The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces
- Hong-Kun Xu^{1, 2}Email author,
- Maryam A Alghamdi^{3} and
- Naseer Shahzad^{2}
https://doi.org/10.1186/s13663-015-0282-9
© Xu et al.; licensee Springer. 2015
Received: 12 December 2014
Accepted: 9 February 2015
Published: 25 March 2015
Abstract
The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces is established. The strong convergence of this technique is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit solves an additional variational inequality. Applications to variational inequalities, hierarchical minimization problems, and nonlinear evolution equations are included.
Keywords
MSC
1 Introduction
The viscosity technique for nonexpansive mappings in Hilbert spaces was introduced by Moudafi [1], following the ideas of Attouch [2]. Refinements in Hilbert spaces and extensions to Banach spaces were obtained by Xu [3]. 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.
The structure of the paper is set as follows. In Section 2, we introduce the notion of nearest point projections, the demiclosedness principle of nonexpansive mappings, and a convergence lemma. The viscosity implicit midpoint rule for nonexpansive mappings is introduced in Section 3. The main result, that is, the strong convergence of this method, is proved also in this section. Applications to variational inequalities, hierarchical minimization problems and nonlinear evolution equations are presented in the final section, Section 4.
2 Preliminaries
The demiclosedness principle of nonexpansive mappings is quite helpful in verifying the weak convergence of an algorithm to a fixed point of a nonexpansive mapping.
Lemma 2.1
[11] (The demiclosedness principle)
Let H be a Hilbert space, C a closed convex subset of H, and \(T: C\to C\) a nonexpansive mapping with \(\operatorname{Fix}(T)\neq\emptyset\). If \(\{x_{n}\}\) is a sequence in C such that (i) \(\{x_{n}\}\) weakly converges to x and (ii) \(\{(I-T)x_{n}\}\) converges strongly to 0, then \(x=Tx\).
In proving the strong convergence of a sequence \(\{x_{n}\}\) to a point \(\bar{x}\), we always consider the real sequence \(\{\|x_{n}-\bar{x}\|^{2}\}\) and then apply the following convergence lemma.
Lemma 2.2
[12]
- (i)
\(\sum_{n=1}^{\infty}\gamma_{n}=\infty\), and
- (ii)
either \(\limsup_{n\to\infty}\delta_{n}/\gamma_{n}\le0\) or \(\sum_{n=1}^{\infty}|\delta_{n}|<\infty\).
3 The viscosity technique for implicit midpoint rule
- (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 \(\lim_{n\to\infty}\frac{\alpha_{n+1}}{\alpha_{n}}=1\).
The main result of this paper is the following result, the proof of which seems nontrivial.
Theorem 3.1
Proof
We divide the proof into several steps.
4 Applications
4.1 Application to variational inequalities
- (A1)A is L-Lipschitzian for some \(L>0\), that is,$$\|Ax-Ay\|\le L\|x-y\|,\quad x,y\in H. $$
- (A2)A is μ-inverse strongly monotone (μ-ism) for some \(\mu>0\), namely,$$\langle Ax-Ay,x-y\rangle\ge\mu\|Ax-Ay\|^{2},\quad x,y \in H. $$
Under the conditions (A1) and (A2), it is well known [13] that the operator \(T=P_{C}(I-\lambda A)\) is nonexpansive provided \(0<\lambda<2\mu\). It turns out that for this range of values of λ, fixed point algorithms can be applied to solve the VI (4.1). Applying Theorem 3.1 we get the result below.
Theorem 4.1
4.2 Application to hierarchical minimization
We next consider a hierarchical minimization problem (see [14] and references therein).
Theorem 4.2
4.3 Application to nonlinear evolution equation
- (B1)
\(A(t)\) and \(f(t,u)\) are periodic in t of period \(\xi> 0\).
- (B2)For each t and each pair \(u, v \in H\),$$\bigl\langle f(t,u) - f(t,v), u - v\bigr\rangle \le0. $$
- (B3)
For each t and each \(u\in D(A(t))\), \(\langle A(t)u,u\rangle\ge0\).
- (B4)There exists a mild solution u of (4.7) on \(\mathbb{R}^{+}\) for each initial value \(v \in H\). Recall that u is a mild solution of (4.7) with initial value \(u(0)=v\) if, for each \(t>0\),where \(\{U(t,s)\}_{t\ge s\ge0}\) is the evolution system for the homogeneous linear system$$u(t)=U(t,0)v+\int_{0}^{t} U(t,s)f\bigl(s,u(s) \bigr)\,ds, $$$$ \frac{du}{dt}+A(t)u=0\quad (t>s). $$(4.8)
- (B5)There exists some \(R > 0\) such thatfor \(\|u\|= R\) and all \(t \in[0,\xi]\).$$\bigl\langle f(t,u),u\bigr\rangle < 0 $$
Declarations
Acknowledgements
The authors are grateful to the anonymous referees for their helpful comments and suggestions, which improved the presentation of this manuscript. This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (49-130-35-HiCi). The authors, therefore, acknowledge technical and financial support of KAU.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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