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Convergence and stability theorems for the PicardMann hybrid iterative scheme for a general class of contractivelike operators
 Hudson Akewe^{1} and
 Godwin Amechi Okeke^{2}Email author
https://doi.org/10.1186/s1366301503154
© Akewe and Okeke; licensee Springer. 2015
 Received: 27 November 2014
 Accepted: 21 April 2015
 Published: 6 May 2015
Abstract
In this paper we use the general class of contractivelike operators introduced by Bosede and Rhoades (J. Adv. Math. Stud. 3(2):13, 2010) to prove strong convergence and stability results for PicardMann hybrid iterative schemes considered in a real normed linear space. We establish the strong convergence and stability of the Picard iterative scheme as a corollary. Our results generalize and improve a multitude of results in the literature, including the recent results of Chidume (Fixed Point Theory Appl. 2014:233, 2014).
Keywords
 strong convergence result
 PicardMann hybrid iterative schemes
 contractivelike operators
MSC
 47H09
 47H10
 54H25
1 Introduction and preliminary definitions
Fixed point iterative schemes are designed to be applied in solving equations arising in physical formulation but there is no systematic study of the numerical aspects of these iterative schemes. In computational mathematics, it is of vital interest to know which of the given iterative procedures converge faster to a desired solution, commonly known as the rate of convergence. Some of the notable authors whose research is in this direction are Hussain et al. [1], Phuengrattana and Suantai [2] and Khan [3]. Harder and Hicks [4, 5] revealed the importance of investigating the stability of various iteration procedures for various classes of nonlinear mappings. Harder [6] established applications of stability results to firstorder differential equations.
We will now consider some of these schemes, as they are relevant to this work.
If each \(\alpha_{n}=1\) in (1.2), we have the Picard iterative scheme (1.1).
Rhoades [8], perhaps for the first time, used computer programs to compare the rate of convergence of Mann and Ishikawa iterative procedures. He illustrated the difference in the rate of convergence for increasing and decreasing functions through examples.
In [2], Phuengrattana and Suantai defined the SP iterative scheme and proved that this scheme is equivalent to, and faster than, Mann [9], Ishikawa and Noor iterative schemes for increasing functions. Recently, Chugh and Kumar [10] introduced the CR iterative scheme and proved some convergence results. In the light of the iterative schemes mentioned above, it is clear that the study of the rate of convergence of several iterative schemes has attracted the interest of several wellknown mathematicians. Rhoades and Soltuz [11] introduced some multistep iterative schemes and proved some equivalence results. Akewe et al. [12] introduced the Kirkmultistep type iterative schemes and proved strong convergence and stability results, with some numerical examples to back up their work. These various results are worth emulating.
He showed that the hybrid scheme (PicardMann scheme (1.3)) converges faster than all of Picard (1.1), Mann (1.2) and Ishikawa [13] iterative schemes in the sense of Berinde [14] for contractions. He also proved the strong convergence and weak convergence theorems with the help of his iterative process (1.3) for the class of nonexpansive mappings in general Banach spaces and applied it to obtain results in uniformly convex Banach spaces.
Motivated by the work of Khan [3], we prove the strong convergence of the PicardMann iterative scheme for a general class of operators in a real normed space.

for each \(x,y \in X\), there exist \(\delta\in[0,1)\) and a monotone increasing function \(\varphi: R^{+} \rightarrow R^{+}\) with \(\varphi(0)=0\) such that$$ d(Tx,Ty)\leq\delta d(x,y)+ \varphi\bigl(d(x,Tx)\bigr). $$(1.5)
In 2014 Chidume and Olaleru [19] gave several examples to show that the class of mappings satisfying (1.7) is more general than that of (1.5) and (1.4), provided the fixed point exists. The authors [19] proved that every contraction map with a fixed point satisfies inequality (1.7) in the following example.
Several wellknown mathematicians have established some interesting fixed points results for certain classes of nonlinear mappings (see, e.g. Bosede [20], Rhoades [21], Zamfirescu [22]).
Example 1.1
[19]
 (i)
T is continuous and maps K into itself.
 (ii)
\(Tp=p\) implies \(p=0\).
 (iii)$$\begin{aligned} \Txp\_{\infty} =&\frac{9}{10}\bigl\ \bigl(0,x_{1}^{2},x_{2}^{2},x_{3}^{2}, \ldots\bigr)\bigr\ _{\infty}\\ \leq&\frac{9}{10}\bigl\ (0,x_{1},x_{2},x_{3}, \ldots)\bigr\ _{\infty}\\ =&\frac{9}{10}\xp\_{\infty}\quad\forall x\in K\ (\mbox{since } p=0). \end{aligned}$$
 (iv)
T is not a contraction map. To see this, let \(x=(\frac {3}{4},\frac{3}{4},\frac{3}{4},\ldots)\) and \(y=(\frac{1}{2},\frac {1}{2},\frac{1}{2},\ldots)\).
Clearly, \(x,y\in K\), \(\xy\_{\infty}=\frac{1}{4}\), and \(\TxTy\ _{\infty}=\frac{45}{160}\).
Suppose there exists \(a\in[0,1)\) such that \(\TxTy\_{\infty}\leq a\ xy\_{\infty}\), \(\forall x,y\in K\), we must then have thatfor the above choices of x and y. But then this implies that \(a\geq \frac{180}{160}>1\). So, T is not a contraction map.$$\TxTy\_{\infty}=\frac{45}{160}\leq\frac{a}{4}=\xy \_{\infty}$$  (v)
It is clear that every contraction map with a fixed point satisfies inequality (1.7). This completes our example.
We now give the following example.
Example 1.2

