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

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.

Throughout this paper, 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 ∈ 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 Example 1.1 (See [22]) Let X = R (with the usual norm), B = [-1 π , 1 π ] and |t| < 1. For each u ∈ 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.
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 = R (with the usual norm), B = [0, 1] and let ϕ be the Cantor ternary function. If we define T : B → X by then T n u → 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 multistep Noor iterative procedure with errors for some special type of asymptotically nonexpansive mappings (asymptotically non-expansive mapping in the intermediate sense and asymptotically quasi-non-expansive mapping) in Banach spaces: For a given u 1 ∈ B, and a fixed m ∈ N (set of all positive integers), the iterative sequences . . , 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][26][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 ∈ B, the iterative sequences {u n } ∞ n=0 defined by  [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 ∈ B, and a fixed m ∈ N, we compute the iterative sequences {u n } ∞ n=0 by  [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. [4,34]) The mapping T is called pseudocontractive if the inequality

Definition 2.1 (See
holds for each q, r ∈ 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(qr) ∈ J(qr) such that for all q, r ∈ B. T is called strongly pseudocontractive if there exists a t > 1 such that for all q, r ∈ D T and t > 0. T is called local strongly pseudocontractive if, for each q ∈ D T , there exists a t q > 1 such that for all q, r ∈ D T and t > 0.  T) . Now, if lim n→∞ δ n = 0 implies that lim n→∞ 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 ∞ n=0 δ n < ∞ implies that lim n→∞ 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. . Now, if lim n→∞ μ n = 0 implies that lim n→∞ 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 ∞ n=0 μ n < ∞ implies that lim n→∞ v n = q, then the iterative procedure defined by u n+1 = f (u n , T, S) is said to be an almost commonstable on B. Now, we recall some lemmas which are essential to prove the main results of this paper.

Lemma 2.5 (See
Lemma 2.6 (See [35]) Let x, y ∈ X. Then x ≤ x + ry for every r > 0 if and only if there

Lemma 2.7 (See [4]) Let T : D T ⊆ X → X be an operator with F T = ϕ. Then T is strictly hemicontractive if and if only if there exists a t > 1 such that for all x ∈ D T and q ∈ F T there exists h ∈ J(xq) satisfying
Lemma 2.8 (See [12]) Let X be an arbitrary norm linear space and T : D T ⊆ X → X be an operator.

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 λ = σ -1 σ ∈ (0, 1), where σ > 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
. . , m} are any appropriate real sequences in [0, 1] satisfying the following conditions: where θ is a constant in (0, λ) and λ ∈ (0, 1). Assume an iterative sequence {u n } ∞ n=0 defined by (1.5). Let {w n } ∞ n=0 be any sequence in B and {μ n } ∞ n=0 be a sequence defined by Then (i) the iterative sequence {u n } ∞ n=0 given by (1.5) converges strongly to the common fixed r of T and S and the following inequality holds: Proof (i) From the condition (2), we obtain c (m) n = δ n b (m) n , and δ n → 0 as n → ∞. By an application of Lemma 2.8, we see that F T ∩ F S is singleton, and let F T ∩ F S = {r} for some r ∈ 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 ∈ B is the fixed point of T, it follows that Now, for all n ≥ 0 from (3.4) and (3.5), we have Again, from (1.5), we get  Now, substituting (3.10) in (3.9), we have But, if we replace m by m -1 in (3.10), then we have Continuing the above procedure up to second iterative step of (1.5), we obtain n + · · · + L m- 3 Now, from the last equation of (1.5), we have Substituting (3.16) in (3.6), we have Now, if we put α n = u nr , in (3.17), then, by condition (3), we observe that α n+1 ≤ (1ω n )α n + ω n β n + γ n , n ≥ 0, with {ω n } ∞ n=0 ⊂ [0, 1], ∞ n=0 ω n = ∞, ∞ n=0 γ n < ∞ and lim n→∞ β n = 0. That is lim n→∞ u nr = 0. This ensures that the sequence {u n } ∞ n=0 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 ≥ 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 (3.21) Substituting (3.21) in (3.20), we have Now, from the third equation of (3.1), we get Continuing the above procedure up to second iterative step of (3.1), we obtain Now, from the last equation of (3.1), we have (3.25) Combining (3.24) and (3.25), we obtain  Convergence behavior corresponding to u 0 = 0.5 for 1000 iterative steps 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.    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].

Conclusion
In this study, we established the convergence, almost common-stability and commonstability 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][8][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.