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New approximation methods for solving elliptic boundary value problems via PicardMann iterative processes with mixed errors
 Tengfei Li^{1, 2} and
 Hengyou Lan^{1}Email authorView ORCID ID profile
 Received: 8 August 2017
 Accepted: 29 November 2017
 Published: 13 December 2017
Abstract
In this paper, we introduce and study a class of new PicardMann iterative methods with mixed errors for common fixed points of two different nonexpansive and contraction operators. We also give convergence and stability analysis of the new PicardMann iterative approximation and propose numerical examples to show that the new PicardMann iteration converges more effectively than the Picard iterative process, Mann iterative process, PicardMann iterative process due to Khan and other related iterative processes. Furthermore, as an application, we explore iterative approximation of solutions for an elliptic boundary value problem in Hilbert spaces by using the new PicardMann iterative methods with mixed errors for contraction operators.
Keywords
 new PicardMann approximation
 common fixed point
 nonexpansion and contraction
 elliptic boundary value problem
 convergence and stability
MSC
 47H10
 54H25
 47H09
1 Introduction
In particular, many problems in physics and other applications cannot be formulated as equations but have some more complicated structure, and usually the socalled complementarity problem, which is equivalent to a variational inequality. Further, the applicability of variational inequality theory, which was initially developed to cope with equilibrium problems (e.g., the Signorini problem, which was first posed by Antonio Signorini in 1959), has been extended to involve problems in economics, finance, electrodynamics, mechanics, engineering science, optimization and game theory. Hence, the variational method is very important in optimal control theory, and such generalization is often needed in optimalcontrol theory of elliptic problems. In fact, optimal control problems in control theory are searching for a kind of control mode which can transform the initial state of the control object to the terminal state and make sure that the objective function can reach the maximum or minimum. For more details on variational inequalities in the context of their optimal control, one can refer to [2–5] and the references therein, and the following examples.
Example 1.1
([8])
Based on the above analysis of dualization and optimality condition for the optimal control problems (1.2) and (1.3), Liu and Sun [8] introduced and studied an iterative nonoverlapping domain decomposition method for (1.3)(1.6) and proved convergence of the sequence generated by the iterative method. Furthermore, by using an iterative algorithm due to the penalized gradient projection method, adaptive finite element method, edge stabilization Galerkin method, variational iteration method, etc., such kind of problems as (1.2) or (1.3) were considered by many authors and researchers. See, for example, [9–16] and the references therein. Especially noteworthy, Zhou and Li [17] pointed out ‘though much achievement has been achieved, application of the variational iteration method to Cauchy problems has not yet been dealt with’.
On the other hand, in order to compare to Picard, Mann and Ishikawa iterations for approximating fixed points and to solve equation systems, Khan [18] introduced and studied a PicardMann hybrid iterative process and showed that the PicardMann hybrid iterative process converges faster than all of Picard, Mann and Ishikawa iterative processes for contractions. Following on the works of Khan [18], by using an uptodate method for approximating common fixed points of countable families of nonlinear operators, Deng [19] introduced a modified PicardMann hybrid iterative algorithm for a sequence of nonexpansive mappings and established strong convergence and weak convergence of the iterative sequence generated by the modified hybrid iterative algorithm in a convex Banach space. Okeke and Abbas [20] introduced and studied PicardKrasnoselskii hybrid iterations, which converge faster than Picard, and gave an application to delay differential equations Mann, Krasnoselskii and Ishikawa iterative processes for contractive nonlinear operators. However, one can know that the PicardKrasnoselskii hybrid iteration is a special case of the PicardMann hybrid iterative process due to Khan [18], it is because \(\alpha_{n}\in(0,1)\) of (1.4) in [18] includes \(\lambda\in(0,1)\) in (1.7) of [20] (see [20, Example 2.2, p. 25]). Jiang et al. [21] proved convergence of Mann iterative sequences for approximating solutions of a higher order nonlinear neutral delay differential equation and proposed advantages of the presented results through three extraordinary examples. However, how to establish the error estimates between the approximate solutions and the exact solutions for partial differential equations is not reported in the literature.
Moreover, Roussel [22] pointed out that equilibria are not always stable. Since stable and unstable equilibria play quite different roles in the dynamics of a system, it is useful to be able to classify equilibrium points based on their stability. Thus, there are many scholars and researchers who have discussed stability of the iterative sequence generated by the algorithm for solving the investigated problems. See, for example, [23–27] and the references therein. Especially, stimulated by the work of Bosede and Rhoades [28], Akewe and Okeke [27] obtained stability results for the PicardMann hybrid iterative scheme due to Khan [18] for a general class of contractivelike operators introduced by Bosede and Rhoades [28]. However, how does one obtain stability analysis when the PicardMann hybrid iterative scheme due to Khan [18] is generalized for two different nonexpansive and contraction operators and one involves errors or mixed errors? This is a significant and challenging research work.
Motivated and inspired by the above works, we aim in this paper to introduce and study a class of new PicardMann iterative methods with mixed errors for common fixed points of two different nonexpansive and contraction operators. Then convergence and stability analysis of the new PicardMann iterative approximation are given. Finally, two numerical examples to verify effectiveness of the new PicardMann iteration are presented, and a new iterative approximation of solutions for an elliptic boundary value problem in Hilbert spaces is investigated by using the new PicardMann iterative methods with mixed errors for nonexpansive operators, which are different from the method proposed in [1].
2 New PicardMann approximation methods
In this section, we shall introduce and study a class of new PicardMann iterative methods with mixed errors for common fixed points of two different nonexpansive and contraction operators and prove convergence and stability of the new PicardMann iterative approximation.
We need the following definitions and lemmas for our main results.
Definition 2.1
 (i)nonexpansive if$$\begin{aligned} \Vert TuTv\Vert \le \Vert uv\Vert , \quad \forall u, v\in K; \end{aligned}$$(2.1)
 (ii)contraction if there exists a constant \(k\in[0, 1)\) such that$$\begin{aligned} \Vert TuTv\Vert \le k\Vert uv\Vert , \quad \forall u, v\in K. \end{aligned}$$(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 Sstable 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])
Now, we establish a class of new PicardMann 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.
 (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 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

