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.
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1 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 [2–5] 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 [1–3, 6–21] and the references cited therein. Among the articles cited in [1–3, 6–21], 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, 12–15].
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
The mapping T is called Lipschitzian if there exists a \(L > 0\) such that
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
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 \(p,q \in B\),
T is called an asymptotically non-expansive mapping in the intermediate sense if T is uniformly continuous and
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 \(p \in B\), \(q \in F_{T}\),
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
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
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
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
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
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
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, 25–27] 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
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
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.
2 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
The mapping T is called pseudocontractive if the inequality
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
for all \(q, r \in B\). T is called strongly pseudocontractive if there exists a \(t > 1\) such that
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
for all \(q, r \in D_{T}\) and \(t' > 0\).
Definition 2.2
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
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
(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
Lemma 2.8
(See [12])
Let X be an arbitrary norm linear space and \(T:D_{T} \subseteq X \to X\) be an operator.
-
(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.
-
(b)
If T is strictly hemicontractive, then \(F_{T}\) is a singleton.
3 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)
\(a_{n}^{(i)} + b_{n}^{(i)} + c_{n}^{(i)} = 1\), for each \(i \in \{ 1,2, 3,\ldots, m\}\),
-
(2)
\(c_{n}^{ ( m )} = \mathrm{o} ( b_{n}^{ ( m )} )\),
-
(3)
\(\lim_{n \to \infty } c_{n}^{ ( j )} = 0\), for each \(j \in \{ 1, 2, 3,\ldots, m - 1 \} \),
-
(4)
\(\sum_{n = 0}^{\infty } b_{n}^{(j)} = \infty \) for each \(j \in \{ 2, 3,\ldots, m\}\),
-
(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
where
Then
-
(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:
$$\begin{aligned}& \Vert u_{n + 1} - r \Vert \\& \quad \le \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert u_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + c_{n}^{(m - 1)}Lb_{n}^{(m)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + c_{n}^{(m - 2)}L^{2}b_{n}^{(m)}b_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + c_{n}^{(m - 3)}L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + c_{n}^{(2)}L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{} + c_{n}^{(1)}L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(2)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr], \quad n \ge 0, \end{aligned}$$ -
(ii)
$$\begin{aligned}& \Vert w_{n + 1} - r \Vert \\& \quad \le \bigl( 1 - \theta b_{n}^{(m)} \bigr) \Vert w_{n} - r \Vert \\& \quad\quad{} + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + L \bigl[ b_{n}^{(m)}c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \\& \quad\quad{} + L \bigl[ b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] \bigr] \bigr] \bigr] \cdots \bigr] \bigr] + \mu _{n}, \quad n \ge 0, \end{aligned}$$
-
(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,
-
(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
Since T is strictly hemicontractive, from Lemma 2.7, we obtain
Now, from (3.2) and Lemma 2.6, we have
Also, from the first equation of (1.5), we get
and since \(r \in B\) is the fixed point of T, it follows that
Now, for all \(n \ge 0\) from (3.4) and (3.5), we have
which implies that
Again, from (1.5), we get
But
Substituting (3.8) in (3.7), we have
But, if we replace m by \(m - 1\) in (3.8), then we have
Now, substituting (3.10) in (3.9), we have
But, if we replace m by \(m - 1\) in (3.10), then we have
Substituting (3.12) in (3.11), we have
Continuing the above procedure up to second iterative step of (1.5), we obtain
Now, from the last equation of (1.5), we have
Substituting (3.15) in (3.14), we have
Substituting (3.16) in (3.6), we have
Now, if we put
in (3.17), then, by condition (3), we observe that
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
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
Now, for all \(n \ge 0\) combining (3.5) and (3.18), we obtain
But, applying the second equation of (3.1), we have
But after a simple calculation we get
Substituting (3.21) in (3.20), we have
Now, from the third equation of (3.1), we get
Substituting (3.23) in (3.22), we have
Continuing the above procedure up to second iterative step of (3.1), we obtain
Now, from the last equation of (3.1), we have
Combining (3.24) and (3.25), we obtain
Inserting (3.26) in (3.19), we find
Now, using (3.27), we obtain
(iii) If we put
in (3.28), then we obtain
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
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:
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:
where h is a constant.
