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Convergence and stability of modified multi-step Noor iterative procedure with errors for strictly hemicontractive-type mappings in Banach spaces

Abstract

In this paper, we introduce and study a modified multi-step Noor iterative procedure with errors for two Lipschitz strictly hemicontractive-type mappings in arbitrary Banach spaces and constitute its convergence and stability. The obtained results in this paper generalize and extend the corresponding result of Hussain et al. (Fixed Point Theory Appl. 2012:160, 2012) and some analogous results of several authors in the literature. Finally, a numerical example is included to illustrate our analytical results and to display the efficiency of our proposed novel iterative procedure with errors.

Introduction

In the last few decades, fixed-point theorem-based iterative procedures whose convergence established on the strictly hemicontractive-type mappings earn a great attention for its rigorous applications in the diverse fields of various mathematical problems; see for instance [25] and the references cited therein. Application of strictly hemicontractive-type mapping was initiated by Chidume and Osilike [4] for improving the consequence of Chidume [5]. After Chidume and Osilike [4], several researchers studied strictly hemicontractive-type mapping in many directions; see for instance [13, 621] and the references cited therein. Among the articles cited in [13, 621], Hussain et al. [1] studied Lipschitz strictly hemicontractive-type mapping in arbitrary Banach spaces to extend and improve the equivalent consequences of the monographs [4, 5, 1215].

Throughout this paper, \(\mathbb{R}\) denotes the set of real numbers, B represents a nonempty subset of an arbitrary Banach space X and \(X^{*}\) is a dual space of X. Let T be a single-valued map from B into itself, then \(r \in B\) is called a fixed point of T iff \(T(r) = r\). The symbols \(D_{T}\), \(R_{T}\) and \(F_{T}\) denote the domain of T, the range of T and the set of fixed points of T respectively. Let \(J:X \to 2^{X^{*}}\) be a normalized duality mapping given by

$$ J(r) = \bigl\{ g^{*} \in X^{*}: \bigl\langle r, g^{*} \bigr\rangle = \Vert r \Vert ^{2} = \bigl\Vert g^{*} \bigr\Vert ^{2} \bigr\} . $$

The mapping T is called Lipschitzian if there exists a \(L > 0\) such that

$$ \Vert Tq - Tr \Vert \le L \Vert q - r \Vert $$
(1.1)

for all \(q, r \in B\). If \(L = 1\), then T is called a non-expansive mapping, and if \(0 \le L < 1\), then T is called a contraction mapping.

The mapping T is called a strictly hemicontractive mapping if \(F_{T} \ne \varphi \) and if there exists a constant \(t > 1\) such that

$$ \Vert q - r \Vert \le \bigl\Vert \bigl(1 + t' \bigr) (q - r) - t't(Tq - Tr) \bigr\Vert $$
(1.2)

for all \(q \in D_{T}\), \(r \in F_{T}\) and \(t' > 0\).

If the mapping T satisfies both inequalities (1.1) and (1.2), then it is called a Lipschitz strictly hemicontractive mapping.

The mapping T is called asymptotically non-expansive on B if there exists a sequence \(\{ s_{n} \} \) in \([ 0, \infty )\) with \(\lim_{n \to \infty } s_{n} = 0\) such that, for each \(p,q \in B\),

$$ \bigl\Vert T^{n}p - T^{n}q \bigr\Vert \le ( 1 + s_{n} ) \Vert p - q \Vert ,\quad \forall n \ge 1. $$

T is called an asymptotically non-expansive mapping in the intermediate sense if T is uniformly continuous and

$$ \lim \sup_{n \to \infty } \sup_{p, q \in B} \bigl( \bigl\Vert T^{n}p - T^{n}q \bigr\Vert - \Vert p - q \Vert \bigr) \le 0. $$

The mapping T is called an asymptotically quasi-non-expansive mapping if there exists a sequence \(\{ s_{n} \} \) in \([ 0, \infty )\) with \(\lim_{n \to \infty } s_{n} = 0\) such that, for all \(p \in B\), \(q \in F_{T}\),

$$ \bigl\Vert T^{n}p - q \bigr\Vert \le ( 1 + s_{n} ) \Vert p - q \Vert , \quad \forall n \ge 1. $$

According to the definitions, it is clear that an asymptotically non-expansive mapping must be an asymptotically non-expansive mapping in the intermediate sense and an asymptotically quasi-non-expansive mapping, but the converse is not always true. We may justify this concept by using the following example.

Example 1.1

(See [22])

Let \(X = \mathbb{R}\) (with the usual norm), \(B = [ - \frac{1}{\pi }, \frac{1}{\pi } ]\) and \(\vert t \vert < 1\). For each \(u \in B\), we define

$$ Tu = \textstyle\begin{cases} tu\sin \frac{1}{u} & \text{if } u \ne 0, \\ 0 & \text{if } u = 0. \end{cases} $$

Then T is an asymptotically non-expansive mapping in the intermediate sense and an asymptotically quasi-non-expansive mapping, but is not a Lipschitzian mapping, thus it is not an asymptotically non-expansive mapping as well as it is not a Lipschitz strictly hemicontractive mapping.

Remark 1.2

We note that an asymptotically non-expansive mapping in the intermediate sense or an asymptotically quasi-non-expansive mapping is not always a Lipschitz strictly hemicontractive mapping.

We now provide an example which shows that a Lipschitz strictly hemicontractive mapping is also an asymptotically non-expansive mapping.

Example 1.3

Let \(X = \mathbb{R}\) with the usual norm and \(B = [ 0, 2\pi ]\). Define \(T:B \to B\) by \(Tu = \frac{u\cos u}{2}\) for each \(u \in B\). Clearly \(F_{T} = \{ 0 \} \). For each \(u \in D_{T}\), \(r \in F_{T}\), \(t' > 0\), choose \(t = 2\). Then we have

$$\begin{aligned} \bigl\vert \bigl(1 + t' \bigr) (u - r) - t't(Tu - Tr) \bigr\vert ={}& \bigl\vert \bigl(1 + t' \bigr)u - 2t'Tu \bigr\vert \\ ={}& \biggl\vert \bigl(1 + t' \bigr)u - 2t' \cdot \frac{u\cos u}{2} \biggr\vert \\ \ge {}& \bigl(1 + t' \bigr)u - t'u = u = \vert u - r \vert \end{aligned}$$

and hence T is a strictly hemicontractive mapping.

And, if we consider \(u = \pi \), \(v = 2\pi \), then it is easy to see that \(\vert u - v \vert = \pi \) and hence

$$\begin{aligned} \vert Tu - Tv \vert = {}& \biggl\vert \frac{u\cos u}{2} - \frac{v\cos v}{2} \biggr\vert = \frac{\pi }{2} \vert \cos \pi - 2\cos 2\pi \vert \\ < {}& \frac{\pi }{2} \vert \cos \pi - \cos 2\pi \vert \\ \le {}&\frac{\pi }{2} \vert \pi - 2\pi \vert = \frac{\pi ^{2}}{2} = L \vert u - v \vert , \end{aligned}$$

for all \(u,v \in B\) and \(L = \frac{\pi }{2} > 0\). Thus, T is a Lipschitz strictly hemicontractive mapping.

Furthermore, for a sequence \(\{ \frac{1}{n} \} \) we have

$$\begin{aligned} \bigl\vert T^{n}u - T^{n}v \bigr\vert = {}& \frac{1}{2^{n}} \vert u\cos u - v\cos v \vert = \frac{\pi }{2^{n}} \vert \cos \pi - 2\cos 2\pi \vert \\ < {}& \frac{\pi }{2^{n}} \vert \cos \pi - \cos 2\pi \vert \\ \le {}&\frac{\pi }{2^{n}} \vert \pi - 2\pi \vert = \frac{\pi }{2^{n}} \cdot \pi < \biggl( 1 + \frac{1}{n} \biggr) \cdot \pi = \biggl( 1 + \frac{1}{n} \biggr) \vert u - v \vert , \end{aligned}$$

for all \(u,v \in B\) and \(n \ge 1\). Hence, T is an asymptotically non-expansive mapping. Therefore, a Lipschitz strictly hemicontractive mapping may also be an asymptotically non-expansive mapping.

The following example shows that a strictly hemicontractive mapping is neither a Lipschitzian mapping nor an asymptotically non-expansive mapping.

Example 1.4

(See [23])

Let \(X = \mathbb{R}\) (with the usual norm), \(B = [ 0, 1 ]\) and let φ be the Cantor ternary function. If we define \(T:B \to X\) by

$$ Tu = \textstyle\begin{cases} \frac{u}{2} & \text{if } 0 \le u \le \frac{1}{2}, \\ \varphi ( ( 1 - u ) / 2 ) & \text{if } \frac{1}{2} < u \le 1, \end{cases} $$

then \(T^{n}u \to 0\) uniformly on B and T is a strictly hemicontractive mapping. But we observe that T is neither a Lipschitzian mapping nor an asymptotically non-expansive mapping.

In 2006, Plubtieng and Wangkeeree [24] introduced and studied the following multi-step Noor iterative procedure with errors for some special type of asymptotically non-expansive mappings (asymptotically non-expansive mapping in the intermediate sense and asymptotically quasi-non-expansive mapping) in Banach spaces: For a given \(u_{1} \in B\), and a fixed \(m \in \mathbf{N}\) (set of all positive integers), the iterative sequences \(\{ u_{n}^{(1)} \} , \{ u_{n}^{(2)} \} ,\ldots, \{ u_{n}^{(m)} \} \) defined by

$$ \left . \textstyle\begin{array}{l} u_{n}^{(1)} = a_{n}^{(1)}T^{n}u_{n} + b_{n}^{(1)}u_{n} + c_{n}^{(1)}v_{n}^{(1)} ,\\ u_{n}^{(2)} = a_{n}^{(2)}T^{n}u_{n}^{(1)} + b_{n}^{(2)}u_{n} + c_{n}^{(2)}v_{n}^{(2)}, \\ \cdots \cdots \\ u_{n}^{(m - 1)} = a_{n}^{(m - 1)}T^{n}u_{n}^{(m - 2)} + b_{n}^{(m - 1)}u_{n} + c_{n}^{(m - 1)}v_{n}^{(m - 1)} ,\\ u_{n + 1} = u_{n}^{(m)} = a_{n}^{(m)}T^{n}u_{n}^{(m - 1)} + b_{n}^{(m)}u_{n} + c_{n}^{(m)}v_{n}^{(m)}, \quad n \ge 1, \end{array}\displaystyle \right \} $$
(1.3)

where \(\{ v_{n}^{(1)} \} ,\ldots, \{ v_{n}^{(m)} \} \) are bounded sequences in B and \(\{ a_{n}^{(i)} \} \), \(\{ b_{n}^{(i)} \} \), \(\{ c_{n}^{(i)} \} \) are appropriate real sequences in \([0, 1]\) such that \(a_{n}^{(i)} + b_{n}^{(i)} + c_{n}^{(i)} = 1\) for each \(i \in \{ 1, 2,\ldots, m\}\).

The iterative procedure given by (1.3) is known as the multi-step Noor iterative procedure with errors (MNIPE). After Plubtieng and Wangkeeree [24], a numerous number of research articles have been published on different types of iterative procedures with errors for various kinds of mappings; see for instance [1, 9, 12, 2527] and the references cited therein. Among the above-mentioned articles, Hussain et al. [1] studied the following special type of Ishikawa iterative procedure with errors (STIIPE) for two Lipschitz strictly hemicontractive-type mappings in arbitrary Banach spaces: For a given \(u_{0} \in B\), the iterative sequences \(\{ u_{n} \} _{n = 0}^{\infty }\) defined by

$$ \left . \textstyle\begin{array}{l} u_{n + 1} = u_{n}^{(2)} = a_{n}^{(2)}u_{n} + b_{n}^{(2)}Tu_{n}^{(1)} + c_{n}^{(2)}v_{n}^{(2)} ,\\ u_{n}^{(1)} = a_{n}^{(1)}u_{n} + b_{n}^{(1)}Su_{n} + c_{n}^{(1)}v_{n}^{(1)},\quad n \ge 0, \end{array}\displaystyle \right \} $$
(1.4)

where \(\{ v_{n}^{(1)} \} \), \(\{ v_{n}^{(2)} \} \) are bounded sequences in B and \(\{ a_{n}^{(i)} \} \), \(\{ b_{n}^{(i)} \} \), \(\{ c_{n}^{(i)} \} \) are appropriate real sequences in \([0, 1]\) satisfying \(a_{n}^{(i)} + b_{n}^{(i)} + c_{n}^{(i)} = 1\) for all \(i \in \{ 1, 2\}\).

