A scheme for a solution of a variational inequality for a monotone mapping and a fixed point of a pseudocontractive mapping
- Mohammed Ali Alghamdi^{1},
- Naseer Shahzad^{1}Email author and
- Habtu Zegeye^{2}
https://doi.org/10.1186/s13660-015-0804-3
© Alghamdi et al. 2015
Received: 11 June 2015
Accepted: 28 August 2015
Published: 18 September 2015
Abstract
We introduce an iterative process which converges strongly to a common point of the solution set of a variational inequality problem for a Lipschitzian monotone mapping and the fixed point set of a continuous pseudocontractive mapping in Hilbert spaces. In addition, a numerical example which supports our main result is presented. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.
Keywords
fixed points of mappings monotone mappings pseudocontractive mappings strong convergenceMSC
47H09 47H10 47J20 65J15 47L251 Introduction
We observe that A is monotone if and only if \(T:=I-A\) is pseudocontractive and thus a zero of A, \(N(A):=\{x\in D(A):Ax=0\}\), is a fixed point of T, \(F(T):=\{ x\in D(T):Tx=x\}\). It is now well known that if A is monotone then the solutions of the equation \(Ax=0 \) correspond to the equilibrium points of some evolution systems. Consequently, considerable research efforts have been devoted to iterative methods for approximating fixed points of T when T is nonexpansive or pseudocontractive (see, e.g., [1–10] and the references therein).
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. The classical variational inequality problem is to find a \(u\in C\) such that \(\langle v-u, Au\rangle\geq0\) for all \(v\in C\), where A is a nonlinear mapping. The set of solutions of the variational inequality is denoted by \(VI(C, A)\). In the context of the variational inequality problem, this implies that \(u \in VI(C, A)\) if and only if \(u = P_{C}(u - \lambda Au)\), \(\forall\lambda>0\), where \(P_{C}\) is a metric projection of H into C.
It is now well known that variational inequalities cover disciplines such as partial differential equations, optimal control, optimization, mathematical programming, mechanics and finance. See, for instance, [11–16].
Variational inequalities were introduced and studied by Stampacchia [17] in 1964. Since then, several numerical methods have been developed for solving variational inequalities; see, for instance, [12, 15, 18–23] and the references therein.
In 2006, Nadezhkina and Takahashi [25] introduced the following hybrid method for finding an element of \(F(S) \cap V I(C,A)\) and established the following strong convergence theorem for the sequence generated by this process.
Theorem NT
[25]
Our concern now is the following: can an approximation sequence \(\{x_{n}\}\) be constructed which converges to a common point of the solution set of a variational inequality problem for a monotone mapping and the fixed point set of a continuous pseudocontractive mapping?
In this paper, it is our purpose to introduce an iterative scheme which converges strongly to a common element of the solution set of a variational inequality problem for Lipschitzian monotone mapping and the fixed point set of a continuous pseudocontractive mapping in Hilbert spaces. Our results provide an affirmative answers to our concern. In addition, a numerical example which supports our main result is presented. Our theorems will extend and unify most of the results that have been proved for this important class of nonlinear operators.
2 Preliminaries
Lemma 2.1
[27]
Lemma 2.2
Lemma 2.3
[28]
Lemma 2.4
[11]
Lemma 2.5
[29]
- (1)
\(F_{r}\) is single-valued;
- (2)\(F_{r}\) is firmly nonexpansive type mapping, i.e., for all \(x,y\in H\),$$\|F_{r}x-F_{r}y\|^{2} \leq\langle F_{r}x-F_{r}y,x-y\rangle; $$
- (3)
\(F(F_{r})=F(T) \);
- (4)
\(F(T)\) is closed and convex.
3 Main result
Theorem 3.1
Proof
Now, we consider two cases.
