Stability and convergence of a new composite implicit iterative sequence in Banach spaces
- Liping Yang1Email author and
- Weiming Kong1
https://doi.org/10.1186/s13663-015-0425-z
© Yang and Kong 2015
Received: 3 June 2015
Accepted: 14 September 2015
Published: 22 September 2015
Abstract
The purpose of this paper is to study stability and strong convergence of asymptotically pseudocontractive mappings by using a new composite implicit iteration process in an arbitrary real Banach space. The results in this paper improve and extend the corresponding results in the literature.
Keywords
MSC
1 Introduction
We first recall some definitions and conclusions.
Definition 1.1
- (1)T is said to be asymptotically nonexpansive (see [1]) if there exists a sequence \(\{k_{n}\}\subset [1,\infty)\), \(\lim_{n\to\infty}k_{n}=1\) such that$$ \bigl\Vert T^{n}x-T^{n}y\bigr\Vert \leq k_{n} \|x-y\|, \quad \forall x, y\in D(T), n\geq1; $$
- (2)T is said to be asymptotically pseudocontractive (see [2]) with sequence \(\{k_{n}\}\subset[0, \infty)\), if and only if \(\lim_{n\to\infty }k_{n}=1\), for all \(n\geq1\), \(x, y\in D(T)\) and there exists \(j(x-y)\in J(x-y)\) such that$$ \bigl\langle T^{n}x-T^{n}y, j(x-y)\bigr\rangle \leq k_{n}\|x-y\|^{2}; $$
- (3)T is said to be strictly asymptotically pseudocontractive with sequence \(\{k_{n}\}\subset[0, \infty)\), if and only if \(\lim_{n\to \infty}k_{n}=k\in(0, 1)\), for all \(n\geq1\), \(x, y\in D(T)\) and there exists \(j(x-y)\in J(x-y)\) such that$$ \bigl\langle T^{n}x-T^{n}y, j(x-y)\bigr\rangle \leq k_{n}\|x-y\|^{2}; $$
- (4)T is said to be uniformly L-Lipschitzian (see [3]) if there exists a constant \(L> 0\) such that$$ \bigl\Vert T^{n}x-T^{n}y\bigr\Vert \leq L\|x-y\| \quad \mbox{for all } x, y\in D(T), n\ge1. $$
It is easy to see that every asymptotically nonexpansive mapping is uniformly L-Lipschitzian and asymptotically pseudocontractive. In [4], Rhoades constructed an example to show that the class of asymptotically pseudocontractive mappings properly contains the class of asymptotically nonexpansive mappings.
The class of asymptotically pseudocontractive mappings has been studied by several authors (see [2, 4–7] and the references cited therein) by using the modified Mann iteration process (see [8]) and the modified Ishikawa iteration process (see [9]). Schu [5] proved the following theorem.
Theorem SC
The recursion formula (1.1) is a modification of the well-known Mann iteration process (see, e.g., [8]).
Recently, Chang [6] extended Theorem SC to real uniformly smooth Banach spaces and proved the following theorem.
Theorem CH
Let E be a real uniformly smooth Banach space, K a nonempty bounded closed convex subset of E, \(T: K\to K\) be an asymptotically pseudocontractive mapping with sequence \(\{k_{n}\}\subset[1, \infty)\), \(\lim_{n\to\infty}k_{n}=1\), and \(F(T)=\{x\in K: Tx=x\}\neq\emptyset\). Let \(\{\alpha_{n}\}\subset[0, 1]\) satisfy the following conditions: (i) \(\lim_{n\to\infty}\alpha _{n}=0\), (ii) \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\). For an arbitrary \(x_{0}\in K\), let \(\{x_{n}\}\) be iteratively defined by (1.1). If there exists a strictly increasing function \(\phi: [0, \infty)\to[0, \infty)\), \(\phi(0)=0\) such that \(\langle T^{n}x_{n}-x^{*}, j(x_{n}-x^{*}) \rangle \le k_{n}\lVert x_{n}-x^{*}\rVert^{2}-\phi(\lVert x_{n}-x^{*}\rVert)\), \(\forall n\ge 1\), then \(x_{n}\to x^{*}\in F(T)\).
It is clear that even if K is a nonempty convex subset of E and \(\{u_{n}\}\subset K\) is such that \(\sum_{n=1}^{\infty}\|u_{n}\|<\infty\), then the implicit iterative sequence with errors in the sense of [13] need not be well defined, i.e., \(\{x_{n}\}_{n=1}^{\infty}\) may fail to be in K. More precisely, the conditions imposed on the error terms are not compatible with the randomness of the occurrence of errors.
