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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
- viscosity
- implicit midpoint rule
- nonexpansive mapping
- projection
- variational inequality
- hierarchical minimization
- nonlinear evolution equation
MSC
- 47J25
- 47N20
- 34G20
- 65J15
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
References
- Moudafi, A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46-55 (2000) View ArticleMATHMathSciNetGoogle Scholar
- Attouch, H: Viscosity approximation methods for minimization problems. SIAM J. Optim. 6(3), 769-806 (1996) View ArticleMATHMathSciNetGoogle Scholar
- Xu, HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279-291 (2004) View ArticleMATHMathSciNetGoogle Scholar
- Auzinger, W, Frank, R: Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case. Numer. Math. 56, 469-499 (1989) View ArticleMATHMathSciNetGoogle Scholar
- Bader, G, Deuflhard, P: A semi-implicit mid-point rule for stiff systems of ordinary differential equations. Numer. Math. 41, 373-398 (1983) View ArticleMATHMathSciNetGoogle Scholar
- Deuflhard, P: Recent progress in extrapolation methods for ordinary differential equations. SIAM Rev. 27(4), 505-535 (1985) View ArticleMATHMathSciNetGoogle Scholar
- Schneider, C: Analysis of the linearly implicit mid-point rule for differential-algebra equations. Electron. Trans. Numer. Anal. 1, 1-10 (1993) MATHMathSciNetGoogle Scholar
- Somalia, S: Implicit midpoint rule to the nonlinear degenerate boundary value problems. Int. J. Comput. Math. 79(3), 327-332 (2002) View ArticleMathSciNetGoogle Scholar
- Van Veldhuxzen, M: Asymptotic expansions of the global error for the implicit midpoint rule (stiff case). Computing 33, 185-192 (1984) View ArticleMathSciNetGoogle Scholar
- Alghamdi, MA, Alghamdi, MA, Shahzad, N, Xu, HK: The implicit midpoint rule for nonexpansive mappings. Fixed Point Theory Appl. 2014, 96 (2014) View ArticleMathSciNetGoogle Scholar
- Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990) View ArticleMATHGoogle Scholar
- Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240-256 (2002) View ArticleMATHGoogle Scholar
- Xu, HK: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 150, 360-378 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Cabot, A: Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization. SIAM J. Optim. 15, 555-572 (2005) View ArticleMATHMathSciNetGoogle Scholar
- Browder, FE: Existence of periodic solutions for nonlinear equations of evolution. Proc. Natl. Acad. Sci. USA 53, 1100-1103 (1965) View ArticleMATHMathSciNetGoogle Scholar