Stimulated by the work of Hussain et al. [1, 9], Plubtieng and Wangkeeree [24], Yu et al. [11], Agwu and Igbokwe [17] and Zegeye and Tufa [19] in this paper, we propose and study the following modified multi-step Noor iterative procedure with errors (MMNIPE) for two Lipschitz strictly hemicontractive-type mappings in arbitrary Banach spaces: For a given \(u_{0} \in B\), and a fixed \(m \in \mathbf{N}\), we compute the iterative sequences \(\{ u_{n} \} _{n = 0}^{\infty } \) by

$$ \left . \textstyle\begin{array}{l} u_{n + 1} = u_{n}^{(m)} = a_{n}^{(m)}u_{n} + b_{n}^{(m)}Tu_{n}^{(m - 1)} + c_{n}^{(m)}v_{n}^{(m)} ,\\ u_{n}^{(m - 1)} = a_{n}^{(m - 1)}u_{n} + b_{n}^{(m - 1)}Tu_{n}^{(m - 2)} + c_{n}^{(m - 1)}v_{n}^{(m - 1)} ,\\ \cdots \cdots \\ u_{n}^{(2)} = a_{n}^{(2)}u_{n} + b_{n}^{(2)}Tu_{n}^{(1)} + c_{n}^{(2)}v_{n}^{(2)}, \\ u_{n}^{(1)} = a_{n}^{(1)}u_{n} + b_{n}^{(1)}Su_{n} + c_{n}^{(1)}v_{n}^{(1)}, \quad n \ge 0, \end{array}\displaystyle \right \} $$
(1.5)

where \(\{ v_{n}^{(1)} \} ,\ldots, \{ v_{n}^{(m)} \} \) are bounded sequences in B and \(\{ a_{n}^{(i)} \} \), \(\{ b_{n}^{(i)} \} \), \(\{ c_{n}^{(i)} \} \) are appropriate real sequences in \([0, 1]\) such that \(a_{n}^{(i)} + b_{n}^{(i)} + c_{n}^{(i)} = 1\) for each \(i \in \{ 1, 2,\ldots, m\}\).

Remark 1.5

It is clear that the iterative procedures defined by (1.4) (the STIIPE given by Hussain et al. [1]), the Mann iterative procedure (MIP) given by Mann [28], the Ishikawa iterative procedure (IIP) given by Ishikawa [29], the Noor iterative procedure (NIP) given by Xu and Noor [30], Mann iterative procedures with errors (MIPE) given by Liu [31] and Xu [32], the Ishikawa iterative procedure with errors (IIPE) given by Liu [31] and Xu [32] and the three-step iterative procedure with errors (TIPE) given by Cho et al. [33] are all special cases of the newly proposed MMNIPE given by (1.5). That is, the iterative procedure defined by (1.5) is a general iterative procedure among the above-mentioned iterative procedures.

To the best of our knowledge, there does not exist any work about the convergence and almost common-stability and common-stability of the iterative procedure given by (1.5) for Lipschitz strictly hemicontractive-type mappings in arbitrary Banach spaces. From this context, here we establish the convergence, almost common-stability and common-stability of the newly proposed MMNIPE given by (1.5) for two Lipschitz strictly hemicontractive-type mappings in arbitrary Banach spaces. The rest of this paper is organized as follows:

In Sect. 2, we recall some essential definitions and fundamental results. Sect. 3 is the main part of this paper. Here, we establish convergence, almost common-stability and common-stability of our proposed MMNIPE given by (1.5). In Sect. 4, we discuss a numerical example to verify the main results of this paper. Finally, in Sect. 5, we conclude this paper.

Preliminary notes

This section is devoted to recalling some definitions and fundamental results which are truly needed to establish the main results.

Definition 2.1

(See [4, 34])

The mapping T is called pseudocontractive if the inequality

$$ \Vert q - r \Vert \le \bigl\Vert q - r + t \bigl((I - T)q - (I - T)r \bigr) \bigr\Vert $$
(2.1)

holds for each \(q, r \in B\) and for all \(t > 0\). According to the result of Kato [35], it follows that T is a pseudocontractive if and only if there exists a \(h(q - r) \in J(q - r)\) such that

$$ \bigl\langle Tq - Tr, h(q - r) \bigr\rangle \le \Vert q - r \Vert ^{2} $$
(2.2)

for all \(q, r \in B\). T is called strongly pseudocontractive if there exists a \(t > 1\) such that

$$ \Vert q - r \Vert \le \bigl\Vert \bigl(1 + t' \bigr) (q - r) - t't(Tq - Tr) \bigr\Vert $$
(2.3)

for all \(q, r \in D_{T}\) and \(t' > 0\). T is called local strongly pseudocontractive if, for each \(q \in D_{T}\), there exists a \(t_{q} > 1\) such that

$$ \Vert q - r \Vert \le \bigl\Vert \bigl(1 + t' \bigr) (q - r) - t't_{q}(Tq - Tr) \bigr\Vert $$
(2.4)

for all \(q, r \in D_{T}\) and \(t' > 0\).

Definition 2.2

(See [3638])

Suppose \(u_{0} \in B\) and \(u_{n + 1} = f(u_{n}, T)\) defines an iterative procedure which yields a sequence of points \(\{ u_{n} \} \subset B\). Let \(F_{T} \ne \varphi \) and let \(\{ u_{n} \} \) converge to a fixed point q of T. Let \(\{ v_{n} \} \subset B\) and \(\{ \delta _{n} \} \) be a sequence in \([0, \infty )\), where \(\delta _{n} = \Vert v_{n + 1} - f(v_{n}, T) \Vert \). Now, if \(\lim_{n \to \infty } \delta _{n} = 0\) implies that \(\lim_{n \to \infty } v_{n} = q\), then the iterative procedure defined by \(u_{n + 1} = f(u_{n}, T)\) is said to be T-stable or stable on B with respect to T and if \(\sum_{n = 0}^{\infty } \delta _{n} < \infty \) implies that \(\lim_{n \to \infty } v_{n} = q\), then the iterative procedure defined by \(u_{n + 1} = f(u_{n}, T)\) is said to be an almost T-stable on B with respect to T.

Definition 2.3

(See [1])

Let B be a nonempty convex subset of an arbitrary Banach space X and let T and S be two self-operators on B. Suppose \(u_{0} \in B\) and \(u_{n + 1} = f(u_{n}, T, S)\) defines an iterative procedure which yields a sequence of points \(\{ u_{n} \} \subset B\). Let \(F_{T} \cap F_{S} \ne \varphi \) and let \(\{ u_{n} \} \) converges strongly to a common fixed point q of T and S. Let \(\{ v_{n} \} \) be any bounded sequence in B and \(\{ \mu _{n} \} \) be a sequence in \([0, \infty )\), where \(\mu _{n} = \Vert v_{n + 1} - f(v_{n}, T, S) \Vert \). Now, if \(\lim_{n \to \infty } \mu _{n} = 0\) implies that \(\lim_{n \to \infty } v_{n} = r\), then the iterative procedure defined by \(u_{n + 1} = f(u_{n}, T, S)\) is said to be a common-stable on B and if \(\sum_{n = 0}^{\infty } \mu _{n} < \infty \) implies that \(\lim_{n \to \infty } v_{n} = q\), then the iterative procedure defined by \(u_{n + 1} = f(u_{n}, T, S)\) is said to be an almost common-stable on B.

Now, we recall some lemmas which are essential to prove the main results of this paper.

Lemma 2.4

(See [39])

Let \(\{ \alpha _{n} \} _{n = 0}^{\infty }\), \(\{ \beta _{n} \} _{n = 0}^{\infty }\), \(\{ \gamma _{n} \} _{n = 0}^{\infty }\) and \(\{ \omega _{n} \} _{n = 0}^{\infty }\) be nonnegative real sequences such that

$$ \alpha _{n + 1} \le ( 1 - \omega _{n} )\alpha _{n} + \omega _{n}\beta _{n} + \gamma _{n},\quad n \ge 0, $$
(2.5)

with \(\{ \omega _{n} \} _{n = 0}^{\infty } \subset [0, 1]\), \(\sum_{n = 0}^{\infty } \omega _{n} = \infty \), \(\sum_{n = 0}^{\infty } \gamma _{n} < \infty \) and \(\lim_{n \to \infty } \beta _{n} = 0\). Then \(\lim_{n \to \infty } \alpha _{n} = 0\).

Lemma 2.5

(See [40])

Let \(\{ \alpha _{n} \} _{n = 0}^{\infty }\) and \(\{ \beta _{n} \} _{n = 0}^{\infty }\) be sequences of nonnegative real numbers and \(0 \le \eta < 1\), so that

$$\begin{aligned}& \alpha _{n + 1} \le \eta \alpha _{n} + \beta _{n}, \quad \forall n \ge 0. \end{aligned}$$
(2.6)

(i) If \(\lim_{n \to \infty } \beta _{n} = 0\), then \(\lim_{n \to \infty } \alpha _{n} = 0\).

(ii) If \(\sum_{n = 0}^{\infty } \beta _{n} < \infty \), then \(\sum_{n = 0}^{\infty } \alpha _{n} < \infty \).

Lemma 2.6

(See [35])

Let \(x, y \in X\). Then \(\Vert x \Vert \le \Vert x + ry \Vert \) for every \(r > 0\) if and only if there is \(f \in J(x)\) such that \(\operatorname{Re} (y, f) \ge 0\).

Lemma 2.7

(See [4])

Let \(T:D_{T} \subseteq X \to X\) be an operator with \(F_{T} \ne \varphi \). Then T is strictly hemicontractive if and if only if there exists a \(t > 1\) such that for all \(x \in D_{T}\) and \(q \in F_{T}\) there exists \(h \in J(x - q)\) satisfying

$$ \operatorname{Re} \bigl\langle x - Tx, h(x - q) \bigr\rangle \ge \biggl( 1 - \frac{1}{t} \biggr) \Vert x - q \Vert ^{2}. $$
(2.7)

Lemma 2.8

(See [12])

Let X be an arbitrary norm linear space and \(T:D_{T} \subseteq X \to X\) be an operator.

  1. (a)

    If T is a local strongly pseudocontractive operator and \(F_{T} \ne \varphi \), then \(F_{T}\) is a singleton and T is strictly hemicontractive.

  2. (b)

    If T is strictly hemicontractive, then \(F_{T}\) is a singleton.

Convergence and stability of modified multi-step Noor iterative procedure with errors

In this section, we state and prove the convergence and stability of our proposed MMNIPE for two Lipschitz strictly hemicontractive-type mappings.

Let \(\lambda = \frac{\sigma - 1}{\sigma } \in (0, 1)\), where \(\sigma > 1\), L be a common Lipschitz constant of two strictly hemicontractive-type mappings T, S and I be an identity mapping on the arbitrary Banach space X. In the above-mentioned context, we state and prove the following theorems.

Theorem 3.1

Let B be a nonempty closed convex subset of X and T and S be two Lipschitz strictly hemicontractive-type mappings from B into itself. Suppose that \(\{ v_{n}^{(1)} \} ,\ldots, \{ v_{n}^{(m)} \} \) are arbitrary bounded sequences in B and \(\{ a_{n}^{(i)} \} \), \(\{ b_{n}^{(i)} \} \), \(\{ c_{n}^{(i)} \} \) for each \(i \in \{ 1, 2,\ldots, m\}\) are any appropriate real sequences in \([0, 1]\) satisfying the following conditions:

  1. (1)

    \(a_{n}^{(i)} + b_{n}^{(i)} + c_{n}^{(i)} = 1\), for each \(i \in \{ 1,2, 3,\ldots, m\}\),

  2. (2)

    \(c_{n}^{ ( m )} = \mathrm{o} ( b_{n}^{ ( m )} )\),

  3. (3)

    \(\lim_{n \to \infty } c_{n}^{ ( j )} = 0\), for each \(j \in \{ 1, 2, 3,\ldots, m - 1 \} \),

  4. (4)

    \(\sum_{n = 0}^{\infty } b_{n}^{(j)} = \infty \) for each \(j \in \{ 2, 3,\ldots, m\}\),

  5. (5)
    $$\begin{aligned}& L \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L )^{2}b_{n}^{(m - 1)} + L ( 1 + L )^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad {}+ L^{2} ( 1 + L )^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} + \cdots + L^{m - 3} ( 1 + L )^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \\& \quad {}+ L^{m - 2}(1 + L)^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}b_{n}^{(1)} \\& \quad {}+ \bigl[ c_{n}^{(m)} + ( 1 + L )c_{n}^{(m - 1)} + L ( 1 + L )b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad {}+ L^{2} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} + \cdots + L^{m - 3} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \\& \quad {}+ L^{m - 2} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] + \frac{c_{n}^{(m)}}{b_{n}^{(m)}} \le \lambda ( \lambda - \theta ), \quad n \ge 0, \end{aligned}$$

    where θ is a constant in \((0, \lambda )\) and \(\lambda \in (0, 1)\).

