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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 convergence
MSC
- 47H09
- 47H10
- 47J20
- 65J15
- 47L25
1 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
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