Some results on approximate solutions of variational inequality problems for inverse strongly monotone operators
- Shigeru Iemoto^{1}Email author
https://doi.org/10.1186/s13663-015-0337-y
© Iemoto 2015
Received: 14 February 2015
Accepted: 22 May 2015
Published: 13 June 2015
Abstract
We present results on approximate solutions to variational inequality problems for an injective inverse strongly monotone operator. Our results are based on Edelstein’s theorem (Edelstein in J. Lond. Math. Soc. 37:74-79, 1962).
Keywords
variational inequality problem fixed point projected gradient method hybrid steepest descent method inverse strongly monotone operator cocoercive operator strongly monotone operatorMSC
49J40 47H10 47H09 49K271 Introduction
Theorem 1.1
([2])
As (PGM) requires repetitive use of \(P_{C}\), it works only when the explicit form of \(P_{C}\) is known (e.g., C is a closed ball or a closed cone). The following method, called the hybrid steepest descent method (HSDM) [4], enables us to consider the case in which C has a more complicated form.
Theorem 1.2
([4])
Theorem 1.3
(HSDM [4])
In 2003, Xu and Kim [5] replaced condition (iii) in Theorem 1.3 with \(\lim_{n\to\infty}(c_{n}/ c_{n+1})=1\), which includes the case of \(c_{n}=1/n\). The proofs of Theorems 1.2, 1.3 are based on the Banach contraction mapping principle [6]. In the Hilbert space setting, the Banach contraction mapping principle is as follows.
Theorem 1.4
([6])
Theorem 1.5
([7])
2 Preliminaries
- (i)firmly nonexpansive if for any \(x, y\in C\),$$\Vert Tx-Ty\Vert ^{2}\le \langle x-y, Tx-Ty \rangle; $$
- (ii)nonexpansive if for any \(x, y\in C\),$$\Vert Tx-Ty\Vert \le \Vert x-y\Vert ; $$
- (iii)L-Lipschitz continuous if there exists \(L\in(1, \infty )\) such that for any \(x, y\in C\),$$\Vert Tx-Ty\Vert \le L\Vert x-y\Vert ; $$
- (iv)strictly contractive if there exists \(r\in[0, 1)\) such that for any \(x, y\in C\),$$\Vert Tx-Ty\Vert \le r\Vert x-y\Vert ; $$
- (v)contractive if for any \(x, y\in C\) with \(x\neq y\),$$\Vert Tx-Ty\Vert < \Vert x-y\Vert . $$
Theorem 2.1
Let C be a bounded closed convex subset of a Hilbert space H and S be a nonexpansive self-mapping on C. Then \(\operatorname{Fix}(S)\neq\emptyset\).
Let C be a closed and convex subset of H. Then, for every point \(x\in H\), there exists a unique nearest point in C, denoted by \(P_{C}(x)\), such that \(\Vert x-P_{C}(x)\Vert \le \Vert x-y\Vert \) for all \(y\in C\). \(P_{C}\) is called the metric projection of H onto C. We know that \(P_{C}\) is a firmly nonexpansive mapping of H onto C.
- (i)monotone if for any \(x,y\in C\),$$\langle x-y, Ax-Ay \rangle\geq0; $$
- (ii)β-strongly monotone if there exists \(\beta\in(0, 1)\) such that for any \(x, y\in C\),$$\beta \Vert x-y\Vert ^{2}\le \langle x-y, Ax-Ay \rangle; $$
- (iii)β-inverse strongly monotone if there exists \(\beta \in(0, 1)\) such that for any \(x, y\in C\),$$\beta \Vert Ax-Ay\Vert ^{2}\le \langle x-y, Ax-Ay \rangle. $$
Many methods for solving the variational inequality problem are based on the following (see [8, 10] for instance).
Lemma 2.1
Let C be a closed and convex subset of a Hilbert space H and A be a mapping of C into H and \(a\in(0,\infty)\). Then \(\operatorname {Fix}(P_{C}(I-aA))=VI(C,A)\).
The following lemmas are presented without proof. The first lemma is known as Browder’s demiclosedness principle [11, 15] (see also [8, 10]).
Lemma 2.2
Let C be a closed and convex subset of a Hilbert space H and S be a nonexpansive mapping of C into itself. Let \(\{x_{n}\}\) be a sequence in C which converges weakly to \(u\in C\) and satisfies \(\lim_{n}\|Sx_{n}-x_{n}\|=0\). Then \(u\in\operatorname{Fix}(S)\).
The next lemma is also well known.
Lemma 2.3
([10])
Let \(\{x_{n}\}\) be a sequence in a Hilbert space H. Assume that there is a point \(z\in H\) such that any subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) has a subsequence which converges strongly (resp. weakly) to z. Then \(\{x_{n}\}\) itself converges strongly (resp. weakly) to z.
3 Injective β-inverse strongly monotone operators
As preparation for the next section, here we study the properties of injective β-inverse strongly monotone operators.