Assume an iterative sequence \(\{ u_{n} \} _{n = 0}^{\infty } \) defined by (1.5). Let \(\{ w_{n} \} _{n = 0}^{\infty }\) be any sequence in B and \(\{ \mu _{n} \} _{n = 0}^{\infty } \) be a sequence defined by

$$ \mu _{n} = \Vert w_{n + 1} - x_{n} \Vert , \quad n \ge 0, $$

where

$$ \left . \textstyle\begin{array}{l} x_{n} = x_{n}^{(m)} = a_{n}^{(m)}w_{n} + b_{n}^{(m)}Tx_{n}^{(m - 1)} + c_{n}^{(m)}v_{n}^{(m)} ,\\ x_{n}^{(m - 1)} = a_{n}^{(m - 1)}w_{n} + b_{n}^{(m - 1)}Tx_{n}^{(m - 2)} + c_{n}^{(m - 1)}v_{n}^{(m - 1)} ,\\ \cdots\cdots \\ x_{n}^{(2)} = a_{n}^{(2)}w_{n} + b_{n}^{(2)}Tx_{n}^{(1)} + c_{n}^{(2)}v_{n}^{(2)} ,\\ x_{n}^{(1)} = a_{n}^{(1)}w_{n} + b_{n}^{(1)}Sw_{n} + c_{n}^{(1)}v_{n}^{(1)}, \quad n \ge 0. \end{array}\displaystyle \right \} $$
(3.1)

Then

  1. (i)

    the iterative sequence \(\{ u_{n} \} _{n = 0}^{\infty }\) given by (1.5) converges strongly to the common fixed r of T and S and the following inequality holds:

    $$\begin{aligned}& \Vert u_{n + 1} - r \Vert \\& \quad \le \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)}Lb_{n}^{(m)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + c_{n}^{(m - 2)}L^{2}b_{n}^{(m)}b_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m - 3)}L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + c_{n}^{(2)}L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{} + c_{n}^{(1)}L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr], \quad n \ge 0, \end{aligned}$$
  2. (ii)
    $$\begin{aligned}& \Vert w_{n + 1} - r \Vert \\& \quad \le \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert \\& \quad\quad{} + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + L \bigl[ b_{n}^{(m)}c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{} + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] \bigr] \bigr] \bigr] \cdots \bigr] \bigr] + \mu _{n}, \quad n \ge 0, \end{aligned}$$
  3. (iii)

    \(\sum_{n = 0}^{\infty } \mu _{n} < \infty \) implies that \(\lim_{n \to \infty } w_{n} = r\), so that \(\{ u_{n}^{(m)} \} _{n = 0}^{\infty } \) is almost common-stable on B,

  4. (iv)

    \(\lim_{n \to \infty } w_{n} = r\), implies that \(\lim_{n \to \infty } \mu _{n} = 0\).

Proof

(i) From the condition (2), we obtain \(c_{n}^{ ( m )} = \delta _{n}b_{n}^{ ( m )}\), and \(\delta _{n} \to 0\) as \(n \to \infty \). By an application of Lemma 2.8, we see that \(F_{T} \cap F_{S}\) is singleton, and let \(F_{T} \cap F_{S} = \{ r \} \) for some \(r \in B\). Put

$$ A = \max \Bigl\{ \sup_{n \ge 0} \bigl\{ \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr\} , \sup _{n \ge 0} \bigl\{ \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \bigr\} , \sup_{n \ge 0} \bigl\{ \bigl\Vert v_{n}^{(3)} - r \bigr\Vert \bigr\} ,\ldots, \sup _{n \ge 0} \bigl\{ \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \bigr\} \Bigr\} . $$

Since T is strictly hemicontractive, from Lemma 2.7, we obtain

$$\begin{aligned}& \operatorname{Re} \bigl\langle x - Tx, h(x - r) \bigr\rangle \ge \lambda \Vert x - r \Vert ^{2} \\& \Rightarrow \quad \operatorname{Re} \bigl\langle (I - T - \lambda I)x - (I - T - \lambda I)r, h(x - r) \bigr\rangle \ge 0, \quad \forall x \in B. \end{aligned}$$
(3.2)

Now, from (3.2) and Lemma 2.6, we have

$$ \Vert x - r \Vert \le \bigl\Vert x - r + q \bigl[ (I - T - \lambda I)x - (I - T - \lambda I)r \bigr] \bigr\Vert , \quad \forall x \in B, \text{ and } \forall q > 0. $$
(3.3)

Also, from the first equation of (1.5), we get

$$\begin{aligned}& u_{n + 1} = a_{n}^{(m)}u_{n} + b_{n}^{(m)}Tu_{n}^{(m - 1)} + c_{n}^{(m)}v_{n}^{(m)} \\& \begin{aligned} \Rightarrow\quad \bigl( 1 - b_{n}^{(m)} \bigr)u_{n} &= \bigl( 1 - (1 - \lambda )b_{n}^{(m)} \bigr) u_{n + 1} + b_{n}^{(m)} ( I - T - \lambda I )u_{n + 1} \\ &\quad{} + b_{n}^{(m)} \bigl( Tu_{n + 1} - Tu_{n}^{(m - 1)} \bigr) - c_{n}^{(m)} \bigl( v_{n}^{(m)} - u_{n} \bigr), \end{aligned} \end{aligned}$$
(3.4)

and since \(r \in B\) is the fixed point of T, it follows that

$$ \bigl( 1 - b_{n}^{(m)} \bigr)r = \bigl( 1 - (1 - \lambda )b_{n}^{(m)} \bigr) r + b_{n}^{(m)} ( I - T - \lambda I )r. $$
(3.5)

Now, for all \(n \ge 0\) from (3.4) and (3.5), we have

$$\begin{aligned}& \bigl( 1 - b_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert \\& \quad \ge \bigl\Vert \bigl( 1 - (1 - \lambda )b_{n}^{(m)} \bigr) ( u_{n + 1} - r ) + b_{n}^{(m)} ( I - T - \lambda I ) ( u_{n + 1} - r ) \bigr\Vert \\& \quad\quad{} - b_{n}^{(m)} \bigl\Vert Tu_{n + 1} - Tu_{n}^{(m - 1)} \bigr\Vert - c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - u_{n} \bigr\Vert \\& \quad = \bigl( 1 - (1 - \lambda )b_{n}^{(m)} \bigr) \biggl\Vert u_{n + 1} - r + \frac{b_{n}^{(m)}}{1 - (1 - \lambda )b_{n}^{(m)}} ( I - T - \lambda I ) ( u_{n + 1} - r ) \biggr\Vert \\& \quad\quad{} - b_{n}^{(m)} \bigl\Vert Tu_{n + 1} - Tu_{n}^{(m - 1)} \bigr\Vert - c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - u_{n} \bigr\Vert \\& \quad \ge \bigl( 1 - (1 - \lambda )b_{n}^{(m)} \bigr) \Vert u_{n + 1} - r \Vert - b_{n}^{(m)} \bigl\Vert Tu_{n + 1} - Tu_{n}^{(m - 1)} \bigr\Vert - c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - u_{n} \bigr\Vert \end{aligned}$$

which implies that

$$\begin{aligned} & \Vert u_{n + 1} - r \Vert \\ &\quad \le \frac{1 - b_{n}^{(m)}}{ 1 - (1 - \lambda )b_{n}^{(m)}} \Vert u_{n} - r \Vert + \frac{b_{n}^{(m)}}{ 1 - (1 - \lambda )b_{n}^{(m)}} \bigl\Vert Tu_{n + 1} - Tu_{n}^{(m - 1)} \bigr\Vert \\ &\quad\quad{}+ \frac{c_{n}^{(m)}}{ 1 - (1 - \lambda )b_{n}^{(m)}} \bigl\Vert v_{n}^{(m)} - u_{n} \bigr\Vert \\ &\quad \le \bigl( 1 - \lambda b_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert + \lambda ^{ - 1}b_{n}^{(m)} \bigl\Vert Tu_{n + 1} - Tu_{n}^{(m - 1)} \bigr\Vert + \lambda ^{ - 1}c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - u_{n} \bigr\Vert \\ &\quad \le \bigl( 1 - \lambda b_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert + \lambda ^{ - 1}Lb_{n}^{(m)} \bigl\Vert u_{n + 1} - u_{n}^{(m - 1)} \bigr\Vert + \lambda ^{ - 1}c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - u_{n} \bigr\Vert \\ &\quad \le \bigl( 1 - \lambda b_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert + \lambda ^{ - 1}Lb_{n}^{(m)} \bigl\Vert u_{n + 1} - u_{n}^{(m - 1)} \bigr\Vert + \lambda ^{ - 1}c_{n}^{(m)} \bigl( \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + \Vert u_{n} - r \Vert \bigr) \\ &\quad = \bigl( 1 - \lambda b_{n}^{(m)} + \lambda ^{ - 1}c_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert + \lambda ^{ - 1}Lb_{n}^{(m)} \bigl\Vert u_{n + 1} - u_{n}^{(m - 1)} \bigr\Vert \\ &\quad\quad{}+ \lambda ^{ - 1}c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert . \end{aligned}$$
(3.6)

Again, from (1.5), we get

$$\begin{aligned}& \bigl\Vert u_{n + 1} - u_{n}^{(m - 1)} \bigr\Vert \\& \quad = \bigl\Vert a_{n}^{(m)}u_{n} + b_{n}^{(m)}Tu_{n}^{(m - 1)} + c_{n}^{(m)}v_{n}^{(m)} - a_{n}^{(m - 1)}u_{n} - b_{n}^{(m - 1)}Tu_{n}^{(m - 2)} - c_{n}^{(m - 1)}v_{n}^{(m - 1)} \bigr\Vert \\& \quad = \bigl\Vert \bigl( 1 - b_{n}^{(m)} - c_{n}^{(m)} \bigr)u_{n} + b_{n}^{(m)}Tu_{n}^{(m - 1)} + c_{n}^{(m)}v_{n}^{(m)} - \bigl( 1 - b_{n}^{(m - 1)} - c_{n}^{(m - 1)} \bigr)u_{n} \\& \quad\quad{} - b_{n}^{(m - 1)}Tu_{n}^{(m - 2)} - c_{n}^{(m - 1)}v_{n}^{(m - 1)} \bigr\Vert \\& \quad = \bigl\Vert b_{n}^{(m)} \bigl( Tu_{n}^{(m - 1)} - u_{n} \bigr) + c_{n}^{(m)} \bigl( v_{n}^{(m)} - u_{n} \bigr) + b_{n}^{(m - 1)} \bigl( u_{n} - Tu_{n}^{(m - 2)} \bigr) - c_{n}^{(m - 1)} \bigl( v_{n}^{(m - 1)} - u_{n} \bigr) \bigr\Vert \\& \quad \le \bigl\Vert b_{n}^{(m)} \bigl( Tu_{n}^{(m - 1)} - u_{n} \bigr) + c_{n}^{(m)} \bigl( v_{n}^{(m)} - u_{n} \bigr) \bigr\Vert \\& \quad\quad{}+ \bigl\Vert b_{n}^{(m - 1)} \bigl( u_{n} - Tu_{n}^{(m - 2)} \bigr) - c_{n}^{(m - 1)} \bigl( v_{n}^{(m - 1)} - u_{n} \bigr) \bigr\Vert \\& \quad \le b_{n}^{(m)} \bigl\Vert u_{n} - Tu_{n}^{(m - 1)} \bigr\Vert + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - u_{n} \bigr\Vert + b_{n}^{(m - 1)} \bigl\Vert u_{n} - Tu_{n}^{(m - 2)} \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - u_{n} \bigr\Vert \\& \quad \le b_{n}^{(m)} \bigl( \Vert u_{n} - r \Vert + \bigl\Vert r - Tu_{n}^{(m - 1)} \bigr\Vert \bigr) + c_{n}^{(m)} \bigl( \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + \Vert u_{n} - r \Vert \bigr) \\& \quad\quad{}+ b_{n}^{(m - 1)} \bigl( \Vert u_{n} - r \Vert + \bigl\Vert r - Tu_{n}^{(m - 2)} \bigr\Vert \bigr) + c_{n}^{(m - 1)} \bigl( \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert + \Vert u_{n} - r \Vert \bigr) \\& \quad \le b_{n}^{(m)} \bigl( \Vert u_{n} - r \Vert + L \bigl\Vert u_{n}^{(m - 1)} - r \bigr\Vert \bigr) + c_{n}^{(m)} \bigl( \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + \Vert u_{n} - r \Vert \bigr) \\& \quad\quad{}+ b_{n}^{(m - 1)} \bigl( \Vert u_{n} - r \Vert + L \bigl\Vert u_{n}^{(m - 2)} - r \bigr\Vert \bigr) + c_{n}^{(m - 1)} \bigl( \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert + \Vert u_{n} - r \Vert \bigr) \\& \quad = \bigl[ b_{n}^{(m)} + b_{n}^{(m - 1)} + c_{n}^{(m)} + c_{n}^{(m - 1)} \bigr] \Vert u_{n} - r \Vert + b_{n}^{(m)}L \bigl\Vert u_{n}^{(m - 1)} - r \bigr\Vert + b_{n}^{(m - 1)}L \bigl\Vert u_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert . \end{aligned}$$
(3.7)

