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
MSC
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
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