\(\Txp\_{\infty}= \frac{11}{12}\(0, x_{1}^{2}, x_{2}^{2}, x_{3}^{2},\ldots)\_{\infty}\) if \(\x\_{\infty}\leq1\),

\(\Txp\_{\infty}= \frac{11}{12\x\_{\infty}^{2}}\(0, x_{1}^{2}, x_{2}^{2}, x_{3}^{2},\ldots)\_{\infty}\) if \(\x\_{\infty}> 1\), so that

\(\Txp\_{\infty}= \frac{11}{12}\x\_{\infty}^{2} \leq\frac{11}{12}\x\ _{\infty}\) if \(\x\_{\infty}\leq1\),

\(\Txp\_{\infty}= \frac{11}{12}.1\) if \(\x\_{\infty}> 1\).
Hence, T satisfies contractive condition (1.7). But the map T is not a contraction. To see this, take \(x=(\frac{3}{4}, \frac{3}{4}, \frac {3}{4},\ldots)\); \(y=(\frac{1}{2}, \frac{1}{2}, \frac{1}{2},\ldots)\). Then \(\xy\_{\infty}= \frac{1}{4}\); \(\TxTy\_{\infty}=\frac{11}{12}\(0, \frac{5}{16}, \frac{5}{16},\ldots)\_{\infty}= \frac{55}{192}\).
Suppose there exists \(a \in[0, 1)\) such that \(\TxTy\_{\infty}\leq a\ xy\_{\infty}\) for every \(x,y \in E\), then we must have \(\frac {55}{192}\leq\frac{a}{4}\), which yields that \(a\geq\frac {220}{192}>1\), a contradiction. So, T is not a contraction map.
Example 1.3
Let \(E=[0,1]\). Define \(T:[0,1]\to[0,1]\) by \(Tx=\frac {x}{2}\), where \([0,1]\) has the usual metric. Then T satisfies inequality (1.7) and \(F(T)=[0,1]\).
We shall show that the PicardMann hybrid iterative scheme (1.3) is Tstable.
Now, let \(p=0\). Take \(\alpha_{n}=\frac{1}{2}\), \(y_{n}=\frac{1}{n}\) for each \(n\geq1\).
Then \(\lim_{n\to\infty}y_{n}=0\).
We shall need the following lemma in proving our result.
Lemma 1.1
[23]
Let δ be a real number satisfying \(0\leq\delta< 1\) and \(\{\epsilon_{n}\}^{\infty}_{n=0}\) be a sequence of positive numbers such that \(\lim_{n \rightarrow\infty }\epsilon_{n}=0\). Then, for any sequence of positive numbers \(\{u_{n}\} ^{\infty}_{n=0}\) satisfying \(u_{n+1}\leq\delta u_{n} + \epsilon_{n}\), \(n=0,1,2,\ldots\) , we have \(\lim_{n \rightarrow\infty} u_{n}=0\).
2 Main results
Theorem 2.1
Proof
Theorem 2.1 leads to the following corollary.
Corollary 2.2
(Chidume [7])
3 Stability results for the PicardMann iterative schemes in real normed spaces
In this section we prove stability results for the PicardMann iterative schemes defined by (1.3) for a general class of contractivelike operators introduced by Bosede and Rhoades [18]. The stability of Picard iterative schemes follows as a corollary. The theorem is stated as follows.
Theorem 3.1
Proof
From Theorem 2.1, the sequence \(\{x_{n}\}\), defined by (1.3), converges to p.
Let \(\{z_{n}\}^{\infty}_{n=0}\), \(\{u_{n}\}^{\infty}_{n=0}\) be real sequences in E.
Then we shall prove that \(\lim_{n\rightarrow\infty} z_{n}=p\) for mappings satisfying condition (3.1).
Theorem 3.1 yields the following corollary.
Corollary 3.2
(Bosede and Rhoades [18])
Declarations
Acknowledgements
The authors wish to thank the anonymous referees for their suggestions and corrections which led to the improvement of this paper. The second author is grateful to the Covenant University Centre for Research and Development (CUCRID) for supporting his research.
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
 Hussain, N, Chugh, R, Kumar, V, Rafiq, A: On the rate of convergence of Kirktype iterative schemes. J. Appl. Math. 2012, Article ID 526503 (2012) View ArticleMathSciNetGoogle Scholar