But

$$\begin{aligned} \bigl\Vert u_{n}^{(m - 1)} - r \bigr\Vert &= \bigl\Vert a_{n}^{(m - 1)}u_{n} + b_{n}^{(m - 1)}Tu_{n}^{(m - 2)} + c_{n}^{(m - 1)}v_{n}^{(m - 1)} - r \bigr\Vert \\ &= \bigl\Vert \bigl( 1 - b_{n}^{(m - 1)} - c_{n}^{(m - 1)} \bigr)u_{n} + b_{n}^{(m - 1)}Tu_{n}^{(m - 2)} + c_{n}^{(m - 1)}v_{n}^{(m - 1)} - r \bigr\Vert \\ &= \bigl\Vert u_{n} - r + b_{n}^{(m - 1)} \bigl( Tu_{n}^{(m - 2)} - r + r - u_{n} \bigr) + c_{n}^{(m - 1)} \bigl( v_{n}^{(m - 1)} - r + r - u_{n} \bigr) \bigr\Vert \\ &\le \Vert u_{n} - r \Vert + b_{n}^{(m - 1)} \bigl\Vert Tu_{n}^{(m - 2)} - r \bigr\Vert + b_{n}^{(m - 1)} \Vert u_{n} - r \Vert + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\ &\quad\quad{} + c_{n}^{(m - 1)} \Vert u_{n} - r \Vert \\ & \le \Vert u_{n} - r \Vert + b_{n}^{(m - 1)}L \bigl\Vert u_{n}^{(m - 2)} - r \bigr\Vert + b_{n}^{(m - 1)} \Vert u_{n} - r \Vert + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\ &\quad\quad{} + c_{n}^{(m - 1)} \Vert u_{n} - r \Vert \\ &= \bigl[ 1 + b_{n}^{(m - 1)} + c_{n}^{(m - 1)} \bigr] \Vert u_{n} - r \Vert + b_{n}^{(m - 1)}L \bigl\Vert u_{n}^{(m - 2)} - r \bigr\Vert \\ &\quad{} + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert . \end{aligned}$$
(3.8)

Substituting (3.8) in (3.7), we have

$$\begin{aligned}& \bigl\Vert u_{n + 1} - u_{n}^{(m - 1)} \bigr\Vert \\& \quad \le \bigl[ b_{n}^{(m)} + b_{n}^{(m - 1)} + c_{n}^{(m)} + c_{n}^{(m - 1)} \bigr] \Vert u_{n} - r \Vert \\& \quad\quad{}+ b_{n}^{(m)}L \bigl[ \bigl[ 1 + b_{n}^{(m - 1)} + c_{n}^{(m - 1)} \bigr] \Vert u_{n} - r \Vert + b_{n}^{(m - 1)}L \bigl\Vert u_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \bigr] \\& \quad\quad{}+ b_{n}^{(m - 1)}L \bigl\Vert u_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad = \bigl[ b_{n}^{(m)} + b_{n}^{(m - 1)} + b_{n}^{(m)}L \bigl[ 1 + b_{n}^{(m - 1)} + c_{n}^{(m - 1)} \bigr] + c_{n}^{(m)} + c_{n}^{(m - 1)} \bigr] \Vert u_{n} - r \Vert \\& \quad\quad{}+ \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}L^{2} + b_{n}^{(m - 1)}L \bigr] \bigl\Vert u_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \\& \quad\quad{}+ \bigl[ b_{n}^{(m)}Lc_{n}^{(m - 1)} + c_{n}^{(m - 1)} \bigr] \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad = \bigl[ ( 1 + L )b_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} \bigr] \Vert u_{n} - r \Vert \\& \quad\quad{}+ Lb_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert u_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert . \end{aligned}$$
(3.9)

But, if we replace m by \(m - 1\) in (3.8), then we have

$$ \begin{aligned}[b] \bigl\Vert u_{n}^{(m - 2)} - r \bigr\Vert = {}&\bigl[ 1 + b_{n}^{(m - 2)} + c_{n}^{(m - 2)} \bigr] \Vert u_{n} - r \Vert + b_{n}^{(m - 2)}L \bigl\Vert u_{n}^{(m - 3)} - r \bigr\Vert \\&{}+ c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert . \end{aligned} $$
(3.10)

Now, substituting (3.10) in (3.9), we have

$$ \begin{gathered} \bigl\Vert u_{n + 1} - u_{n}^{(m - 1)} \bigr\Vert \\ \quad \le \bigl[ ( 1 + L )b_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} \bigr] \Vert u_{n} - r \Vert \\ \quad \quad{} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} \bigl[ \bigl[ 1 + b_{n}^{(m - 2)} + c_{n}^{(m - 2)} \bigr] \Vert u_{n} - r \Vert + b_{n}^{(m - 2)}L \bigl\Vert u_{n}^{(m - 3)} - r \bigr\Vert \\ \quad \quad {}+ c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \bigr] + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\ \quad \le \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} + c_{n}^{(m)} \\ \quad \quad{} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigr] \Vert u_{n} - r \Vert \\ \quad \quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl\Vert u_{n}^{(m - 3)} - r \bigr\Vert + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \\ \quad \quad{} + c_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert + c_{n}^{(m - 2)}Lb_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert . \end{gathered} $$
(3.11)

But, if we replace m by \(m - 1\) in (3.10), then we have

$$ \begin{aligned}[b] \bigl\Vert u_{n}^{(m - 3)} - r \bigr\Vert = {}&\bigl[ 1 + b_{n}^{(m - 3)} + c_{n}^{(m - 3)} \bigr] \Vert u_{n} - r \Vert + b_{n}^{(m - 3)}L \bigl\Vert u_{n}^{(m - 4)} - r \bigr\Vert \\&{}+ c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert . \end{aligned} $$
(3.12)

Substituting (3.12) in (3.11), we have

$$\begin{aligned}& \bigl\Vert u_{n + 1} - u_{n}^{(m - 1)} \bigr\Vert \\& \quad \le \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} + c_{n}^{(m)} \\& \quad\quad{}+ \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigr] \Vert u_{n} - r \Vert \\& \quad\quad{}+ L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl[ \bigl[ 1 + b_{n}^{(m - 3)} + c_{n}^{(m - 3)} \bigr] \Vert u_{n} - r \Vert \\& \quad\quad{} + Lb_{n}^{(m - 3)} \bigl\Vert u_{n}^{(m - 4)} - r \bigr\Vert + c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \bigr] + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert + c_{n}^{(m - 2)}L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad = \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + L ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} + c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} \\& \quad\quad{} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigr] \Vert u_{n} - q \Vert \\& \quad\quad{}+ L^{3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \bigl\Vert u_{n}^{(m - 4)} - r \bigr\Vert \\& \quad\quad{} + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m - 2)}L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{} + c_{n}^{(m - 3)}L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert . \end{aligned}$$
(3.13)

Continuing the above procedure up to second iterative step of (1.5), we obtain

$$\begin{aligned}& \bigl\Vert u_{n + 1} - u_{n}^{(m - 1)} \bigr\Vert \\& \quad \le \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + L ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} + \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \\& \quad\quad{}+ \bigl[ c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad\quad{}+ L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \\& \quad\quad{}+ \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}c_{n}^{(3)} \bigr] \bigr] \Vert u_{n} - r \Vert \\& \quad\quad{}+ L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \bigl\Vert u_{n}^{(1)} - r \bigr\Vert + \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert + c_{n}^{(m - 2)}L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m - 3)}L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + c_{n}^{(2)}L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \bigr]. \end{aligned}$$
(3.14)

Now, from the last equation of (1.5), we have

$$\begin{aligned} \begin{aligned} \bigl\Vert u_{n}^{(1)} - r \bigr\Vert &= \bigl\Vert a_{n}^{(1)}u_{n} + b_{n}^{(1)}Su_{n} + c_{n}^{(1)}v_{n}^{(1)} - r \bigr\Vert \\ &\le \Vert u_{n} - r \Vert + b_{n}^{(1)} \Vert Su_{n} - r \Vert + b_{n}^{(1)} \Vert u_{n} - r \Vert + c_{n}^{(1)} \Vert u_{n} - r \Vert + c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \\ &\le \bigl( 1 + b_{n}^{(1)} + c_{n}^{(1)} \bigr) \Vert u_{n} - r \Vert + b_{n}^{(1)}L \Vert u_{n} - r \Vert + c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \\ &= \bigl[ 1 + (1 + L)b_{n}^{(1)} + c_{n}^{(1)} \bigr] \Vert u_{n} - r \Vert + c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert . \end{aligned} \end{aligned}$$
(3.15)

Substituting (3.15) in (3.14), we have

$$\begin{aligned}& \bigl\Vert u_{n + 1} - u_{n}^{(m - 1)} \bigr\Vert \\& \quad \le \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + L ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} + \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \\& \quad\quad{} + \bigl[ c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \\& \quad\quad{} + \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}c_{n}^{(3)} \bigr] \bigr] \Vert u_{n} - r \Vert \\& \quad\quad{} + L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \bigl[ \bigl( 1 + (1 + L)b_{n}^{(1)} + c_{n}^{(1)} \bigr) \Vert u_{n} - r \Vert \\& \quad\quad{} + c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + c_{n}^{(m - 2)}L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{} + c_{n}^{(m - 3)}L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + c_{n}^{(2)}L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad = \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + L ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \\& \quad\quad{} + \cdots + L^{m - 3} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} \\& \quad\quad{} + L^{m - 2}(1 + L) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}b_{n}^{(1)} \\& \quad\quad{} + \bigl[ c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \\& \quad\quad{} + \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}c_{n}^{(2)} \\& \quad\quad{}+ L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] \Vert u_{n} - r \Vert \\& \quad\quad{} + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + c_{n}^{(m - 2)}L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{} + c_{n}^{(m - 3)}L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + c_{n}^{(2)}L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{} + c_{n}^{(1)}L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert . \end{aligned}$$
(3.16)

Substituting (3.16) in (3.6), we have

$$\begin{aligned}& \Vert u_{n + 1} - r \Vert \\& \quad \le \bigl( 1 - \lambda b_{n}^{(m)} + \lambda ^{ - 1}c_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert \\& \quad\quad{} + \lambda ^{ - 1}Lb_{n}^{(m)} \bigl[ \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} \\& \quad\quad{} + L ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad\quad{}+ L^{2} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \\& \quad\quad{} + \cdots + L^{m - 3} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} \\& \quad\quad{}+ L^{m - 2}(1 + L) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}b_{n}^{(1)} \\& \quad\quad{}+ \bigl[ c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} + \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}c_{n}^{(2)} \\& \quad\quad{}+ L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] \Vert u_{n} - r \Vert \\& \quad\quad{}+ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m - 2)}L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m - 3)}L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{}+ \cdots + c_{n}^{(2)}L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(1)}L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] + \lambda ^{ - 1}c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \\& \quad = \bigl( 1 - \lambda b_{n}^{(m)} + \lambda ^{ - 1}c_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert \\& \quad\quad{}+ \lambda ^{ - 1}Lb_{n}^{(m)} \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} \\& \quad\quad{}+ L ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad\quad{}+ L^{2} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \\& \quad\quad{} + \cdots + L^{m - 3} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} \\& \quad\quad{}+ L^{m - 2}(1 + L) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}b_{n}^{(1)} \\& \quad\quad{}+ \bigl[ c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \\& \quad\quad{}+ \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}c_{n}^{(2)} \\& \quad\quad{} + L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] \Vert u_{n} - r \Vert \\& \quad\quad{} + c_{n}^{(m)}\lambda ^{ - 1} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)}\lambda ^{ - 1}L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m - 2)}\lambda ^{ - 1}L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m)}b_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{}+ \cdots + c_{n}^{(2)}\lambda ^{ - 1}L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(1)}\lambda ^{ - 1}L^{m - 1} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \\& \quad \le \bigl[ 1 - b_{n}^{(m)} \bigl[ \lambda - \lambda ^{ - 1}L \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L )^{2}b_{n}^{(m - 1)} + L ( 1 + L )^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad\quad{}+ L^{2} ( 1 + L )^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} + \cdots + L^{m - 3} ( 1 + L )^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} \\& \quad\quad{}+ L^{m - 2}(1 + L)^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)}b_{n}^{(1)} \\& \quad\quad{}+ c_{n}^{(m)} + ( 1 + L )c_{n}^{(m - 1)} + L ( 1 + L )b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad\quad{}+ L^{2} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} + \cdots + L^{m - 3} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \\& \quad\quad{} + L^{m - 2} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] - \lambda ^{ - 1}\delta _{n} \bigr] \bigr] \Vert u_{n} - r \Vert \\& \quad\quad{}+ \lambda ^{ - 1} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - q \bigr\Vert + c_{n}^{(m - 1)}Lb_{n}^{(m)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m - 2)}L^{2}b_{n}^{(m)}b_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m - 3)}L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{}+ \cdots + c_{n}^{(2)}L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(1)}L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] \\& \quad \le \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - q \bigr\Vert + c_{n}^{(m - 1)}Lb_{n}^{(m)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(m - 2)}L^{2}b_{n}^{(m)}b_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m - 3)}L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{}+ \cdots + c_{n}^{(2)}L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{}+ c_{n}^{(1)}L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] \\& \quad \le \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert + \delta _{n}\lambda ^{ - 1} ( 1 + L )b_{n}^{(m)}A \bigl[ 1 + Lb_{n}^{(m)}b_{n}^{(m - 1)} + L^{2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad\quad{}+ L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} + \cdots + L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)} \bigr] \\& \quad\quad{}+ L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} ]. \end{aligned}$$
(3.17)

Now, if we put

$$\begin{aligned}& \alpha _{n} = \Vert u_{n} - r \Vert , \\& \omega _{n} = \theta b_{n}^{(m)}, \\& \begin{aligned} \beta _{n} = {}&\theta ^{ - 1}\delta _{n}\lambda ^{ - 1} ( 1 + L )b_{n}^{(m)}A \bigl[ 1 + Lb_{n}^{(m)}b_{n}^{(m - 1)} \\ &{}+ L^{2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} + L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \\ &{}+ \cdots + L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}+ L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} \bigr], \end{aligned} \\& \gamma _{n} = 0, \end{aligned}$$

in (3.17), then, by condition (3), we observe that

$$ \alpha _{n + 1} \le ( 1 - \omega _{n} )\alpha _{n} + \omega _{n}\beta _{n} + \gamma _{n}, \quad n \ge 0, $$

with \(\{ \omega _{n} \} _{n = 0}^{\infty } \subset [0, 1]\), \(\sum_{n = 0}^{\infty } \omega _{n} = \infty \), \(\sum_{n = 0}^{\infty } \gamma _{n} < \infty \) and \(\lim_{n \to \infty } \beta _{n} = 0\).