Then
-
(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
$$\begin{aligned} D = {}&\lambda ^{ - 1} ( 1 + L ) \Bigl[ \sup_{n \ge 0} \bigl\{ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert \bigr\} + Lb_{n}^{(m)}\sup_{n \ge 0} \bigl\{ c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \bigr\} \\ &{}+ L^{2}b_{n}^{(m)}b_{n}^{(m - 1)} \sup_{n \ge 0} \bigl\{ c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert \bigr\} \\ &{}+ L^{3}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \sup_{n \ge 0} \bigl\{ c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \bigr\} \\ &{}+ \cdots + L^{m - 2}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)} \cdots b_{n}^{(3)}\sup_{n \ge 0} \bigl\{ c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert \bigr\} \\ &{}+ L^{m - 1}b_{n}^{(m)}b_{n}^{(m - 1)}b_{n}^{(m - 2)}b_{n}^{(m - 3)} \cdots b_{n}^{(2)}\sup_{n \ge 0} \bigl\{ c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr\} \Bigr], \end{aligned}$$ -
(ii)
$$\begin{aligned}& \Vert w_{n + 1} - r \Vert \\& \quad \le ( 1 - \theta h ) \Vert w_{n} - r \Vert + \lambda ^{ - 1} ( 1 + L ) \bigl[ c_{n}^{(m)} \bigl\Vert v_{n}^{(m)} - r \bigr\Vert + L \bigl[ c_{n}^{(m - 1)} \bigl\Vert v_{n}^{(m - 1)} - r \bigr\Vert \\& \quad\quad{} + L \bigl[ c_{n}^{(m - 2)} \bigl\Vert v_{n}^{(m - 2)} - r \bigr\Vert + L \bigl[ c_{n}^{(m - 3)} \bigl\Vert v_{n}^{(m - 3)} - r \bigr\Vert \\& \quad\quad{} + \cdots + L \bigl[ c_{n}^{(2)} \bigl\Vert v_{n}^{(2)} - r \bigr\Vert + L \bigl[ c_{n}^{(1)} \bigl\Vert v_{n}^{(1)} - r \bigr\Vert \bigr] \bigr] \bigr] \bigr] \cdots \bigr] \bigr] + \mu _{n}, \quad \forall n \ge 0, \end{aligned}$$
-
(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
Now, putting
we get \(0 \le \eta < 1\) and \(\lim_{n \to \infty } \beta _{n} = 0\). Hence, by an application of Lemma 2.5, we obtain
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
(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
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
we get \(0 \le \eta < 1\) and \(\lim_{n \to \infty } \beta _{n} = 0\). Hence, by an application of Lemma 2.5, we obtain
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:
-
(6)
\(\alpha _{n}^{(i)} + \beta _{n}^{(i)} = 1\), for each \(i \in \{ 1,2, 3,\ldots, m\}\),
-
(7)
\(\sum_{n = 0}^{\infty } \alpha _{n}^{(j)} = \infty \) for each \(j \in \{ 2, 3,\ldots, m\}\),
-
(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
and let \(\{ w_{n} \} _{n = 0}^{\infty }\) be any sequence in B and \(\{ \mu _{n} \} _{n = 0}^{\infty } \) be a sequence defined by
where
Then
-
(i)
the iterative sequence \(\{ u_{n} \} _{n = 0}^{\infty }\) given by (3.29) converges strongly to the common fixed r of T and S,
-
(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,
-
(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:
where h is a constant.
Then
-
(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:
$$\begin{aligned}& \Vert u_{n + 1} - r \Vert \le ( 1 - \theta h ) \Vert u_{n} - r \Vert , \quad \forall n \ge 0, \end{aligned}$$ -
(ii)
\(\Vert w_{n + 1} - r \Vert \le ( 1 - \theta h ) \Vert w_{n} - r \Vert + \mu _{n}\), \(\forall n \ge 0\),
-
(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.
4 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:
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
Hence, both T and S are Lipschitzian mappings on B.
Likewise, using (1.1) we have
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
Then, for \(n = 0\), \(m = 10\), we obtain
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
For the STIIPE given by Hussain et al. [1], we consider \(v_{n}^{ ( i )} = \frac{1}{n + 1}\), where \(i = 1, 2\) and
For the IIPE given by Liu [31], we consider \(v_{n}^{ ( i )} = \frac{1}{n + 1}\), where \(i = 1, 2\) and
For the TIPE given by Cho et al. [33], we consider \(v_{n}^{ ( i )} = \frac{1}{n + 1}\), where \(i = 1, 2, 3\) and
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].
Convergence behavior corresponding to \(u_{0} = 0.5\) for 1000 iterative steps
5 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, 7–9], 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.
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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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DOI: https://doi.org/10.1186/s13663-021-00692-6