 Phuengrattana, W, Suantai, S: On the rate of convergence of Mann, Ishikawa, Noor and SPiterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math. 235(9), 30063014 (2011) View ArticleMATHMathSciNetGoogle Scholar
 Khan, SH: A PicardMann hybrid iterative process. Fixed Point Theory Appl. 2013, 69 (2013) View ArticleGoogle Scholar
 Harder, AM, Hicks, TL: A stable iteration procedure for nonexpansive mappings. Math. Jpn. 33, 687692 (1988) MATHMathSciNetGoogle Scholar
 Harder, AM, Hicks, TL: Stability results for fixed point iteration procedures. Math. Jpn. 33, 693706 (1988) MATHMathSciNetGoogle Scholar
 Harder, AM: Fixed point theory and stability results for fixed point iteration procedures. Ph. D. thesis, University of MissouriRolla (1987) Google Scholar
 Chidume, CE: Strong convergence and stability of Picard iteration sequences for a general class of contractivetype mappings. Fixed Point Theory Appl. 2014, 233 (2014) View ArticleMathSciNetGoogle Scholar
 Rhoades, BE: Fixed point iteration using infinite matrices. Trans. Am. Math. Soc. 196, 161176 (1974) View ArticleMATHMathSciNetGoogle Scholar
 Mann, WR: Mean value methods in iterations. Proc. Am. Math. Soc. 44, 506510 (1953) View ArticleGoogle Scholar
 Chugh, R, Kumar, V: Strong convergence of SP iterative scheme for quasicontractive operators. Int. J. Comput. Appl. 31(5), 2127 (2011) Google Scholar
 Rhoades, BE, Soltuz, SM: The equivalence between MannIshikawa iterations and multistep iteration. Nonlinear Anal. 58, 219228 (2004) View ArticleMATHMathSciNetGoogle Scholar
 Akewe, H, Okeke, GA, Olayiwola, A: Strong convergence and stability of Kirkmultisteptype iterative schemes for contractivetype operators. Fixed Point Theory Appl. 2014, 45 (2014) View ArticleMathSciNetGoogle Scholar
 Ishikawa, S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147150 (1974) View ArticleMATHMathSciNetGoogle Scholar
 Berinde, V: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare (2002) MATHGoogle Scholar
 Osilike, MO: Stability results for Ishikawa fixed point iteration procedure. Indian J. Pure Appl. Math. 26(10), 937941 (1995) MATHMathSciNetGoogle Scholar
 Rhoades, BE: Fixed point theorems and stability results for fixed point iteration procedures. Indian J. Pure Appl. Math. 21, 19 (1990) MATHMathSciNetGoogle Scholar
 Imoru, CO, Olatinwo, MO: On the stability of Picard and Mann iteration. Carpath. J. Math. 19, 155160 (2003) MATHMathSciNetGoogle Scholar
 Bosede, AO, Rhoades, BE: Stability of Picard and Mann iteration for a general class of functions. J. Adv. Math. Stud. 3(2), 13 (2010) MathSciNetGoogle Scholar
 Chidume, CE, Olaleru, JO: Picard iteration process for a general class of contractive mappings. J. Niger. Math. Soc. 33, 1923 (2014) MathSciNetGoogle Scholar
 Bosede, AO: Noor iterations associated with Zamfirescu mappings in uniformly convex Banach spaces. Fasc. Math. 42, 2938 (2009) MATHMathSciNetGoogle Scholar
 Rhoades, BE: A comparison of various definition of contractive mapping. Trans. Am. Math. Soc. 226, 257290 (1977) View ArticleMATHMathSciNetGoogle Scholar
 Zamfirescu, T: Fixed point theorems in metric spaces. Arch. Math. (Basel) 23, 292298 (1972) View ArticleMATHMathSciNetGoogle Scholar
 Berinde, V: On the stability of some fixed point procedures. Bul. Ştiinţ.  Univ. Baia Mare, Ser. B Fasc. Mat.Inform. 18(1), 714 (2002) MATHMathSciNetGoogle Scholar