Hence, from Lemma 2.4, we have

$$ \lim_{n \to \infty } \alpha _{n} = 0. $$

That is \(\lim_{n \to \infty } \Vert u_{n} - r \Vert = 0\).

This ensures that the sequence \(\{ u_{n} \} _{n = 0}^{\infty }\) of the MMNIPE given by (1.5) converges strongly to the common fixed r of T and S.

(ii) From the first equation of (3.1), we get

$$ \begin{gathered} x_{n} = a_{n}^{(m)}w_{n} + b_{n}^{(m)}Tx_{n}^{(m - 1)} + c_{n}^{(m)}v_{n}^{(m)} \\ \begin{aligned} \Rightarrow \quad \bigl( 1 - b_{n}^{(m)} \bigr)w_{n} = {}& \bigl( 1 - (1 - \lambda )b_{n}^{(m)} \bigr) x_{n} + b_{n}^{(m)} ( I - T - \lambda I )x_{n} \\ &{}+ b_{n}^{(m)} \bigl( Tx_{n} - Tx_{n}^{(m - 1)} \bigr) - c_{n}^{(m)} \bigl( v_{n}^{(m)} - w_{n} \bigr). \end{aligned} \end{gathered} $$
(3.18)

Now, for all \(n \ge 0\) combining (3.5) and (3.18), we obtain

$$\begin{aligned}& \begin{aligned} \bigl( 1 - b_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert &\ge \bigl\Vert \bigl( 1 - (1 - \lambda )b_{n}^{(m)} \bigr) ( x_{n} - r ) + b_{n}^{(m)} ( I - T - \lambda I ) ( x_{n} - r ) \bigr\Vert \\ &\quad{} - b_{n}^{(m)} \bigl\Vert Tx_{n} - Tx_{n}^{(m - 1)} \bigr\Vert - c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - w_{n} \bigr\Vert \\ &= \bigl( 1 - (1 - \lambda )b_{n}^{(m)} \bigr) \biggl\Vert x_{n} - r + \frac{b_{n}^{(m)}}{1 - (1 - \lambda )b_{n}^{(m)}} ( I - T - \lambda I ) ( x_{n} - r ) \biggr\Vert \\ &\quad{} - b_{n}^{(m)} \bigl\Vert Tx_{n} - Tx_{n}^{(m - 1)} \bigr\Vert - c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - w_{n} \bigr\Vert \\ & \ge \bigl( 1 - (1 - \lambda )b_{n}^{(m)} \bigr) \Vert x_{n} - r \Vert - b_{n}^{(m)} \bigl\Vert Tx_{n} - Tx_{n}^{(m - 1)} \bigr\Vert - c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - w_{n} \bigr\Vert \end{aligned} \\& \begin{aligned}[b] \Rightarrow \quad \Vert x_{n} - r \Vert &\le \frac{ ( 1 - b_{n}^{(m)} )}{ ( 1 - (1 - \lambda )b_{n}^{(m)} )} \Vert w_{n} - r \Vert + \frac{b_{n}^{(m)}}{ ( 1 - (1 - \lambda )b_{n}^{(m)} )} \bigl\Vert Tx_{n} - Tx_{n}^{(m - 1)} \bigr\Vert \\ &\quad{} + \frac{c_{n}^{(m)}}{ ( 1 - (1 - \lambda )b_{n}^{(m)} )} \bigl\Vert v_{n}^{(m)} - w_{n} \bigr\Vert \\ & \le \bigl( 1 - \lambda b_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert + \lambda ^{ - 1}Lb_{n}^{(m)} \bigl\Vert x_{n} - x_{n}^{(m - 1)} \bigr\Vert + \lambda ^{ - 1}c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - w_{n} \bigr\Vert \\ & \le \bigl( 1 - \lambda b_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert + \lambda ^{ - 1}Lb_{n}^{(m)} \bigl\Vert x_{n} - x_{n}^{(m - 1)} \bigr\Vert \\ &\quad{} + \lambda ^{ - 1}c_{n}^{(m)} \bigl( \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + \Vert w_{n} - r \Vert \bigr) \\ & = \bigl( 1 - \lambda b_{n}^{(m)} + \lambda ^{ - 1}c_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert + \lambda ^{ - 1}Lb_{n}^{(m)} \bigl\Vert x_{n} - x_{n}^{(m - 1)} \bigr\Vert \\ &\quad{} + \lambda ^{ - 1}c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert . \end{aligned} \end{aligned}$$
(3.19)

But, applying the second equation of (3.1), we have

$$\begin{aligned}& \bigl\Vert x_{n} - x_{n}^{(m - 1)} \bigr\Vert \\& \quad \le \bigl\Vert b_{n}^{(m)} \bigl( Tx_{n}^{(m - 1)} - w_{n} \bigr) + c_{n}^{(m)} \bigl( v_{n}^{(m)} - w_{n} \bigr) \bigr\Vert \\& \quad\quad{} + \bigl\Vert b_{n}^{(m - 1)} \bigl( w_{n} - Tx_{n}^{(m - 2)} \bigr) - c_{n}^{(m - 1)} \bigl( v_{n}^{(m - 1)} - w_{n} \bigr) \bigr\Vert \\& \quad \le b_{n}^{(m)} \bigl\Vert w_{n} - Tx_{n}^{(m - 1)} \bigr\Vert + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - w_{n} \bigr\Vert + b_{n}^{(m - 1)} \bigl\Vert w_{n} - Tx_{n}^{(m - 2)} \bigr\Vert \\& \quad\quad{} + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - w_{n} \bigr\Vert \\& \quad \le b_{n}^{(m)} \bigl( \Vert w_{n} - r \Vert + \bigl\Vert r - Tx_{n}^{(m - 1)} \bigr\Vert \bigr) + c_{n}^{(m)} \bigl( \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + \Vert w_{n} - r \Vert \bigr) \\& \quad\quad{} + b_{n}^{(m - 1)} \bigl( \Vert w_{n} - r \Vert + \bigl\Vert r - Tx_{n}^{(m - 2)} \bigr\Vert \bigr) + c_{n}^{(m - 1)} \bigl( \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert + \Vert w_{n} - r \Vert \bigr) \\& \quad \le b_{n}^{(m)} \bigl( \Vert w_{n} - r \Vert + L \bigl\Vert x_{n}^{(m - 1)} - r \bigr\Vert \bigr) + c_{n}^{(m)} \bigl( \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + \Vert w_{n} - r \Vert \bigr) \\& \quad\quad{}+ b_{n}^{(m - 1)} \bigl( \Vert w_{n} - r \Vert + L \bigl\Vert x_{n}^{(m - 2)} - r \bigr\Vert \bigr) + c_{n}^{(m - 1)} \bigl( \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert + \Vert w_{n} - r \Vert \bigr) \\& \quad \le \bigl[ b_{n}^{(m)} + b_{n}^{(m - 1)} + c_{n}^{(m)} + c_{n}^{(m - 1)} \bigr] \Vert w_{n} - r \Vert + Lb_{n}^{(m)} \bigl\Vert x_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{}+ Lb_{n}^{(m - 1)} \bigl\Vert x_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert . \end{aligned}$$
(3.20)

But after a simple calculation we get

$$\begin{aligned} \begin{gathered} \bigl\Vert x_{n}^{(m - 1)} - r \bigr\Vert \\ \quad \le \Vert w_{n} - r \Vert + Lb_{n}^{(m - 1)} \bigl\Vert x_{n}^{(m - 2)} - r \bigr\Vert + b_{n}^{(m - 1)} \Vert w_{n} - r \Vert + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\ \quad\quad{} + c_{n}^{(m - 1)} \Vert w_{n} - r \Vert \\ \quad = \bigl[ 1 + b_{n}^{(m - 1)} + c_{n}^{(m - 1)} \bigr] \Vert w_{n} - r \Vert + Lb_{n}^{(m - 1)} \bigl\Vert x_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert . \end{gathered} \end{aligned}$$
(3.21)

Substituting (3.21) in (3.20), we have

$$\begin{aligned} \begin{gathered} \bigl\Vert x_{n} - x_{n}^{(m - 1)} \bigr\Vert \\ \quad \le \bigl[ b_{n}^{(m)} + b_{n}^{(m - 1)} + c_{n}^{(m)} + c_{n}^{(m - 1)} \bigr] \Vert w_{n} - r \Vert \\ \quad\quad{} + Lb_{n}^{(m)} \bigl[ \bigl[ 1 + b_{n}^{(m - 1)} + c_{n}^{(m - 1)} \bigr] \Vert w_{n} - r \Vert + Lb_{n}^{(m - 1)} \bigl\Vert x_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \bigr] \\ \quad\quad{}+ Lb_{n}^{(m - 1)} \bigl\Vert x_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\ \quad = \bigl[ ( 1 + L )b_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} \bigr] \Vert w_{n} - r \Vert \\ \quad\quad{} + Lb_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert x_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \\ \quad\quad{} + c_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert . \end{gathered} \end{aligned}$$
(3.22)

Now, from the third equation of (3.1), we get

$$ \begin{gathered} \bigl\Vert x_{n}^{(m - 2)} - r \bigr\Vert \\ \quad \le \Vert w_{n} - r \Vert + Lb_{n}^{(m - 2)} \bigl\Vert x_{n}^{(m - 3)} - r \bigr\Vert + b_{n}^{(m - 2)} \Vert w_{n} - r \Vert + c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\ \quad\quad{} + c_{n}^{(m - 2)} \Vert w_{n} - r \Vert \\ \quad = \bigl[ 1 + b_{n}^{(m - 2)} + c_{n}^{(m - 2)} \bigr] \Vert w_{n} - r \Vert + Lb_{n}^{(m - 2)} \bigl\Vert x_{n}^{(m - 3)} - r \bigr\Vert + c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert . \end{gathered} $$
(3.23)

Substituting (3.23) in (3.22), we have

$$\begin{aligned}& \bigl\Vert x_{n} - x_{n}^{(m - 1)} \bigr\Vert \\& \quad \le \bigl[ ( 1 + L )b_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} \bigr] \Vert w_{n} - r \Vert \\& \quad \quad{} + Lb_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl[ \bigl[ 1 + b_{n}^{(m - 2)} + c_{n}^{(m - 2)} \bigr] \Vert w_{n} - r \Vert + Lb_{n}^{(m - 2)} \bigl\Vert x_{n}^{(m - 3)} - r \bigr\Vert \\& \quad \quad{} + c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \bigr] + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)} \bigl( 1 + Lb_{n}^{(m)} \bigr) \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad = \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} + c_{n}^{(m)} \\& \quad \quad{} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigr] \Vert w_{n} - r \Vert \\& \quad \quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl\Vert x_{n}^{(m - 3)} - r \bigr\Vert + c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \\& \quad \quad {}+ \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert . \end{aligned}$$

Continuing the above procedure up to second iterative step of (3.1), we obtain

$$\begin{aligned}& \bigl\Vert x_{n} - x_{n}^{(m - 1)} \bigr\Vert \\& \quad \le \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + L ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} + \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \\& \quad\quad{}+ \bigl[ c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad\quad{}+ L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \\& \quad\quad{} + \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}\cdots b_{n}^{(2)}c_{n}^{(3)} \bigr] \bigr] \Vert w_{n} - r \Vert \\& \quad\quad{} + L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \bigl\Vert x_{n}^{(1)} - r \bigr\Vert + \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \\& \quad\quad{}+ \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{}+ L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{}+ \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}\cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \bigr]. \end{aligned}$$
(3.24)

