In this section, we shall introduce and study a class of new Picard-Mann iterative methods with mixed errors for common fixed points of two different nonexpansive and contraction operators and prove convergence and stability of the new Picard-Mann iterative approximation.
We need the following definitions and lemmas for our main results.
Definition 2.1
Let X be a normed space and \(K\subset X\) be a nonempty subset. Then an operator \(T: K\to K\) is said to be
-
(i)
nonexpansive if
$$\begin{aligned} \Vert Tu-Tv\Vert \le \Vert u-v\Vert , \quad \forall u, v\in K; \end{aligned}$$
(2.1)
-
(ii)
contraction if there exists a constant \(k\in[0, 1)\) such that
$$\begin{aligned} \Vert Tu-Tv\Vert \le k\Vert u-v\Vert , \quad \forall u, v\in K. \end{aligned}$$
(2.2)
Remark 2.1
The constant k in Definition 2.1(ii) is called the Lipschitz constant of T. Contractive operators are sometimes called Lipschitzian operators. If the above condition is instead satisfied for \(k\le1\), then the operator T is said to be nonexpansive.
Definition 2.2
Let S be a selfmap of the normed space X, \(x_{0}\in X\), and let \(x_{n+1}=h(S, x_{n})\) define an iteration procedure which yields a sequence of points \(\{x_{n}\}\subset X\). Suppose that \(\{x\in X: Sx=x\}\neq\emptyset\) and \(\{x_{n}\}\) converges to a fixed point \(x^{*}\) of S. Let \(\{w_{n}\}\subset X\) and let \(\varepsilon_{n}=\Vert w_{n+1}-h(S, w_{n})\Vert \). If \(\lim \epsilon_{n}=0\) implies that \(w_{n}\to x^{*}\), then the iteration procedure defined by \(x_{n+1}=h(S, x_{n})\) is said to be S-stable or stable with respect to S.
Lemma 2.1
([29])
Let
X
be a normed space and
C
be a nonempty closed convex bounded subset of X. Then each nonexpansive operator
\(T: C\to C\)
has a fixed point in C.
Lemma 2.2
([30])
Let
\(\{a_{n}\}\), \(\{b_{n}\}\), \(\{ c_{n}\}\)
be three nonnegative real sequences satisfying
$$\begin{aligned} a_{n+1}\leq ( 1-t_{n} ) a_{n}+t_{n}b_{n}+c_{n}, \end{aligned}$$
(2.3)
where
\(t_{n}\in[0, 1]\), \(\sum_{n=0}^{\infty}t_{n}=\infty\), \(\lim_{n\to\infty}b_{n}=0\), \(\sum_{n=0}^{\infty}c_{n} <\infty\). Then
\(a_{n}\to0\) (\(n\to\infty\)).
Now, we establish a class of new Picard-Mann iterations with mixed errors for common fixed points of two different nonlinear operators (in short, (PMMD)) as follows.
Algorithm 2.1
Step 1. Choose \(x_{0}\) in a normed space X.
Step 2. Let
$$\begin{aligned} \textstyle\begin{cases} x_{n+1}=T_{1}y_{n}+h_{n},\\ y_{n}=(1-\alpha_{n})x_{n}+\alpha_{n}T_{2}x_{n}+\alpha_{n}d_{n}+e_{n}, \end{cases}\displaystyle \end{aligned}$$
(2.4)
where \(T_{1}, T_{2}: X\to X\) are two nonlinear operators, and \(h_{n}, d_{n}, e_{n} \in X\) are errors to take into account a possible inexact computation of the operator points.
Step 3. Choose sequences \(\{\alpha_{n}\}\), \(\{h_{n}\}\), \(\{d_{n}\}\) and \(\{e_{n}\}\) such that for \(n\ge0\), \(\{\alpha_{n}\}\subset[0, 1]\) and \(\{h_{n}\}\), \(\{d_{n}\}\), \(\{e_{n}\}\) are three sequences in X satisfying the following conditions P:
-
(i)
\(d_{n}=d_{n}^{\prime}+d_{n}^{\prime\prime}\);
-
(ii)
\(\lim_{n\to\infty} \Vert d_{n}^{\prime} \Vert =0\);
-
(iii)
\(\sum_{n=0}^{\infty} \Vert h_{n}\Vert <\infty\), \(\sum_{n=0}^{\infty} \Vert d_{n}^{\prime\prime} \Vert <\infty\), \(\sum_{n=0}^{\infty} \Vert e_{n}\Vert <\infty\).