Furthermore, since \(\{x_{n+1} \}\) is bounded subset of H which is reflexive, we can choose a subsequence \(\{x_{n_{i}+1}\}\) of \(\{ x_{n+1}\}\) such that \(x_{n_{i}+1}\rightharpoonup z\) and \(\limsup_{n\to\infty}\langle u-x^{*}, x_{n+1}-x^{*}\rangle=\lim_{i\to\infty }\langle u-x^{*},x_{n_{i}+1}-x^{*}\rangle\). This implies from (3.11) that \(x_{n_{i}}\rightharpoonup z\).
If, in Theorem 3.1, we assume that \(T=I\), the identity mapping on C, we obtain the following corollary.
Corollary 3.2
If, in Theorem 3.1, we assume that \(A=0\), we obtain the following corollary, which is Theorem 3.1 of [29].
Corollary 3.3
If, in Theorem 3.1, we assume that A is α-inverse strongly monotone then A is Lipschitzian and we obtain the following corollary.
Corollary 3.4
If, in Theorem 3.1, we assume that \(C=H\), a real Hilbert space, then \(P_{C}\) becomes identity mapping and \(VI(C,A)=A^{-1}(0)\), and hence we get the following corollary.
Corollary 3.5
We also note that the method of proof of Theorem 3.1 provides the following theorem for approximating the common minimum-norm point of the solution set of a variational inequality problem for monotone mapping and fixed point set of a continuous pseudocontractive mapping.
Theorem 3.6
Remark 3.7
Theorem 3.1 extends Theorem 3.1 of Takahashi and Toyoda [24] and Theorem 3.2 of Yao et al. [22], Theorem 3.1 of Iiduka and Takahashi [19] and the results of Nadezhkina and Takahashi [25] in the sense that our scheme provides a common point of the solution set of variational inequalities for a more general class of monotone mappings and/or the fixed point set of a more general class of continuous pseudocontractive mappings. Our results provide an affirmative answer to our concern.
4 Applications to minimization problems
In this section, we study the problem of finding a minimizer of a continuously Fréchet differentiable convex functional in Hilbert spaces. Let f be a continuously Fréchet differentiable convex functionals of H into \((-\infty,\infty)\) such that the gradient of f, \((\bigtriangledown f)\) is continuous and monotone. For \(\gamma>0\), and \(x\in H\), let \(T_{r_{n}}x:=\{z\in H:\langle y-z,(I- (\bigtriangledown f))z\rangle -\frac{1}{\gamma}\langle y-z, (1+\gamma)z-x\rangle\leq0, \forall y\in H\}\). Then the following theorem holds.
Theorem 4.1
Proof
We note that \(T:=(I-\bigtriangledown f)\) is continuous pseudocontractive mapping with \(F(T)= (\bigtriangledown f)^{-1}(0)\) and from the convexity and Frećhet differentiability of f we see that the zero of ▽f is given by \(\mathcal{N}=\arg\min_{y\in C}f(y)\). Thus, the conclusion follows from Corollary 3.3. □
5 Numerical example
In this section, we give an example of a continuous pseudocontractive mapping T and a Lipschitzian monotone mapping with all the conditions of Theorem 3.1 and some numerical experiment results to explain the conclusion of the theorem.
