- Research
- Open Access
Modified semi-implicit midpoint rule for nonexpansive mappings
- Yonghong Yao^{1},
- Naseer Shahzad^{2}Email author and
- Yeong-Cheng Liou^{3, 4}
https://doi.org/10.1186/s13663-015-0414-2
© Yao et al. 2015
- Received: 22 May 2015
- Accepted: 28 August 2015
- Published: 17 September 2015
Abstract
The purpose of the paper is to construct iterative methods for finding the fixed points of nonexpansive mappings. We present a modified semi-implicit midpoint rule with the viscosity technique. We prove that the suggested method converges strongly to a special fixed point of nonexpansive mappings under some different control conditions. Some applications are also included.
Keywords
- modified semi-implicit midpoint rule
- nonexpansive mapping
- strong convergence
MSC
- 47J25
- 47N20
- 34G20
- 65J15
1 Introduction
The implicit midpoint rule is one of the powerful numerical methods for solving ordinary differential equations and differential algebraic equations. For related works, please refer to [1–9].
If we write the function f in the form \(f(t)=g(t)-t\), then differential equation (1.1) becomes \(x'=g(t)-t\). Then the equilibrium problem associated with the differential equation is the fixed point problem \(t=g(t)\).
Xu et al. [11] showed the following strong convergence theorem.
Theorem 1.1
- (C1):
\(\lim_{n\to\infty}\alpha_{n}=0\);
- (C2):
\(\sum_{n=0}^{\infty}\alpha_{n}=\infty\);
- (C3):
either \(\sum_{n=0}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty \) or \(\lim_{n\to\infty}\frac{\alpha_{n+1}}{\alpha_{n}}=1\).
Remark 1.2
Remark 1.3
Note that the proof of Theorem 1.1 in [11] is technical. However, Step 6 in the proof of Theorem 1.1 is also complicated.
Fixed point method has attracted so much attention. Now we briefly recall some related historic approaches.
Browder [12] introduced an implicit scheme as follows. Fix \(u\in C\) and, for each \(t\in(0,1)\), let \(x_{t}\) be the unique fixed point in C of the contraction \(T_{t}\) which maps C into C: \(T_{t}x=tu+(1-t)Tx\), \(x\in C\). Browder proved that \(s\mbox{-} \lim_{t\downarrow0}x_{t}=P_{\operatorname{Fix}(T)}u\). That is, the strong limit of \(\{x_{t}\}\) as \(t\to0^{+}\) is the fixed point of T which is nearest from \(\operatorname{Fix}(T)\) to u.
Refinements in Hilbert spaces and extensions to Banach spaces were obtained by Xu [16]. This technique uses (strict) contractions to regularize a nonexpansive mapping for the purpose of selecting a particular fixed point of the nonexpansive mapping, for instance, the fixed point of minimal norm, or of a solution to another variational inequality.
Motivated and inspired by the above work, in this paper we aim to construct a unified iterative algorithm for finding the fixed points of nonexpansive mappings. We present a modified semi-implicit midpoint rule with the viscosity technique for nonexpansive mappings. We prove that the suggested algorithm converges strongly to a special fixed point of nonexpansive mappings under some different conditions. Some applications are also included.
2 Tools
2.1 Some notations
Let H be a real Hilbert space with inner product \(\langle\cdot,\cdot\rangle\) and norm \(\|\cdot\|\), respectively. Let C be a nonempty closed convex subset of H.
2.2 Existing algorithm and convergence result
Let C be a nonempty closed convex subset of a real Hilbert space H. Let \(Q:C\to C\) be an α-contraction and \(T:C\to C\) be a nonexpansive mapping with \(\operatorname{Fix}(T)\ne\emptyset\).
Algorithm 2.1
3 Some lemmas
The following demiclosedness principles for nonexpansive mappings are well known.
Lemma 3.1
([17])
Let C be a nonempty closed convex subset of a Hilbert space H, and let \(T:C\to C\) be a nonexpansive mapping with \(\operatorname{Fix}(T)\ne\emptyset\). Assume that \(\{y_{n}\}\) is a sequence in C such that \(y_{n}\rightharpoonup x^{\dagger}\) and \((I-T)y_{n}\to0\). Then \(x^{\dagger}\in \operatorname{Fix}(T)\).