Now, from the last equation of (3.1), we have

$$\begin{aligned} \begin{aligned} \bigl\Vert x_{n}^{(1)} - r \bigr\Vert &= \bigl\Vert a_{n}^{(1)}w_{n} + b_{n}^{(1)}Sw_{n} + c_{n}^{(1)}v_{n}^{(1)} - r \bigr\Vert \\ &\le \Vert w_{n} - r \Vert + b_{n}^{(1)} \Vert Sw_{n} - r \Vert + b_{n}^{(1)} \Vert w_{n} - r \Vert + c_{n}^{(1)} \Vert w_{n} - r \Vert + c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \\ &= \bigl( 1 + b_{n}^{(1)} + c_{n}^{(1)} \bigr) \Vert w_{n} - r \Vert + b_{n}^{(1)} \Vert Sw_{n} - r \Vert + c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \\ &\le \bigl( 1 + b_{n}^{(1)} + c_{n}^{(1)} \bigr) \Vert w_{n} - r \Vert + b_{n}^{(1)}L \Vert w_{n} - r \Vert + c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \\ &= \bigl( 1 + (1 + L)b_{n}^{(1)} + c_{n}^{(1)} \bigr) \Vert w_{n} - r \Vert + c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert . \end{aligned} \end{aligned}$$
(3.25)

Combining (3.24) and (3.25), we obtain

$$\begin{aligned}& \bigl\Vert x_{n} - x_{n}^{(m - 1)} \bigr\Vert \\& \quad \le \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} + L ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \\& \quad\quad{} + \cdots + L^{m - 3} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \\& \quad\quad{}+ L^{m - 2}(1 + L) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}b_{n}^{(1)} \\& \quad\quad{}+ \bigl[ c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad\quad{}+ L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} + \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}\cdots b_{n}^{(3)}c_{n}^{(2)} \\& \quad\quad{}+ L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] \Vert w_{n} - r \Vert \\& \quad\quad{}+ \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{}+ L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{}+ \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}\cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{}+ L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr]. \end{aligned}$$
(3.26)

Inserting (3.26) in (3.19), we find

$$\begin{aligned}& \Vert x_{n} - r \Vert \\& \quad \le \bigl[ 1 - \lambda b_{n}^{(m)} + \lambda ^{ - 1}c_{n}^{(m)} + \lambda ^{ - 1}Lb_{n}^{(m)} \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} \\& \quad\quad{} + L ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} + L^{2} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \\& \quad\quad{} + \cdots + L^{m - 3} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \\& \quad\quad{} + L^{m - 2}(1 + L) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}b_{n}^{(1)} \\& \quad\quad{} + \bigl[ c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} + \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}\cdots b_{n}^{(3)}c_{n}^{(2)} \\& \quad\quad{} + L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] \bigr] \Vert w_{n} - r \Vert \\& \quad\quad{} + \lambda ^{ - 1}Lb_{n}^{(m)} \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}\cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{} + L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] + \lambda ^{ - 1}c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \\& \quad = \bigl[ 1 - b_{n}^{(m)} \bigl[ \lambda - \lambda ^{ - 1}L \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)} \\& \quad\quad{} + L ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)} + L^{2} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \\& \quad\quad{} + \cdots + L^{m - 3} ( 1 + L ) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \\& \quad\quad{} + L^{m - 2}(1 + L) \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}b_{n}^{(1)} \\& \quad\quad{} + \bigl[ c_{n}^{(m)} + \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m - 1)} + L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} + \cdots + L^{m - 3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}\cdots b_{n}^{(3)}c_{n}^{(2)} \\& \quad\quad{} + L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] - \lambda ^{ - 1}\delta _{n} \bigr] \bigr] \Vert w_{n} - r \Vert \\& \quad\quad{} + \lambda ^{ - 1} \bigl( 1 + Lb_{n}^{(m)} \bigr)c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + \lambda ^{ - 1}L \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m)}c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + \lambda ^{ - 1}L^{2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m)}b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{} + \lambda ^{ - 1}L^{3} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + \lambda ^{ - 1}L^{m - 2} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{} + \lambda ^{ - 1}L^{m - 1} \bigl( 1 + Lb_{n}^{(m)} \bigr)b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \\& \quad \le \bigl[ 1 - b_{n}^{(m)} \bigl[ \lambda - \lambda ^{ - 1}L \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L )^{2}b_{n}^{(m - 1)} \\& \quad\quad{} + L ( 1 + L )^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)} + L^{2} ( 1 + L )^{3}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \\& \quad\quad{} + \cdots + L^{m - 3} ( 1 + L )^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \\& \quad\quad{} + L^{m - 2}(1 + L)^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}b_{n}^{(1)} \\& \quad\quad{} + \bigl[ c_{n}^{(m)} + ( 1 + L )c_{n}^{(m - 1)} + L ( 1 + L )b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad\quad{} + L^{2} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} + \cdots + L^{m - 3} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \\& \quad\quad{} + L^{m - 2} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] - \lambda ^{ - 1}\delta _{n} \bigr] \bigr] \Vert w_{n} - r \Vert \\& \quad\quad{} + \lambda ^{ - 1} ( 1 + L )c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + \lambda ^{ - 1}L ( 1 + L )b_{n}^{(m)}c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + \lambda ^{ - 1}L^{2} ( 1 + L )b_{n}^{(m)}b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{} + \lambda ^{ - 1}L^{3} ( 1 + L )b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + \lambda ^{ - 1}L^{m - 2} ( 1 + L )b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{} + \lambda ^{ - 1}L^{m - 1} ( 1 + L )b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \\& \quad \le \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert \\& \quad\quad{} + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + L \bigl[ b_{n}^{(m)}c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{} + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] \bigr] \bigr] \bigr] \cdots \bigr] \bigr]. \end{aligned}$$
(3.27)

Now, using (3.27), we obtain

$$\begin{aligned}& \Vert w_{n + 1} - r \Vert \\& \quad \le \Vert w_{n + 1} - x_{n} \Vert + \Vert x_{n} - r \Vert \\& \quad \le \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert \\& \quad\quad{} + \lambda ^{ - 1} ( 1 + L ) [ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + L [ b_{n}^{(m)}c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad \le \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)}A + L \bigl[ b_{n}^{(m)}c_{n}^{(m - 1)}A \\& \quad\quad{}+ L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}c_{n}^{(m - 2)}A + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)}A \\& \quad\quad{}+ \cdots + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} A \\& \quad\quad{}+ L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)}A \bigr] \bigr] \bigr] \bigr] \cdots \bigr] \bigr] + \mu _{n} \\& \quad = \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L )Ab_{n}^{(m)} \bigl[ \delta _{n} + L \bigl[ c_{n}^{(m - 1)} \\& \quad\quad{}+ L \bigl[ b_{n}^{(m - 1)}c_{n}^{(m - 2)} + L \bigl[ b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} + \cdots + L \bigl[ b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \\& \quad\quad{}+ L \bigl[ b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] \bigr] \bigr] \cdots \bigr] \bigr] + \mu _{n}, \quad n \ge 0. \end{aligned}$$
(3.28)

(iii) If we put

$$\begin{aligned}& \alpha _{n} = \Vert w_{n} - r \Vert , \\& \omega _{n} = \theta b_{n}^{(m)}, \\& \begin{aligned} \beta _{n} &= \theta ^{ - 1}\lambda ^{ - 1} ( 1 + L )A \bigl[ \delta _{n} + L \bigl[ c_{n}^{(m - 1)} + L \bigl[ b_{n}^{(m - 1)}c_{n}^{(m - 2)} + L \bigl[ b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \\ &\quad{} + \cdots + L \bigl[ b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} + L \bigl[ b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] \bigr] \bigr] \cdots \bigr] \bigr], \end{aligned} \\& \gamma _{n} = \mu _{n}, \quad \forall n \ge 0, \end{aligned}$$

in (3.28), then we obtain

$$ \alpha _{n + 1} \le ( 1 - \omega _{n} )\alpha _{n} + \omega _{n}\beta _{n} + \gamma _{n}, \quad n \ge 0, $$

where \(\{ \omega _{n} \} _{n = 0}^{\infty } \subset [0, 1]\), \(\sum_{n = 0}^{\infty } \omega _{n} = \infty \), \(\sum_{n = 0}^{\infty } \gamma _{n} < \infty \) and \(\lim_{n \to \infty } \beta _{n} = 0\).

Hence, from Lemma 2.4, we have \(\lim_{n \to \infty } \alpha _{n} = 0\).

That is \(\lim_{n \to \infty } \Vert w_{n} - r \Vert = 0\). Hence, \(\lim_{n \to \infty } w_{n} = r\) and this ensures that \(\{ u_{n}^{(m)} \} _{n = 0}^{\infty } \) is almost common-stable on B.

(iv) Considering \(\lim_{n \to \infty } w_{n} = r\), we have \(\lim_{n \to \infty } w_{n + 1} = r\).

Now, using (3.27), we get

$$\begin{aligned} \mu _{n} = {}& \Vert w_{n + 1} - x_{n} \Vert \le \Vert w_{n + 1} - r \Vert + \Vert x_{n} - r \Vert \\ \le {}& \Vert w_{n + 1} - r \Vert + \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert \\ &{} + \lambda ^{ - 1} ( 1 + L )Ab_{n}^{(m)} \bigl[ \delta _{n} + L \bigl[ c_{n}^{(m - 1)} + L \bigl[ b_{n}^{(m - 1)}c_{n}^{(m - 2)} + L \bigl[ b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \\ &{}+ \cdots + L \bigl[ b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} + Lb_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] \bigr] \cdots \bigr] \bigr] \to 0, \end{aligned}$$

as \(n \to \infty \), this means that \(\lim_{n \to \infty } \mu _{n} = 0 \).

This completes the proof. □

Theorem 3.2

Let B, X, T, S, θ, \(\{ u_{n} \} _{n = 0}^{\infty }\), \(\{ x_{n} \} _{n = 0}^{\infty } \), \(\{ w_{n} \} _{n = 0}^{\infty } \), \(\{ v_{n}^{ ( 1 )} \} ,\ldots, \{ v_{n}^{ ( m )} \} \), and \(\{ \mu _{n} \} _{n = 0}^{\infty } \) be as in Theorem 3.1and \(\{ a_{n}^{(i)} \} \), \(\{ b_{n}^{(i)} \} \), \(\{ c_{n}^{(i)} \} \) for each \(i \in \{ 1, 2,\ldots, m \} \) be any appropriate real sequences in \([0, 1]\) satisfying the conditions (1), (3), (4), (5) of Theorem 3.1with the following property:

$$ \sum_{n = 0}^{\infty } c_{n}^{ ( m )} < \infty . $$

Then the results of Theorem 3.1hold.

Proof

The proof of this theorem is similar to the proof of Theorem 3.1, so here we omit it. □

Theorem 3.3

Let B, X, T, S, θ, \(\{ u_{n} \} _{n = 0}^{\infty }\), \(\{ x_{n} \} _{n = 0}^{\infty } \), \(\{ w_{n} \} _{n = 0}^{\infty } \), \(\{ v_{n}^{ ( 1 )} \} ,\ldots, \{ v_{n}^{ ( m )} \} \), and \(\{ \mu _{n} \} _{n = 0}^{\infty } \) be as in Theorem 3.1and \(\{ a_{n}^{(i)} \} \), \(\{ b_{n}^{(i)} \} \), \(\{ c_{n}^{(i)} \} \) for each \(i \in \{ 1, 2,\ldots, m \} \) be any appropriate real sequences in \([0, 1]\) satisfying the conditions (1), (3) and (5) of Theorem 3.1with the following property:

$$ \lim_{n \to \infty } c_{n}^{ ( m )} = 0, \quad \textit{and}\quad b_{n}^{ ( m )} \ge h > 0, \quad \forall n \ge 0, $$

where h is a constant.

Then

  1. (i)

    the iterative sequence \(\{ u_{n} \} _{n = 0}^{\infty }\) given by (1.5) converges strongly to the common fixed r of T and S and the following inequality holds:

    $$ \Vert u_{n + 1} - r \Vert \le ( 1 - \theta h ) \Vert u_{n} - r \Vert + D, \quad \forall n \ge 0, $$

    where

    $$\begin{aligned} D = {}&\lambda ^{ - 1} ( 1 + L ) \Bigl[ \sup_{n \ge 0} \bigl\{ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \bigr\} + Lb_{n}^{(m)}\sup_{n \ge 0} \bigl\{ c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \bigr\} \\ &{}+ L^{2}b_{n}^{(m)}b_{n}^{(m - 1)} \sup_{n \ge 0} \bigl\{ c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \bigr\} \\ &{}+ L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \sup_{n \ge 0} \bigl\{ c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \bigr\} \\ &{}+ \cdots + L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}\sup_{n \ge 0} \bigl\{ c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \bigr\} \\ &{}+ L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}\sup_{n \ge 0} \bigl\{ c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr\} \Bigr], \end{aligned}$$
  2. (ii)
    $$\begin{aligned}& \Vert w_{n + 1} - r \Vert \\& \quad \le ( 1 - \theta h ) \Vert w_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + L \bigl[ c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + L \bigl[ c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + L \bigl[ c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + L \bigl[ c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert + L \bigl[ c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] \bigr] \bigr] \bigr] \cdots \bigr] \bigr] + \mu _{n}, \quad \forall n \ge 0, \end{aligned}$$
  3. (iii)

    \(\lim_{n \to \infty } w_{n} = r\), if and only if \(\lim_{n \to \infty } \mu _{n} = 0\).