Example 5.1
Declarations
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (5-130-36-RG). The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are grateful to the reviewers for their meticulous reading of 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
- Browder, FE, Petryshyn, WV: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197-228 (1967) MATHMathSciNetView ArticleGoogle Scholar
- Chidume, CE, Mutangadura, S: An example on the Mann iteration methods for Lipschitzian pseudocontractions. Proc. Am. Math. Soc. 129, 2359-2363 (2001) MATHMathSciNetView ArticleGoogle Scholar
- Chidume, CE, Zegeye, H, Shahzad, N: Convergence theorems for a common fixed point of a finite family of nonself nonexpansive mappings. Fixed Point Theory Appl. 2005, 233-241 (2005) MATHMathSciNetGoogle Scholar
- Halpern, B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957-961 (1967) MATHView ArticleGoogle Scholar
- Ishikawa, S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147-150 (1974) MATHMathSciNetView ArticleGoogle Scholar
- Ishikawa, S: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. Am. Math. Soc. 59, 65-71 (1976) MATHMathSciNetView ArticleGoogle Scholar
- Lions, PL: Approximation de points fixes de contractions. C. R. Acad. Sci. Paris Sér. A-B 284, 1357-1359 (1977) MATHGoogle Scholar
- Matinez-Yanes, C, Xu, HK: Strong convergence of the CQ method for fixed point processes. Nonlinear Anal. 64, 2400-2411 (2006) MathSciNetView ArticleGoogle Scholar
- Reich, S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274-276 (1979) MATHMathSciNetView ArticleGoogle Scholar
- Qihou, L: The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings. J. Math. Anal. Appl. 148, 55-62 (1990) MATHMathSciNetView ArticleGoogle Scholar
- Mainge, PE: Strong convergence of projected subgradient methods for nonsmooth and non-strictly convex minimization. Set-Valued Anal. 16, 899-912 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Noor, MA: A class of new iterative methods for solving mixed variational inequalities. Math. Comput. Model. 31, 11-19 (2000) MATHView ArticleGoogle Scholar
- Yamada, I: The hybrid steepest-descent method for variational inequality problems over the intersection of the 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
- Yao, Y, Xu, H-K: Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications. Optimization 60, 645-658 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Zegeye, H, Shahzad, N: Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces. Optim. Lett. 5, 691-704 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Zegeye, H, Shahzad, N: Strong convergence theorems for a common zero of a countably infinite family of λ-inverse strongly accretive mappings. Nonlinear Anal. 71, 531-538 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Stampacchia, G: Formes bilineaires coercivites sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413-4416 (1964) MATHMathSciNetGoogle Scholar
- Cai, G, Bu, S: An iterative algorithm for a general system of variational inequalities and fixed point problems in q-uniformly smooth Banach spaces. Optim. Lett. 7, 267-287 (2013) MATHMathSciNetView ArticleGoogle Scholar
- Iiduka, H, Takahashi, W: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 61, 341-350 (2005) MATHMathSciNetView ArticleGoogle Scholar
- Iiduka, H, Takahashi, W, Toyoda, M: Approximation of solutions of variational inequalities for monotone mappings. Panam. Math. J. 14, 49-61 (2004) MATHMathSciNetGoogle Scholar
- Kinderlehrer, D, Stampaccia, G: An Iteration to Variational Inequalities and Their Applications. Academic Press, New York (1990) Google Scholar
- Yao, Y, Liou, YC, Kang, SM: Algorithms construction for variational inequalities. Fixed Point Theory Appl. 2011, Article ID 794203 (2011) MathSciNetGoogle Scholar
- Zegeye, H, Shahzad, N: A hybrid approximation method for equilibrium, variational inequality and fixed point problems. Nonlinear Anal. Hybrid Syst. 4, 619-630 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Takahashi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417-428 (2003) MATHMathSciNetView ArticleGoogle Scholar
- Nadezhkina, N, Takahashi, W: Strong convergence theorem by a hybrid method for non-expansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230-1241 (2006) MATHMathSciNetView ArticleGoogle Scholar
- Takahashi, W: Nonlinear Functional Analysis. Kindikagaku, Tokyo (1988) Google Scholar
- Zegeye, H, Shahzad, N: Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62, 4007-4014 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Xu, HK: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109-113 (2002) MATHView ArticleGoogle Scholar
- Zegeye, H: An iterative approximation method for a common fixed point of two pseudocontractive mappings. ISRN Math. Anal. 2011, Article ID 621901 (2011) MathSciNetGoogle Scholar
- Rockafellar, RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75-88 (1970) MATHMathSciNetView ArticleGoogle Scholar