Step 4. If \(x_{n+1}\), \(y_{n}\), \(\alpha_{n}\), \(h_{n}\), \(d_{n}\) and \(e_{n}\) satisfy (2.4) to sufficient accuracy, go to Step 5; otherwise, set \(n: =n+1\) and return to Step 2.
Step 5. Let \(\{w_{n}\}\) be any sequence in X and define \(\{ \varepsilon_{n}\}\) by
$$ \textstyle\begin{cases} \varepsilon_{n}=\Vert w_{n+1}- ( T_{1}\xi_{n}+h_{n} ) \Vert , \\ \xi_{n}=(1-\alpha_{n})w_{n}+\alpha_{n} T_{2} w_{n}+\alpha_{n}d_{n}+e_{n}. \end{cases} $$
(2.5)
Step 6. If \(\varepsilon_{n}\), \(w_{n+1}\), \(\xi_{n}\), \(\alpha_{n}\), \(h_{n}\), \(d_{n}\) and \(e_{n}\) satisfy (2.5) to sufficient accuracy, stop; otherwise, set \(n:=n+1\) and return to Step 3.
Remark 2.2
For special choices of the operators \(T_{1}\) and \(T_{2}\), the space X, and the errors \(h_{n}\), \(d_{n}\) and \(e_{n}\) in (2.4), one can obtain a large number of Picard iterative process, Mann iterative process, Picard-Mann iterative process due to Khan [18] and other related iterations. Now we list some special cases of iteration (2.4) as follows.
-
Special Case I If \(h_{n}=d_{n}=e_{n}=0\), the iterative process (2.4) becomes the following Picard-Mann iteration for two different operators (in short, (PMD)): For any given \(x_{0}\in X\),
$$\begin{aligned} \textstyle\begin{cases} x_{n+1}=T_{1}y_{n}, \\ y_{n}=(1-\alpha_{n})x_{n}+\alpha_{n}T_{2}x_{n}. \end{cases}\displaystyle \end{aligned}$$
(2.6)
-
Special Case II When \(T_{1}=T_{2}=T\), for any given \(x_{0}\in X\), iteration (2.4) reduces to the sequence \(\{x_{n}\}\) defined by
$$\begin{aligned} \textstyle\begin{cases} x_{n+1}=Ty_{n}+h_{n}, \\ y_{n}=(1-\alpha_{n})x_{n}+\alpha_{n}Tx_{n}+\alpha_{n}d_{n}+e_{n}. \end{cases}\displaystyle \end{aligned}$$
(2.7)
We note that the iterative processes (PMD) and the Picard-Mann iteration with mixed errors (2.7) (in short, (PMM)) are new and not studied in the literature.
-
Special Case III If \(T_{1}=T_{2}=T\), then (2.6) reduces to
$$\begin{aligned} \textstyle\begin{cases} x_{n+1}=Ty_{n}, \\ y_{n}=(1-\alpha_{n})x_{n}+\alpha_{n}Tx_{n}, \end{cases}\displaystyle \end{aligned}$$
(2.8)
which was the Picard-Mann iterative process (in short, (PM)) studied by Khan [18] when \(\alpha_{n}\in(0, 1)\). We note that (PM) can be obtained from (2.7) if \(h_{n}=d_{n}=e_{n}=0\) for all \(n\ge0\). Further, the iterative process (2.8) reduces to the Picard-Krasnoselskii hybrid iterations studied by Okeke and Abbas [20] when \(\alpha_{n}=\lambda\in(0, 1)\). As Khan [18] pointed out, the iteration (2.8) is independent of all Picard and Mann iterative processes if \(\{\alpha_{n}\}\subset(0, 1)\). But one can easily see that the iterative process (2.8) will reduce to Picard and a special case of Ishikawa iterative process when \(\alpha_{n}=0\) and \(\alpha_{n}=1\), respectively.
-
Special Case IV When \(T_{1}=I\), the identity operator, for any given \(x_{0}\in X\), the iteration (PMD) defined by (2.6) can be written as
$$\begin{aligned} x_{n+1}=(1-\alpha_{n})x_{n}+\alpha_{n}T_{2}x_{n}, \end{aligned}$$
(2.9)
which is the Mann iterative process (in short, (MI)) for \(\alpha _{n}\in[0, 1]\).