Special Case I If \(h_{n}=d_{n}=e_{n}=0\), the iterative process (2.4) becomes the following PicardMann 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 byWe note that the iterative processes (PMD) and the PicardMann iteration with mixed errors (2.7) (in short, (PMM)) are new and not studied in the literature.$$\begin{aligned} \textstyle\begin{cases} x_{n+1}=Ty_{n}+h_{n}, \\ y_{n}=(1\alpha_{n})x_{n}+\alpha_{n}Tx_{n}+\alpha_{n}d_{n}+e_{n}. \end{cases}\displaystyle \end{aligned}$$(2.7)

Special Case III If \(T_{1}=T_{2}=T\), then (2.6) reduces towhich was the PicardMann 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 PicardKrasnoselskii 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.$$\begin{aligned} \textstyle\begin{cases} x_{n+1}=Ty_{n}, \\ y_{n}=(1\alpha_{n})x_{n}+\alpha_{n}Tx_{n}, \end{cases}\displaystyle \end{aligned}$$(2.8)

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 aswhich is the Mann iterative process (in short, (MI)) for \(\alpha _{n}\in[0, 1]\).$$\begin{aligned} x_{n+1}=(1\alpha_{n})x_{n}+\alpha_{n}T_{2}x_{n}, \end{aligned}$$(2.9)
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 PicardMann iterative processes with mixed errors generated by Algorithm 2.1.
Theorem 2.1
 (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 ratewhere \(\hat{\alpha}=\limsup_{n\to\infty}\alpha_{n}\in(0, 1]\);$$\begin{aligned} \vartheta=1\hat{\alpha}(1\theta)< 1, \end{aligned}$$(2.10)
 (ii)if, in addition, for any sequence \(\{z_{n}\}\subset X\), there exists \(\alpha>0\) such that \(\alpha_{n}\ge\alpha\) for all \(n\geq 0\), thenwhere \(\varepsilon_{n}\) is defined by (2.5).$$\begin{aligned} \lim_{n\to\infty}w_{n}=x^{*} \quad\textit{if and only if}\quad\lim_{n\to\infty}\varepsilon_{n}=0, \end{aligned}$$(2.11)
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, PicardMann 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 PicardMann hybrid iterative scheme for a general class of contractivelike 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.
From Theorem 2.1 and Remark 2.1, we have the following result.
Theorem 2.2
 (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 \(n\geq 0\), thenwhere \(\epsilon_{n}\) is defined by$$\begin{aligned} \lim_{n\to\infty}z_{n}=p \quad\Longleftrightarrow\quad\lim _{n\to \infty}\epsilon_{n}=0, \end{aligned}$$(2.18)$$\begin{aligned} \textstyle\begin{cases} \epsilon_{n}=\Vert z_{n+1} ( Ts_{n}+h_{n} ) \Vert , \\ s_{n}=(1\alpha_{n})z_{n}+\alpha_{n} Tz_{n}+\alpha_{n}d_{n}+e_{n}. \end{cases}\displaystyle \end{aligned}$$(2.19)
3 Numerical simulations and an application
In order to verify our main results presented in the above section, in this section, we give some numerical simulations and consider approximation of the elliptic boundary value problem (1.1) by using the new PicardMann iterative methods with mixed errors for contractive operators.
3.1 Numerical examples
We first give the following examples and their numerical simulations to show verification of Theorem 2.1 and Remark 2.3(iii) and to display effectiveness of the new PicardMann iterative methods with mixed errors.
Example 3.1
A comparison of the iterative processes (PMMD), (PMD) and (MI)
Iteration number  (PMMD)  (PMD)  (MI) 