Stability results established in metric space, normed linear space, and Banach space settings are available in the literature. There are several authors whose contributions are of colossal value in the study of stability of the fixed point iterative procedures: Imoru and Olatinwo [17], Olatinwo and Postolache [18], Akewe and Okeke [19].
Harder and Hicks [20] mentioned that the study of the stability of iterative schemes is useful for both theoretical and numerical investigations. Consequently, several authors have studied the stability of iterative schemes for various types of nonlinear mappings (see, e.g., [20–25] and the references cited therein).
The purpose of this paper is to study the stability and convergence of the composite implicit iterative sequence for an asymptotically pseudocontractive mapping in arbitrary real Banach spaces.
2 Preliminaries
In the sequel, \(F(T)=\{x\in K: Tx=x\}\) denotes the set of fixed points of T. We give the stability definition for the sequence \(\{x_{n}\}\) defined by (1.4).
Definition 2.1
The following lemmas will be needed in proving our main results.
Lemma 2.2
(Lemma 2 of [26])
We denote Φ := {\(\phi \mid \phi: [0, \infty)\to[0, \infty)\) be a nondecreasing function such that \(\phi(t)=0\) if and only if \(t=0\)}.
Lemma 2.3
(Lemma 2.1 of [27])
Lemma 2.4
(Lemma 1.1 of [28])
Lemma 2.5
First we give two auxiliary lemmas.
Lemma 2.6
Proof
3 Main results
Theorem 3.1
- (1)
\(\{x_{n}\}\) converges strongly to the unique common fixed point p of T;
- (2)
\(\{x_{n}\}\) is both almost T-stable and weakly T-stable.
Proof
If \(\varepsilon _{n}/\alpha_{n}\to0\), taking \(a_{n}= \lVert s_{n}-p \rVert \), \(t_{n}=\alpha_{n}\delta\), \(c_{n}=0\) in Lemma 2.2, from (3.10), we have \(s_{n}\to p\) (\(n\to\infty\)), i.e., \(\{x_{n}\}\) is weakly T-stable. This completes the proof. □
Example 3.2
Example 3.3
The iteration chart with initial value \(\pmb{x_{1}=0.25}\)
n | \(\boldsymbol{x_{n}}\) |
---|---|
1 | 0.25 |
2 | 0.134228 |
3 | 0.090397 |
4 | 0.067943 |
5 | 0.054373 |
6 | 0.045313 |
7 | 0.038840 |
8 | 0.033985 |
9 | 0.030209 |
10 | 0.027188 |
Therefore, the conditions of Theorem 3.1 are fulfilled.
Theorem 3.4
Proof
If \(\{\beta_{n}\}=\{0\}\) for all \(n\ge1\) in (1.4), it follows from Theorem 3.4 that we have the following result.
Theorem 3.5
Remark 3.6
Remark 3.7
We remark that if the error terms are added in (1.4) and are assumed to be bounded, then the results of this paper still hold. On carefully reading Thakur’s work [16], we discovered that there are gaps in the proof of Lemma 2.1 in [16]. In (2.2) and (2.3) of Lemma 2.1, one cannot deduce \(\sum_{n=1}^{\infty}\sigma_{n}<\infty\) from \(\sum_{n=1}^{\infty}\mu_{n}<\infty\), \(\sum_{n=1}^{\infty}d_{n}<\infty\). Thus, his main results would not hold.