Lemma 3.2
([18])
Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be bounded sequences in a Banach space E and \(\{\beta_{n}\}\) be a sequence in \([0,1]\) with \(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq \limsup_{n\rightarrow \infty}\beta_{n}<1\). Suppose that \(x_{n+1}=(1-\beta_{n})x_{n}+\beta_{n}z_{n}\) for all \(n\geq0\) and \(\limsup_{n\rightarrow \infty}(\|z_{n+1}-z_{n}\|-\|x_{n+1}-x_{n}\|)\leq0\). Then \(\lim_{n\rightarrow\infty}\|z_{n}-x_{n}\|=0\).
Lemma 3.3
([19])
- (i)
\(\{\alpha_{n}\}_{n\in\mathbb{N}}\subset[0,1]\) and \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
- (ii)
\(\limsup_{n\to\infty}\sigma_{n}\le0\);
- (iii)
\(\sum_{n=1}^{\infty}\delta_{n}<\infty\).
4 Main results
In this section, we firstly present the following unified algorithm.
Let C be a nonempty closed convex subset of a real Hilbert space H. Let \(T:C\to C\) be a nonexpansive mapping with \(\operatorname{Fix}(T)\ne\emptyset\). Let \(Q:C\to C\) be an α-contraction.
Algorithm 4.1
Remark 4.2
Now, we show the boundedness of the sequence \(\{x_{n}\}\).
Conclusion 4.3
The sequence \(\{x_{n}\}\) generated by (4.1) is bounded.
Proof
Now we give the following result.
Theorem 4.4
Proof
Remark 4.5
The proof of Theorem 4.4 is very simple.
Remark 4.6
In (4.1), if we choose \(\beta_{n}\equiv0\) for all n, then (4.1) is reduced to (1.4). Thus, our Algorithm 4.1 includes Algorithm 1.4 as a special case, and Theorem 1.1 is also a special case of our Theorem 4.4.
Next, we can define the following algorithm.
Algorithm 4.7
Proposition 4.8
The sequence \(\{y_{n}\}\) generated by (4.4) converges strongly to \(q=P_{\operatorname{Fix}(T)}Q(q)\) provided \(\lim_{n\to\infty}\alpha_{n}=0\).
In fact, we can rewrite (4.4) as \(y_{n}=\frac{\alpha_{n}}{1-\beta _{n}}Q(y_{n})+(1-\frac{\alpha_{n}}{1-\beta_{n}})Ty_{n}\) for all n. Thus, Proposition 4.8 can be deduced from Theorem 2.2.
Next we use Proposition 4.8 to show the convergence analysis of Algorithm 4.1 under other control conditions.
Theorem 4.9
Remark 4.10
Note that conditions (C1), (C2), and (C4) were presented by Lions in [14]. At the same time, (C7) is different from (C6). In fact, we can choose \(\beta_{n}=\beta\in(0,1)\) in (C7).
Next, we will give another control condition instead of (C4) and (C7).
Theorem 4.11
Proof
Remark 4.12
Note that condition (C8) has been used in a large number of references. Theorems 4.4, 4.9, and 4.11 demonstrate the strong convergence of Algorithm 4.1 under different control conditions on parameters \(\{\alpha_{n}\}\) and \(\{\beta _{n}\}\). Our algorithm and results provide a unified framework for the class problem of algorithmic approach to the fixed point of nonlinear operators.
5 Applications
5.1 Application to variational inequalities
Theorem 5.1
5.2 Application to hierarchical minimization
Theorem 5.2
5.3 Periodic solution of a nonlinear evolution equation
We assume that \(A(t)\) and \(f(t,u)\) are periodic in t with a common period \(\xi>0\).
An interesting result on the existence of periodic solutions of equation (5.8) is due to Browder [20].
Theorem 5.3
- (i)For each t and each pair \(u,v\in H\),$$\operatorname{Re}\bigl\langle f(t,u)-f(t,v), u-v\bigr\rangle \le0. $$
- (ii)
For each t and each \(u\in D(A(t))\), \(\operatorname{Re}\langle A(t)u,u\rangle\ge0\).
- (iii)
There exists a mild solution u of equation (5.1) on \(\mathbb{R}^{+}\) for each initial value \(v\in H\).
- (iv)There exists some \(R>0\) such thatfor \(\|u\|=R\) and all \(t\in[0,\xi]\).$$\operatorname{Re}\bigl\langle f(t,u),u\bigr\rangle < 0 $$
Consequently, T has a fixed point which we denote by v, and the corresponding solution u of (5.8) with the initial condition \(u(0)=v\) is a desired periodic solution of (5.8) with period ξ. In other words, to find a periodic solution u of (5.8) is equivalent to finding a fixed point of T.
5.4 Fredholm integral equation
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for technical and financial support.
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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