Lemma 3.1
Proof
Corollary 3.1
Proof
- (a)
If \(Sx=Sy\), it follows that \(\Vert U_{a}x-U_{a}y\Vert =0\). So, \(\Vert U_{a}x-U_{a}y\Vert <\Vert x-y\Vert \) for \(x, y\in H\) with \(x\neq y\).
- (b)If \(Sx\neq Sy\), \(ASx\neq ASy\). So, it follows from (2) and \(a\in(0, 2\beta)\) that$$\Vert U_{a}x-U_{a}y\Vert ^{2}\le \Vert x-y \Vert ^{2}-a(2\beta -a)\Vert ASx-ASy\Vert < \Vert x-y\Vert ^{2}. $$
Lemma 3.2
Let C be a closed convex subset of a Hilbert space H. Furthermore, let A be an injective β-inverse strongly monotone operator sending C into H and \(P_{C}\) be the metric projection of H onto C. Then, for \(a\in(0, 2\beta)\), \(P_{C}(I-aA)\) is contractive on C.
Proof
Lemma 3.3
Let C be a closed convex subset of a Hilbert space H and A be an injective β-inverse strongly monotone operator sending C into H with \(VI(C, A)\neq\emptyset\). Then \(VI(C, A)\) is a singleton.
Proof
Finally, we give an example of an injective inverse strongly monotone operator in \(\mathbb {R}\) which is not strongly monotone.
Example 1
In the above example, it is difficult to apply Newton’s method and (HSDM), which are valid for continuous and differentiable mappings, or strongly monotone and Lipschitz continuous operators. Hence, there will be many injective inverse strongly monotone operators which are not strongly monotone. Our main results in the next section are effective for such operators.
4 Main results
In this section, we present iterative algorithms for solving the variational inequality problem for an injective inverse strongly monotone operator and their convergence analyses.
Theorem 4.1
Proof
The following are direct consequences of Theorem 4.1.
Corollary 4.1
Proof
Since \(P_{C}(I-aA)\) is nonexpansive for \(a\in(0, 2\beta)\), it follows from Theorem 2.1 and Lemma 2.1 that \(\emptyset\neq\operatorname {Fix}(P_{C}(I-aA))=VI(C,A)\). From Theorem 4.1, we reach the conclusion. □
Corollary 4.2
Proof
The following theorem is derived directly from Theorem 1.5 and Lemma 3.1.
Theorem 4.2
Proof
From Lemma 3.1, \(P_{C}(I-aA)\) is a contractive self-mapping on C. Then, by Theorem 1.5, \(\{u_{n}\}\) converges strongly to the unique fixed point v in \(\operatorname{Fix}(P_{C}(I-aA))\). From Lemma 2.1, we know \(\operatorname{Fix}(P_{C}(I-aA))=VI(C,A)\). Thus, v is the unique point in \(VI(C,A)\). □
Finally, the following theorem due to Yamada [4] is connected with Theorem 1.2.
Theorem 4.3
Proof
Corollary 4.3
Proof
Declarations
Acknowledgements
The author would like to thank the referees for their valuable suggestions on the paper.
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
References
- Lions, JL, Stampacchia, G: Variational inequalities. Commun. Pure Appl. Math. 20, 493-519 (1967) MATHMathSciNetView ArticleGoogle Scholar
- Goldstein, AA: Convex programming in Hilbert space. Bull. Am. Math. Soc. 70, 709-710 (1964) MATHView ArticleGoogle Scholar
- Zeidler, E: Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Optimization. Springer, Berlin (1985); Translated from the German by Leo F. Boron MATHView ArticleGoogle Scholar
- Yamada, I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D, Censor, Y, Reich, S (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473-504. North-Holland, Amsterdam (2001) View ArticleGoogle Scholar
- Xu, HK, Kim, TH: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185-201 (2003) MATHMathSciNetView ArticleGoogle Scholar
- Banach, S: Sur les opérations dans les ensembles abstraits et leurs applications. Fundam. Math. 3, 133-181 (1922) MATHGoogle Scholar
- Edelstein, M: On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 37, 74-79 (1962) MATHMathSciNetView ArticleGoogle Scholar
- Bauschke, HH, Combettes, PL: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Ouvrages de Mathématiques de la SMC. Springer, Berlin (2011); With a foreword by Hédy Attouch MATHView ArticleGoogle Scholar
- Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990) MATHView ArticleGoogle Scholar
- Takahashi, W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama (2000) MATHGoogle Scholar
- Browder, FE: Fixed-point theorems for noncompact mappings in Hilbert space. Proc. Natl. Acad. Sci. USA 53, 1272-1276 (1965) MATHMathSciNetView ArticleGoogle Scholar
- Browder, FE: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54, 1041-1044 (1965) MATHMathSciNetView ArticleGoogle Scholar
- Göhde, D: Zum Prinzip der kontraktiven Abbildung. Math. Nachr. 30, 251-258 (1965) MATHMathSciNetView ArticleGoogle Scholar
- Kirk, WA: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72, 1004-1006 (1965) MATHMathSciNetView ArticleGoogle Scholar
- Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591-597 (1967) MATHMathSciNetView ArticleGoogle Scholar