Based on Lemma 2.1 and the existence of fixed point for a contraction operator, in the sequel, we will prove convergence and stability of the new Picard-Mann iterative processes with mixed errors generated by Algorithm 2.1.
Theorem 2.1
Let
X
be a normed space and
\(C\subset X\)
be a nonempty closed convex bounded subset. Let
\(T_{1}: C\to C\)
be nonexpansive and
\(T_{2}: C\to C\)
be a contraction operator with constant
\(\theta\in[0, 1)\). Suppose that
\(F(T_{1}\cap T_{2}):=\{x\in C: T_{i}x=x, i=1, 2\}\neq\emptyset\)
and
\(\sum_{n=0}^{\infty}\alpha_{n}=\infty\). Then
-
(i)
the iterative sequence
\(\{x_{n}\}\)
generated by (PMMD) in Algorithm
2.1
converges to
\(x^{*}\in F(T_{1}\cap T_{2})\)
with convergence rate
$$\begin{aligned} \vartheta=1-\hat{\alpha}(1-\theta)< 1, \end{aligned}$$
(2.10)
where
\(\hat{\alpha}=\limsup_{n\to\infty}\alpha_{n}\in(0, 1]\);
-
(ii)
if, in addition, for any sequence
\(\{z_{n}\}\subset X\), there exists
\(\alpha>0\)
such that
\(\alpha_{n}\ge\alpha\)
for all
\(n\geq 0\), then
$$\begin{aligned} \lim_{n\to\infty}w_{n}=x^{*} \quad\textit{if and only if}\quad\lim_{n\to\infty}\varepsilon_{n}=0, \end{aligned}$$
(2.11)
where
\(\varepsilon_{n}\)
is defined by (2.5).
Proof
It follows from (2.4) that
$$\begin{aligned} &\bigl\Vert x_{n+1}-x^{*}\bigr\Vert \\ &\quad \le\bigl\Vert y_{n}-x^{*}\bigr\Vert +\Vert h_{n}\Vert \\ &\quad \le(1-\alpha_{n})\bigl\Vert x_{n}-x^{*}\bigr\Vert +\alpha_{n}\bigl\Vert T_{2} x_{n}-x^{*} \bigr\Vert \\ &\quad \quad{}+\alpha_{n}\bigl(\bigl\Vert d_{n}^{\prime} \bigr\Vert +\bigl\Vert d_{n}^{\prime\prime}\bigr\Vert \bigr)+ \Vert e_{n}\Vert +\Vert h_{n}\Vert \\ &\quad \le(1-\alpha_{n})\bigl\Vert x_{n}-x^{*}\bigr\Vert +\alpha _{n}\theta\bigl\Vert x_{n}-x^{*} \bigr\Vert \\ &\quad \quad{}+\alpha_{n}\bigl\Vert d_{n}^{\prime}\bigr\Vert +\bigl(\bigl\Vert d_{n}^{\prime\prime}\bigr\Vert +\Vert e_{n}\Vert +\Vert h_{n}\Vert \bigr) \\ &\quad =\vartheta_{n}\bigl\Vert x_{n}-x^{*}\bigr\Vert +(1-\theta)\alpha _{n}\cdot\frac{1}{1-\theta}\bigl\Vert d_{n}^{\prime}\bigr\Vert \\ &\quad \quad{}+\bigl(\bigl\Vert d_{n}^{\prime\prime}\bigr\Vert +\Vert e_{n}\Vert +\Vert h_{n}\Vert \bigr), \end{aligned}$$
(2.12)
where \(\vartheta_{n}=1-(1-\theta)\alpha_{n}\). Since \(\sum_{n=0}^{\infty}\alpha_{n}=\infty\), by Lemma 2.2 and (2.12), now we know that \(\Vert x_{n}-x^{*}\Vert \to 0\) (\(n\to\infty\)). Thus, the sequence \(\{x_{n}\}\) converges to \(x^{*}\) for \(\vartheta_{n}\).
Further, by (2.12), we have
$$\begin{aligned} \limsup_{n\to\infty}\vartheta_{n}=1-\hat{\alpha}(1-\theta), \end{aligned}$$
(2.13)
where \(\hat{\alpha}=\limsup_{n\to\infty}\alpha_{n}\).
