>(*p*∗) − <sup>Ω</sup>(*p*∗∗) = *Q*(*p*∗ − *p*∗∗), we conclude that *p*∗ = *p*∗∗.