Proof

(i) Following the proof of Theorem 3.1, we have

$$\begin{aligned}& \Vert u_{n + 1} - r \Vert \\& \quad \le \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L )c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + \lambda ^{ - 1}L ( 1 + L )b_{n}^{(m)}c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + \lambda ^{ - 1}L^{2} ( 1 + L )b_{n}^{(m)}b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \\& \quad\quad{} + \lambda ^{ - 1}L^{3} ( 1 + L )b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad \quad{} + \cdots + \lambda ^{ - 1}L^{m - 2} ( 1 + L )b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{} + \lambda ^{ - 1}L^{m - 1} ( 1 + L )b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \\& \quad \le ( 1 - \theta h ) \Vert u_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + Lb_{n}^{(m)}c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + L^{2}b_{n}^{(m)}b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad \quad{} + L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] \\& \quad \le ( 1 - \theta h ) \Vert u_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L ) \Bigl[ \sup_{n \ge 0} \bigl\{ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \bigr\} + Lb_{n}^{(m)}\sup_{n \ge 0} \bigl\{ c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \bigr\} \\& \quad\quad{} + L^{2}b_{n}^{(m)}b_{n}^{(m - 1)} \sup_{n \ge 0} \bigl\{ c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \bigr\} \\& \quad\quad{} + L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \sup_{n \ge 0} \bigl\{ c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \bigr\} \\& \quad \quad{} + \cdots + L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}\sup_{n \ge 0} \bigl\{ c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \bigr\} \\& \quad\quad{} + L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}\sup_{n \ge 0} \bigl\{ c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr\} \Bigr] \\& \quad = ( 1 - \theta h ) \Vert u_{n} - r \Vert + D, \quad n \ge 0. \end{aligned}$$

Now, putting

$$\begin{aligned}& \alpha _{n} = \Vert u_{n} - r \Vert ,\quad\quad \eta = 1 - \theta h\quad \text{and} \\& \begin{aligned} \beta _{n} = {}&\lambda ^{ - 1} ( 1 + L ) \Bigl[ \sup_{n \ge 0} \bigl\{ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r^{*} \bigr\Vert \bigr\} + Lb_{n}^{(m)}\sup_{n \ge 0} \bigl\{ c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r^{*} \bigr\Vert \bigr\} \\ &{} + L^{2}b_{n}^{(m)}b_{n}^{(m - 1)} \sup_{n \ge 0} \bigl\{ c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r^{*} \bigr\Vert \bigr\} \\ &{}+ L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \sup_{n \ge 0} \bigl\{ c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r^{*} \bigr\Vert \bigr\} \\ & {}+ \cdots + L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}\sup_{n \ge 0} \bigl\{ c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r^{*} \bigr\Vert \bigr\} \\ & {}+ L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}\sup_{n \ge 0} \bigl\{ c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r^{*} \bigr\Vert \bigr\} \Bigr], \quad \forall n \ge 0, \end{aligned} \end{aligned}$$

we get \(0 \le \eta < 1\) and \(\lim_{n \to \infty } \beta _{n} = 0\). Hence, by an application of Lemma 2.5, we obtain

$$ \lim_{n \to \infty } \Vert u_{n} - r \Vert = 0. $$

This ensures that the sequence \(\{ u_{n} \} _{n = 0}^{\infty }\) converges strongly to the common fixed r of T and S.

(ii) Again, from (3.27), we find

$$\begin{aligned}& \Vert w_{n + 1} - r \Vert \\& \quad \le \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + L \bigl[ b_{n}^{(m)}c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad \quad{} + \cdots + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad \quad{} + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] \bigr] \bigr] \bigr] \cdots \bigr] \bigr] + \mu _{n} \\& \quad \le ( 1 - \theta h ) \Vert w_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + L \bigl[ c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad \quad{} + L \bigl[ c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + L \bigl[ c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + L \bigl[ c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert + L \bigl[ c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] \bigr] \bigr] \bigr] \cdots \bigr] \bigr] + \mu _{n}. \end{aligned}$$

(iii) Considering \(\lim_{n \to \infty } w_{n} = r\), we have \(\lim_{n \to \infty } w_{n + 1} = r\).

Now, using (3.27), we get

$$\begin{aligned} \mu _{n} &= \Vert w_{n + 1} - x_{n} \Vert \\ &\le \Vert w_{n + 1} - r \Vert + \Vert x_{n} - r \Vert \\ &\le \Vert w_{n + 1} - r \Vert + ( 1 - \theta h ) \Vert w_{n} - r \Vert \\ &\quad{} + \lambda ^{ - 1} ( 1 + L )b_{n}^{(m)} \bigl[ \delta _{n} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + L \bigl[ c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\ & \quad{} + L \bigl[ b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + L \bigl[ b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\ &\quad{} + \cdots + L \bigl[ b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\ & \quad{} + L \bigl[ b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] \bigr] \bigr] \bigr] \cdots \bigr] \bigr] \to 0, \end{aligned}$$

as \(n \to \infty \), this means that \(\lim_{n \to \infty } \mu _{n} = 0 \).

Conversely, suppose that \(\lim_{n \to \infty } \mu _{n} = 0\). Now, by setting

$$\begin{aligned}& \alpha _{n} = \Vert w_{n} - r \Vert , \quad\quad \eta = 1 - \theta h\quad \text{and} \\& \begin{aligned} \beta _{n} &= \lambda ^{ - 1} ( 1 + L ) \Bigl[ \sup_{n \ge 0} \bigl\{ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r^{*} \bigr\Vert \bigr\} + Lb_{n}^{(m)}\sup_{n \ge 0} \bigl\{ c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r^{*} \bigr\Vert \bigr\} \\ & \quad{} + L^{2}b_{n}^{(m)}b_{n}^{(m - 1)} \sup_{n \ge 0} \bigl\{ c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r^{*} \bigr\Vert \bigr\} \\ &\quad{} + L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \sup_{n \ge 0} \bigl\{ c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r^{*} \bigr\Vert \bigr\} \\ &\quad{} + \cdots + L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}\sup_{n \ge 0} \bigl\{ c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r^{*} \bigr\Vert \bigr\} \\ &\quad{} + L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}\sup_{n \ge 0} \bigl\{ c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r^{*} \bigr\Vert \bigr\} \Bigr] + \mu _{n}, \quad \forall n \ge 0, \end{aligned} \end{aligned}$$

we get \(0 \le \eta < 1\) and \(\lim_{n \to \infty } \beta _{n} = 0\). Hence, by an application of Lemma 2.5, we obtain

$$ \lim_{n \to \infty } \Vert w_{n} - r \Vert = 0. $$

This completes the proof. □

The following corollaries show the convergence, almost common-stability and common-stability for the corresponding modified multi-step Noor iterative procedure without errors for two Lipschitz strictly hemicontractive-type mappings in arbitrary Banach spaces.

Corollary 3.4

Let B be a nonempty closed convex subset of X and T and S be two Lipschitz strictly hemicontractive-type mappings from B into itself. Suppose that \(\{ \alpha _{n}^{(i)} \} _{n = 0}^{\infty }\), \(\{ \beta _{n}^{(i)} \} _{n = 0}^{\infty }\) for each \(i \in \{ 1, 2,\ldots, m\}\) are any appropriate real sequences in \([0, 1]\) satisfying the following conditions:

  1. (6)

    \(\alpha _{n}^{(i)} + \beta _{n}^{(i)} = 1\), for each \(i \in \{ 1,2, 3,\ldots, m\}\),

  2. (7)

    \(\sum_{n = 0}^{\infty } \alpha _{n}^{(j)} = \infty \) for each \(j \in \{ 2, 3,\ldots, m\}\),

  3. (8)
    $$\begin{aligned}& L \bigl[ ( 1 + L )\beta _{n}^{(m)} + ( 1 + L )^{2} \bigl[ \beta _{n}^{(m - 1)} + L ( 1 + L )^{2}\beta _{n}^{(m - 1)}\beta _{n}^{(m - 2)} + L^{2} \beta _{n}^{(m - 1)}\beta _{n}^{(m - 2)}\beta _{n}^{(m - 3)} \\& \quad\quad {} + \cdots + L^{m - 3}\beta _{n}^{(m - 1)}\beta _{n}^{(m - 2)} \cdots \beta _{n}^{(2)} + L^{m - 2}\beta _{n}^{(m - 1)}\beta _{n}^{(m - 2)} \cdots \beta _{n}^{(2)}\beta _{n}^{(1)} \bigr] \bigr] \\& \quad \le \lambda ( \lambda - \theta ), \quad n \ge 0, \end{aligned}$$

    where θ is a constant in \((0, \lambda )\) and \(\lambda \in (0, 1)\).

For \(u_{0} \in B\), we assume an iterative sequence \(\{ u_{n} \} _{n = 0}^{\infty } \) defined by

$$ \left . \textstyle\begin{array}{l} u_{n + 1} = u_{n}^{(m)} = \alpha _{n}^{(m)}u_{n} + \beta _{n}^{(m)}Tu_{n}^{(m - 1)} ,\\ u_{n}^{(m - 1)} = \alpha _{n}^{(m - 1)}u_{n} + \beta _{n}^{(m - 1)}Tu_{n}^{(m - 2)} ,\\ \cdots \cdots \\ u_{n}^{(2)} = \alpha _{n}^{(2)}u_{n} + \beta _{n}^{(2)}Tu_{n}^{(1)}, \\ u_{n}^{(1)} = \alpha _{n}^{(1)}u_{n} + \beta _{n}^{(1)}Su_{n}, \quad n \ge 0 \end{array}\displaystyle \right \} $$
(3.29)

and let \(\{ w_{n} \} _{n = 0}^{\infty }\) be any sequence in B and \(\{ \mu _{n} \} _{n = 0}^{\infty } \) be a sequence defined by

$$ \mu _{n} = \Vert w_{n + 1} - x_{n} \Vert , \quad n \ge 0, $$

where

$$ \left . \textstyle\begin{array}{l} x_{n} = x_{n}^{(m)} = \alpha _{n}^{(m)}w_{n} + \beta _{n}^{(m)}Tx_{n}^{(m - 1)} ,\\ x_{n}^{(m - 1)} = \alpha _{n}^{(m - 1)}w_{n} + \beta _{n}^{(m - 1)}Tx_{n}^{(m - 2)} ,\\ \cdots \cdots \\ x_{n}^{(2)} = \alpha _{n}^{(2)}w_{n} + \beta _{n}^{(2)}Tx_{n}^{(1)} ,\\ x_{n}^{(1)} = \alpha _{n}^{(1)}w_{n} + \beta _{n}^{(1)}Sw_{n}, \quad n \ge 0. \end{array}\displaystyle \right \} $$
(3.30)

Then

  1. (i)

    the iterative sequence \(\{ u_{n} \} _{n = 0}^{\infty }\) given by (3.29) converges strongly to the common fixed r of T and S,

  2. (ii)

    \(\sum_{n = 0}^{\infty } \mu _{n} < \infty \) implies that \(\lim_{n \to \infty } w_{n} = r\), so that \(\{ u_{n}^{(m)} \} _{n = 0}^{\infty } \) is almost common-stable on B,

  3. (iii)

    \(\lim_{n \to \infty } w_{n} = r\), implies that \(\lim_{n \to \infty } \mu _{n} = 0\).

Proof

The proof follows from the proof of Theorem 3.1 and, for brevity, here we omit it. □

Corollary 3.5

Let B, X, T, S, \(\{ u_{n} \} _{n = 0}^{\infty }\), \(\{ x_{n} \} _{n = 0}^{\infty } \), \(\{ w_{n} \} _{n = 0}^{\infty } \) and \(\{ \mu _{n} \} _{n = 0}^{\infty }\) be as in Corollary 3.4and θ be as in Theorem 3.1. Suppose that \(\{ \alpha _{n}^{(i)} \} _{n = 0}^{\infty }\), \(\{ \beta _{n}^{(i)} \} _{n = 0}^{\infty }\) for each \(i \in \{ 1, 2,\ldots, m \} \) are any appropriate real sequences in \([0, 1]\) satisfying the similar condition of condition (3) of Theorem 3.1and the conditions (6), (7) and (8) of Corollary 3.4along with the following property:

$$ \beta _{n}^{ ( m )} \ge h > 0, \quad \forall n \ge 0, $$

where h is a constant.

Then

  1. (i)

    the iterative sequence \(\{ u_{n} \} _{n = 0}^{\infty }\) given by (3.29) converges strongly to the common fixed r of T and S and the following inequality holds:

    $$\begin{aligned}& \Vert u_{n + 1} - r \Vert \le ( 1 - \theta h ) \Vert u_{n} - r \Vert , \quad \forall n \ge 0, \end{aligned}$$
  2. (ii)

    \(\Vert w_{n + 1} - r \Vert \le ( 1 - \theta h ) \Vert w_{n} - r \Vert + \mu _{n}\), \(\forall n \ge 0\),

  3. (iii)

    \(\lim_{n \to \infty } w_{n} = r\), if and only if \(\lim_{n \to \infty } \mu _{n} = 0\).

Proof

The proof follows from the proof of Theorem 3.3 and, for brevity, here we omit it. □

Remark 3.6

(a):

If we put \(m = 2\) in our Theorem 3.1, Theorem 3.2 and Theorem 3.3, then we can easily establish Theorem 9, Theorem 10 and Theorem 11 of Hussain et al. [1], respectively. Therefore, we can comment that the results of Hussain et al. [1] are special case of our results.