Next, we prove the conclusion (ii). Since \(0<\alpha\leq\alpha_{n}\), it follows from the proof of inequality (2.12) and (2.5) that
$$\begin{aligned} &\bigl\Vert T_{1}\xi_{n}+h_{n}-x^{*} \bigr\Vert \\ &\quad \le \bigl[ 1-(1-\theta)\alpha_{n} \bigr] \bigl\Vert w_{n}-x^{*}\bigr\Vert +\alpha_{n}\bigl\Vert d_{n}^{\prime}\bigr\Vert +\bigl(\bigl\Vert d_{n}^{\prime\prime}\bigr\Vert +\Vert e_{n}\Vert + \Vert h_{n}\Vert \bigr), \end{aligned}$$
(2.14)
and
$$\begin{aligned} &\bigl\Vert w_{n+1}-x^{*}\bigr\Vert \\ &\quad \le\bigl\Vert T_{1}\xi_{n}+h_{n}-x^{*} \bigr\Vert +\varepsilon _{n} \\ &\quad \le \bigl[ 1-(1-\theta)\alpha_{n} \bigr] \bigl\Vert w_{n}-x^{*}\bigr\Vert +\alpha_{n}\bigl\Vert d_{n}^{\prime}\bigr\Vert +\varepsilon_{n} \\ &\quad \quad{}+\bigl(\bigl\Vert d_{n}^{\prime\prime}\bigr\Vert +\Vert e_{n}\Vert +\Vert h_{n}\Vert \bigr) \\ &\quad \le \bigl[ 1-(1-\theta)\alpha_{n} \bigr] \bigl\Vert w_{n}-x^{*}\bigr\Vert +(1-\theta)\alpha_{n}\cdot \frac{1}{1-\theta } \biggl( \bigl\Vert d_{n}^{\prime}\bigr\Vert +\frac{\varepsilon _{n}}{\alpha} \biggr) \\ &\quad \quad{}+\bigl(\bigl\Vert d_{n}^{\prime\prime}\bigr\Vert +\Vert e_{n}\Vert +\Vert h_{n}\Vert \bigr). \end{aligned}$$
(2.15)
Let \(\lim_{n\to\infty}\varepsilon_{n}=0\). Then, by \(\sum_{n=0} ^{\infty}\alpha_{n}=\infty\), Lemma 2.2 and (2.15), we know that \(\lim_{n\to\infty}w_{n}=x^{*}\).
Conversely, if \(\lim_{n\rightarrow\infty}w_{n}=x^{*}\), then it follows from (2.14) and \(\alpha_{n}\le1\) that, for all \(n\ge0\),
$$\begin{aligned} \varepsilon_{n}&=\bigl\Vert w_{n+1}- ( T_{1} \xi_{n}+h_{n} ) \bigr\Vert \\ &\le\bigl\Vert w_{n+1}-x^{*}\bigr\Vert +\bigl\Vert T_{1}\xi _{n}+h_{n}-x^{*}\bigr\Vert \\ &\le\bigl\Vert w_{n+1}-x^{*}\bigr\Vert + \bigl[ 1-(1- \theta)\alpha _{n} \bigr] \bigl\Vert w_{n}-x^{*} \bigr\Vert \\ &\quad{}+\alpha_{n}\bigl\Vert d_{n}^{\prime}\bigr\Vert + \bigl( \bigl\Vert d_{n}^{\prime\prime}\bigr\Vert +\Vert e_{n}\Vert +\Vert h_{n}\Vert \bigr) \\ &\le\bigl\Vert w_{n+1}-x^{*}\bigr\Vert +\bigl\Vert w_{n}-x^{*}\bigr\Vert + \bigl( \bigl\Vert d_{n}^{\prime}\bigr\Vert +\bigl\Vert d_{n}^{\prime\prime} \bigr\Vert +\Vert e_{n}\Vert +\Vert h_{n}\Vert \bigr) , \end{aligned}$$
(2.16)
this implies that \(\varepsilon_{n} \to0\) as \(n\to\infty\). This completes the proof. □
Remark 2.3
(i) Since the errors in Algorithm 2.1 exist objectively when the inexact calculation of operator points is considered, the iterative process (2.4) (i.e., (PMMD)) is more truthful than the Picard iteration, Mann iteration, Picard-Mann iteration due to Khan [18] and so on. One can easily observe in the next numerical simulations visually.
(ii) We note that the stability analysis in Theorem 2.1 is little discussed in the literature. Akewe and Okeke [27] gave the stability theorems for the Picard-Mann hybrid iterative scheme for a general class of contractive-like operators. However, comparing with the stability analysis in [27], we use a different method to analyze the stability and also extend the application of stability for iterations.