0  25.0000  25.0000  25.0000 
5  7.9781  7.8794  21.0937 
10  7.9944  7.9102  20.0644 
15  7.9975  7.9249  19.4604 
20  7.9987  7.9340  19.0346 
25  7.9992  7.9403  18.7069 
30  7.9995  7.9451  18.4413 
35  7.9996  7.9489  18.2183 
40  7.9997  7.9519  18.0264 
45  7.9998  7.9545  17.8583 
50  7.9999  7.9567  17.7088 
55  7.9999  7.9585  17.5743 
60  7.9999  7.9602  17.4521 
65  7.9999  7.9617  17.3403 
70  8.0000  7.9630  17.2373 
75  8.0000  7.9642  17.1418 
80  8.0000  7.9652  17.0529 
85  8.0000  7.9662  16.9697 
90  8.0000  7.9671  16.8915 
95  8.0000  7.9679  16.8179 
100  8.0000  7.9687  16.7483 
105  8.0000  7.9694  16.6823 
Remark 3.1
If these mixed errors can be used properly, the property of (2.4) will be better than the other algorithms. From Figure 1 and Table 1, it is easy to see that the iterative process (PMMD) is effective and the sequence \(\{ x_{n}\}\) generated by (PMMD) converges much faster.
Next, we verify Theorem 2.2 by the following numerical example.
Example 3.2
A comparison of the iterative processes (PMM) and (PM)
Iteration number  (PMM)  (PM) 

0  25.0000  25.0000 
1  6.6065  19.8945 
2  5.3218  15.8549 
3  5.0625  12.4783 
4  5.0131  9.6469 
5  5.0031  7.4247 
6  5.0008  5.9644 
7  5.0002  5.2762 
8  5.0001  5.0623 
9  5.0000  5.0128 
10  5.0000  5.0026 
11  5.0000  5.0005 
12  5.0000  5.0001 
13  5.0000  5.0000 
3.2 An application to the elliptic boundary value problem
From Theorem 2.2, we have the following existence results of solutions for problem (3.4).
Theorem 3.1
 (i)
the iterative sequence \(\{u_{n}\}\) generated by (2.7) converges to a weak solution \(u^{*}\in F(T)\) of problem (3.4) with convergence rate \(\vartheta=1\hat{\alpha}(1\theta)<1\), where \(\hat {\alpha}=\limsup_{n\to\infty}\alpha_{n}\in(0, 1]\) and \(\theta =\sup_{u\in C}\Vert (I^{\prime}\phi^{\prime\prime})u\Vert \);
 (ii)if, in addition, there exists \(\alpha>0\) such that \(\alpha _{n}\ge\alpha\) for all \(n\geq0\), thenwhere \(\epsilon_{n}\) is defined by (2.19) and \(\{z_{n}\}\) is any sequence.$$\begin{aligned} \lim_{n\to\infty}z_{n}=u^{*} \quad \Longleftrightarrow\quad\lim_{n\to\infty}\epsilon_{n}=0, \end{aligned}$$(3.7)
Proof
From the proof of [1, Theorem 6], it follows that \(C\subset H_{0}^{1}(\Omega)\) is a closed convex and bounded subset, and \(\Vert (I^{\prime}\phi^{\prime\prime})u\Vert <1\) for some \(u\in C\). By the proof of Theorem 4 in [1], we know that T is a contraction operator. Since a contraction operator has fixed points, the results hold from Theorem 2.2. This completes the proof. □
4 Concluding remarks
However, can our results be obtained when T is only nonexpansive in Theorem 2.2 or \(T_{2}\) is also nonexpansive in Theorem 2.1? These are still open questions that are worth further studying.
Declarations
Acknowledgements
We would like to thank the editors and referees for their valuable comments and suggestions to improve our paper.
Availability of data and materials
Not applicable.
Authors’ information
Further, Mr. TFL is studying for an MA degree. His research interests focus on the theory and algorithm of nonlinear system optimization and control. HYL is a professor in Sichuan University of Science & Engineering. He received his doctoral degree from Sichuan University in 2013. His research interests focus on the structure theory and algorithm of operational research and optimization, nonlinear analysis and applications.
Funding
This work was supported by the Scientific Research Project of Sichuan University of Science & Engineering (2017RCL54), the Innovation Fund of Postgraduate, Sichuan University of Science & Engineering (y2016041).
Authors’ contributions
TFL carried out the proof of the theorems and gave some numerical simulations to show the main results. HYL conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
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Authors’ Affiliations
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