(b):

Since the MIP given by Mann [28], the IIP given by Ishikawa [29], the NIP given by Xu and Noor [30], the MIPE given in Liu [31] and Xu [32], the IIPE given by Liu [31] and Xu [32] and the TIPE given by Cho et al. [33] are all special cases of our newly proposed MMNIPE given by (1.5), by setting the appropriate values of m and \(c_{n}^{ ( m )}\) in our Theorem 3.1, Theorem 3.2 and Theorem 3.3, we can easily obtain the convergence, almost common-stability and common-stability criteria of the above-mentioned iterative procedures for two Lipschitz strictly hemicontractive-type mappings in arbitrary Banach spaces.

Examples

In this section, we provide a numerical example to verify our analytical results and to show a numerical comparison between our newly proposed MMNIPE given by (1.5) and some other most analogous iterative procedures with errors.

Example 4.1

Consider B is a nonempty subset of an arbitrary Banach space X with the usual norm and let \(B = \mathbb{R}\). Suppose that T and S are two self-maps on B which are defined as follows:

$$ Tu = \frac{1}{3}\sin ^{2}u,\quad \text{and} \quad Su = \frac{2}{3}u^{2}. $$

Now, if we let \(L = \frac{2}{3}\), \(\sigma = \frac{3}{2}\), \(\theta = \frac{1}{300}\), then it is obvious that \(F_{T} \cap F_{S} = \{ 0 \} \), \(\lambda = \frac{\sigma - 1}{\sigma } = \frac{3 / 2 - 1}{3 / 2} = \frac{1}{3} \in ( 0, 1 )\) and

$$\begin{aligned}& \begin{gathered} \vert Tu_{1} - Tu_{2} \vert = \frac{1}{3} \bigl\vert \sin ^{2}u_{1} - \sin ^{2}u_{2} \bigr\vert \le \frac{2}{3} \vert \sin u_{1} - \sin u_{2} \vert \vert \sin u_{1} + \sin u_{2} \vert \le L \vert u_{1} - u_{2} \vert , \\ \vert Su_{1} - Su_{2} \vert = \frac{2}{3} \bigl\vert u_{1}^{2} - u_{2}^{2} \bigr\vert = \frac{2}{3} \vert u_{1} - u_{2} \vert \vert u_{1} + u_{2} \vert \le L \vert u_{1} - u_{2} \vert , \quad \forall u_{1}, u_{2} \in \mathbb{R}. \end{gathered} \end{aligned}$$

Hence, both T and S are Lipschitzian mappings on B.

Likewise, using (1.1) we have

$$\begin{aligned}& \bigl\vert \bigl(1 + t' \bigr) (q - r) - t't(Tq - Tr) \bigr\vert \\& \quad \ge \bigl(1 + t' \bigr) \vert q - r \vert - t't \vert Tq - Tr \vert \\& \quad = \vert q - r \vert + t' \bigl( \vert q - r \vert - t \vert Tq - Tr \vert \bigr) \\& \quad \ge \vert q - r \vert + t' \bigl( \vert q - r \vert - tL \vert q - r \vert \bigr) = \bigl( 1 + t' ( 1 - tL ) \bigr) \vert q - r \vert \\& \quad \ge \Vert q - r \Vert \end{aligned}$$

for any \(q, r \in \mathbb{R}\) and \(t' > 0\). Therefore, T is strongly pseudocontractive and hence Lemma 2.8 confirms that T is strictly hemicontractive on B. Also, the similar arguments hold for the mapping S. Hence both T and S are Lipschitz strictly hemicontractive mappings on B.

Now, if we consider

$$\begin{aligned}& b_{n}^{(m)} = \frac{5}{9} \cdot \frac{1}{\sqrt{n} + 100}, \quad\quad c_{n}^{(m)} = \frac{1}{ ( \sqrt{n} + 100 )^{2}}, \quad\quad a_{n}^{(m)} = 1 - \bigl( b_{n}^{(m)} + c_{n}^{(m)} \bigr), \\& b_{n}^{ ( j )} = c_{n}^{ ( j )} = \frac{3}{5} \cdot \frac{1}{n + 100}, \quad\quad a_{n}^{ ( j )} = 1 - \bigl( b_{n}^{ ( j )} + c_{n}^{ ( j )} \bigr), \\& \quad \text{where } j = 1, 2,\ldots, m - 1, \text{ and }\forall n \ge 0. \end{aligned}$$

Then, for \(n = 0\), \(m = 10\), we obtain

$$\begin{aligned}& L \bigl[ ( 1 + L )b_{n}^{(m)} + ( 1 + L )^{2}b_{n}^{(m - 1)} + L ( 1 + L )^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \\& \quad {}+ L^{2} ( 1 + L )^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} + \cdots + L^{m - 3} ( 1 + L )^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)} \\& \quad {}+ L^{m - 2}(1 + L)^{2}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}b_{n}^{(1)} \\& \quad {}+ \bigl[ c_{n}^{(m)} + ( 1 + L )c_{n}^{(m - 1)} + L ( 1 + L )b_{n}^{(m - 1)}c_{n}^{(m - 2)} \\& \quad {}+ L^{2} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} + \cdots + L^{m - 3} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \\& \quad {}+ L^{m - 2} ( 1 + L )b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigr] \bigr] + \frac{c_{n}^{(m)}}{b_{n}^{(m)}} \le 0.042231081 \le 0.11. \end{aligned}$$

Therefore, by an application of Theorem 3.1, we can say that the iterative sequence \(\{ u_{n} \} _{n = 0}^{\infty } \) defined by (1.5) converges strongly to the common fixed 0 of T and S in B and the corresponding MMNIPE given by (1.5) is common-stable as well as almost common-stable on B.

Analogously, by applying Theorem 3.2 and Theorem 3.3, we can easily prove that the iterative sequence \(\{ u_{n} \} _{n = 0}^{\infty } \) defined by (1.5) converges strongly to the common fixed 0 of T and S in B and the corresponding MMNIPE given by (1.5) is common-stable as well as almost common-stable on B.

For the numerical experiment, here we consider our newly proposed MMNIPE given by (1.5) for \(m = 10\), and compared it with the STIIPE given by Hussain et al. [1], the IIPE given by Liu [31], and the TIPE given by Cho et al. [33]. By using MATLAB programming language, we computed the different iterative steps and the numerical comparison is shown in Table 1. Furthermore, the convergence behaviors of these iterative procedures with errors are shown in Fig. 1. For all iterative procedure, we take the initial approximation \(u_{0} = 0.5\). For our proposed MMNIPE given by (1.5), we consider \(v_{n}^{ ( i )} = \frac{1}{n + 1}\), where \(i = 1, 2,3,\ldots, 10\) and

$$\begin{aligned}& b_{n}^{(10)} = \frac{5}{9} \cdot \frac{1}{\sqrt{n} + 100}, \quad\quad c_{n}^{(10)} = \frac{1}{ ( \sqrt{n} + 100 )^{2}}, \\& a_{n}^{(10)} = 1 - \bigl( b_{n}^{(10)} + c_{n}^{(10)} \bigr) = \frac{9n + 1795\sqrt{n} + 89{,}491}{9 ( \sqrt{n} + 100 )^{2}}, \\& b_{n}^{ ( j )} = c_{n}^{ ( j )} = \frac{3}{5} \cdot \frac{1}{n + 100}, \quad\quad a_{n}^{ ( j )} = 1 - \bigl( b_{n}^{ ( j )} + c_{n}^{ ( j )} \bigr) = \frac{5n + 494}{5 ( n + 100 )}, \\& \quad \text{where } j = 1, 2,3,\ldots, 9 \text{ and } \forall n \ge 0. \end{aligned}$$

For the STIIPE given by Hussain et al. [1], we consider \(v_{n}^{ ( i )} = \frac{1}{n + 1}\), where \(i = 1, 2\) and

$$\begin{aligned}& b_{n}^{(2)} = \frac{5}{9} \cdot \frac{1}{\sqrt{n} + 100}, \quad\quad c_{n}^{(2)} = \frac{1}{ ( \sqrt{n} + 100 )^{2}}, \\& a_{n}^{(2)} = 1 - \bigl( b_{n}^{(2)} + c_{n}^{(2)} \bigr) = \frac{9n + 1795\sqrt{n} + 89{,}491}{9 ( \sqrt{n} + 100 )^{2}}, \\& b_{n}^{ ( 1 )} = c_{n}^{ ( 1 )} = \frac{3}{5} \cdot \frac{1}{n + 100}, \quad\quad a_{n}^{ ( 1 )} = 1 - \bigl( b_{n}^{ ( 1 )} + c_{n}^{ ( 1 )} \bigr) = \frac{5n + 494}{5 ( n + 100 )} \quad \text{and}\quad \forall n \ge 0. \end{aligned}$$

For the IIPE given by Liu [31], we consider \(v_{n}^{ ( i )} = \frac{1}{n + 1}\), where \(i = 1, 2\) and

$$\begin{aligned}& b_{n}^{(2)} = \frac{5}{9} \cdot \frac{1}{\sqrt{n} + 100}, \quad\quad a_{n}^{(2)} = 1 - b_{n}^{(2)} = \frac{9\sqrt{n} + 895}{9 ( \sqrt{n} + 100 )}, \\& b_{n}^{ ( 1 )} = \frac{3}{5} \cdot \frac{1}{n + 100}, \quad\quad a_{n}^{ ( 1 )} = 1 - b_{n}^{ ( 1 )} = \frac{5n + 497}{5 ( n + 100 )} \quad \text{and}\quad \forall n \ge 0. \end{aligned}$$

For the TIPE given by Cho et al. [33], we consider \(v_{n}^{ ( i )} = \frac{1}{n + 1}\), where \(i = 1, 2, 3\) and

$$\begin{aligned}& b_{n}^{(3)} = \frac{5}{9} \cdot \frac{1}{\sqrt{n} + 100}, \quad\quad c_{n}^{(3)} = \frac{1}{ ( \sqrt{n} + 100 )^{2}}, \\& a_{n}^{(3)} = 1 - \bigl( b_{n}^{(3)} + c_{n}^{(3)} \bigr) = \frac{9n + 1795\sqrt{n} + 89{,}491}{9 ( \sqrt{n} + 100 )^{2}}, \\& b_{n}^{ ( j )} = c_{n}^{ ( j )} = \frac{3}{5} \cdot \frac{1}{n + 100}, \quad\quad a_{n}^{ ( j )} = 1 - \bigl( b_{n}^{ ( j )} + c_{n}^{ ( j )} \bigr) = \frac{5n + 494}{5 ( n + 100 )},\\& \quad \text{where } j = 1, 2 \text{ and } \forall n \ge 0. \end{aligned}$$

The comparison table (Table 1) confirms that the rate of convergence of our proposed MMNIPE given by (1.5) is better than that of the STIIPE given by Hussain et al. [1], the IIPE given by Liu [31] and the TIPE given by Cho et al. [33].

Figure 1
figure1

Convergence behavior corresponding to \(u_{0} = 0.5\) for 1000 iterative steps

Table 1 Numerical results corresponding to \(u_{0}=0.5\) for 1000 iterative steps

Conclusion

In this study, we established the convergence, almost common-stability and common-stability criteria of our proposed MMNIPE given by (1.5) for two Lipschitz strictly hemicontractive-type mappings in arbitrary Banach spaces. The obtained results of this paper provided easy and straightforward techniques for proving the convergence, almost common-stability and common-stability criteria of the proposed MMNIPE given by (1.5). Furthermore, the results of this paper extended the corresponding results of Hussain et al. [1, 79], Zegeye et al. [2], Meche et al. [3], Chidume and Osilike [4], Chidume [5], Liu et al. [12], Zeng [13], Yu et al. [11], Yang [25], Chidume [36], Deng [37, 38] and Liu [39]. According to the Remark 3.6, our results generalized and unify the corresponding results of Hussain et al. [1], Mann [28], Ishikawa [29], Xu and Noor [30], Liu [31] and Xu [32] and Cho et al. [33] in the case of establishing the fixed-point theorem-based iterative procedures for two Lipschitz strictly hemicontractive-type mappings. At the end of this work, we discussed a computational numerical example which verify our main results and compare the performance of our proposed MMNIPE given by (1.5) with other most analogous iterative procedures with errors. From the comparison table (Table 1), we conclude that our proposed MMNIPE given by (1.5) superior over the STIIPE given by Hussain et al. [1] and the TIPE given by Cho et al. [33] in the case of convergence at the common fixed point of two Lipschitz strictly hemicontractive-type mappings.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Asaduzzaman, M. Convergence and stability of modified multi-step Noor iterative procedure with errors for strictly hemicontractive-type mappings in Banach spaces. Fixed Point Theory Algorithms Sci Eng 2021, 6 (2021). https://doi.org/10.1186/s13663-021-00692-6

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MSC

  • 47J25
  • 47H09
  • 47H10
  • 54H25

Keywords

  • Modified multi-step Noor iterative procedure with errors
  • Lipschitz strictly hemicontractive-type mapping
  • Convergence
  • Common stability
  • Almost common-stability
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