(iii) According to inequality (2.12), one can obtain
$$\begin{aligned} &\bigl\Vert x_{n+1}-x^{*}\bigr\Vert \\ &\quad \le\vartheta_{n}\bigl\Vert x_{n}-x^{*}\bigr\Vert +\bigl(\Vert d_{n}\Vert +\Vert e_{n}\Vert + \Vert h_{n}\Vert \bigr) \\ &\quad \le\vartheta_{n}\vartheta_{n-1}\bigl\Vert x_{n-1}-x^{*}\bigr\Vert +\vartheta_{n}\bigl(\Vert d_{n-1}\Vert +\Vert e_{n-1}\Vert +\Vert h_{n-1} \Vert \bigr) \\ &\quad \quad{}+\bigl(\Vert d_{n}\Vert +\Vert e_{n}\Vert + \Vert h_{n}\Vert \bigr) \\ &\quad \le\cdots \\ &\quad \le\prod_{i=1}^{n} \vartheta_{i} \bigl\Vert x_{1}-x^{*}\bigr\Vert +\sum _{k=1}^{n-1}\prod_{i=k+1}^{n} \vartheta_{i}\bigl(\Vert d_{k}\Vert +\Vert e_{k}\Vert +\Vert h_{k}\Vert \bigr) \\ &\quad \quad{}+\bigl(\Vert d_{n}\Vert +\Vert e_{n}\Vert + \Vert h_{n}\Vert \bigr), \end{aligned}$$
(2.17)
where \(\prod_{i=1}^{n} \vartheta_{i}=\vartheta_{1}\cdot\vartheta _{2}\cdot \cdots \cdot\vartheta_{n}\) and \(\vartheta_{i}\) is the same as in (2.12) for all \(i=1,2,\ldots,n\). As a matter of fact, \(\sum_{k=1}^{n-1}\prod_{i=k+1}^{n} \vartheta_{i}(\Vert d_{k}\Vert +\Vert e_{k}\Vert +\Vert h_{k}\Vert )+ (\Vert d_{n}\Vert +\Vert e_{n}\Vert +\Vert h_{n}\Vert ) =o(\Vert d_{n}\Vert +\Vert e_{n}\Vert +\Vert h_{n}\Vert )\). Hence, these errors in (2.4) can help to adjust the iteration results to improve the algorithms by using this infinitesimal of a higher order sequence.
From Theorem 2.1 and Remark 2.1, we have the following result.
Theorem 2.2
Let
\(C\subset X\)
be a nonempty closed convex bounded subset of a normed space
X, and let
\(T: C\to C\)
be a contraction operator with constant
\(\theta\in[0, 1)\). If
\(\{\alpha _{n}\}\subset[0, 1]\)
and
\(\sum_{n=0}^{\infty}\alpha_{n}=\infty\)
and
\(\{h_{n}\}\), \(\{d_{n}\}\), \(\{e_{n}\}\)
are three sequences in
X
satisfying the conditions
P, then
-
(i)
the iterative sequence
\(\{x_{n}\}\)
generated by (2.7) (that is, (PMM)) converges to
\(p\in F(T):=\{x\in C: Tx=x\}\)
with convergence rate
\(\vartheta=1-\hat{\alpha}(1-\theta)<1\), where
\(\hat{\alpha }=\limsup_{n\to\infty}\alpha_{n}\in(0, 1]\);
-
(ii)
if, in addition, for any sequence
\(\{z_{n}\}\subset X\), there exists
\(\alpha>0\)
such that
\(\alpha_{n}\ge\alpha\)
for all
\(n\geq 0\), then
$$\begin{aligned} \lim_{n\to\infty}z_{n}=p \quad\Longleftrightarrow\quad\lim _{n\to \infty}\epsilon_{n}=0, \end{aligned}$$
(2.18)
where
\(\epsilon_{n}\)
is defined by
$$\begin{aligned} \textstyle\begin{cases} \epsilon_{n}=\Vert z_{n+1}- ( Ts_{n}+h_{n} ) \Vert , \\ s_{n}=(1-\alpha_{n})z_{n}+\alpha_{n} Tz_{n}+\alpha_{n}d_{n}+e_{n}. \end{cases}\displaystyle \end{aligned}$$
(2.19)