Declarations
Acknowledgements
The authors thank the referees for the comments that helped improve the presentation of this article. This work was supported by the Humanity and Social Science Planning Foundation of Ministry of Education of China (Grant No. 14YJAZH095), the National Natural Science Foundation of China (Grant No. 61374081), the High-level Talents Project in Guangdong Province (Grant No. 2014011), and the Natural Science Foundation of Guangdong Province (Grant Nos. S2013010013034, 2015A030313485).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171-174 (1972) MATHMathSciNetView ArticleGoogle Scholar
- Schu, J: On a theorem of C.E. Chidume concerning the iterative approximation of fixed points. Math. Nachr. 153, 313-319 (1991) MATHMathSciNetView ArticleGoogle Scholar
- Goebel, K, Kirk, WA: A fixed point theorem for transformations whose iterates have uniform Lipschitz constant. Stud. Math. 47, 135-140 (1973) MATHMathSciNetGoogle Scholar
- Rhoades, BE: A comparison of various definition of contractive mappings. Trans. Am. Math. Soc. 226, 257-290 (1977) MATHMathSciNetView ArticleGoogle Scholar
- Schu, J: Iterative construction of fixed point of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 158, 407-413 (1991) MATHMathSciNetView ArticleGoogle Scholar
- Chang, SS: Some results for asymptotically pseudocontractive mappings and asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 129, 845-853 (2000) View ArticleGoogle Scholar
- Ofoedu, EU: Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space. J. Math. Anal. Appl. 321, 722-728 (2006) MATHMathSciNetView ArticleGoogle Scholar
- Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506-510 (1953) MATHView ArticleGoogle Scholar
- Ishikawa, S: Fixed point and iteration of a nonexpansive mapping in a Banach space. Proc. Am. Math. Soc. 59, 65-71 (1976) MATHMathSciNetView ArticleGoogle Scholar
- Xu, HK, Ori, RG: An implicit iteration process for nonexpansive mappings. Numer. Funct. Anal. Optim. 22(5), 767-773 (2001) MATHMathSciNetView ArticleGoogle Scholar
- Osilike, MO: Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps. J. Math. Anal. Appl. 294, 73-81 (2004) MATHMathSciNetView ArticleGoogle Scholar
- Gu, F: The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings. J. Math. Anal. Appl. 329, 766-776 (2007) MATHMathSciNetView ArticleGoogle Scholar
- Chang, SS, Tan, KK, Lee, HWJ, Chan, CK: On the convergence of implicit iteration process with errors for a finite family of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 313, 273-283 (2006) MATHMathSciNetView ArticleGoogle Scholar
- Yang, LP: Convergence of the new composite implicit iteration process with random errors. Nonlinear Anal. TMA 69, 3591-3600 (2008) MATHView ArticleGoogle Scholar
- Su, Y, Li, S: Composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps. J. Math. Anal. Appl. 320, 882-891 (2006) MATHMathSciNetView ArticleGoogle Scholar
- Thakur, BS: Weak and strong of composite implicit iteration process. Appl. Math. Comput. 190, 965-973 (2007) MATHMathSciNetView ArticleGoogle Scholar
- Imoru, CO, Olatinwo, MO: On the stability of Picard and Mann iteration procedures. Carpath. J. Math. 19(2), 155-160 (2003) MATHMathSciNetGoogle Scholar
- Olatinwo, MO, Postolache, M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput. 218(12), 6727-6732 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Akewe, H, Okeke, GA: Convergence and stability theorems for the Picard-Mann hybrid iterative scheme for a general class of contractive-like operators. Fixed Point Theory Appl. 2015, 66 (2015) MathSciNetView ArticleGoogle Scholar
- Harder, AM, Hicks, TL: Stability results for fixed point iteration procedures. Math. Jpn. 33(5), 693-706 (1988) MATHMathSciNetGoogle Scholar
- Zhou, HY, Chang, SS, Cho, YJ: Weak stability of Ishikawa iteration procedures for ϕ-hemicontractive and accretive operators. Appl. Math. Lett. 14, 949-954 (2001) MATHMathSciNetView ArticleGoogle Scholar
- Stević, S: Stability results for ϕ-strongly pseudocontractive mappings. Yokohama Math. J. 50, 71-85 (2003) MATHMathSciNetGoogle Scholar
- Huang, Z: Weak stability of Mann and Ishikawa iterations with errors for ϕ-hemicontractive operators. Appl. Math. Lett. 20, 470-475 (2007) MATHMathSciNetView ArticleGoogle Scholar
- Shahzad, N, Zegeye, H: On stability results for ϕ-strongly pseudocontractive mappings. Nonlinear Anal. TMA 64, 2619-2630 (2006) MATHMathSciNetView ArticleGoogle Scholar
- Osilike, MO: Stability of the Mann and Ishikawa iteration procedures for ϕ-strong pseudocontractions and nonlinear equations of the ϕ-strongly accretive type. J. Math. Anal. Appl. 227, 319-334 (1998) MATHMathSciNetView ArticleGoogle Scholar
- Liu, LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 194, 114-125 (1995) MATHMathSciNetView ArticleGoogle Scholar
- Ćirić, LB, Ume, JS: Iterative processes with errors for nonlinear equations. Bull. Aust. Math. Soc. 69, 117-189 (2005) Google Scholar
- Kato, T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn. 19, 508-520 (1964) View ArticleGoogle Scholar