- Research
- Open Access
Algorithms for the common solution of the split variational inequality problems and fixed point problems with applications
- Panisa Lohawech^{1},
- Anchalee Kaewcharoen^{1}Email author and
- Ali Farajzadeh^{2}
https://doi.org/10.1186/s13660-018-1942-1
© The Author(s) 2018
- Received: 29 August 2018
- Accepted: 10 December 2018
- Published: 29 December 2018
Abstract
In this paper, we establish a new iterative algorithm by combining Nadezhkina and Takahashi’s modified extragradient method and Xu’s algorithm. The mentioned iterative algorithm presents the common solution of the split variational inequality problems and fixed point problems. We show that the sequence produced by our algorithm is weakly convergent. Finally, we give some applications of the main results. This article extends the previous results in this area.
Keywords
- Variational inequality problems
- Extragradient methods
- CQ algorithms
MSC
- 58E35
- 47H10
1 Introduction
Since then, it has been used to study the problems of finding a common solution of VIP and fixed point problem (see [42] and the references therein).
In this paper, we establish a new iterative algorithm by combining Nadezhkina and Takahashi’s modified extragradient method and Xu’s algorithm. The mentioned iterative algorithm presents the common solution of the split variational inequality problems and fixed point problems. We show that the sequence produced by our algorithm is weakly convergent. Finally, we give some applications of the main results. This article extends the results that appeared in [32].
2 Preliminaries
In order to solve the our results, we recall the following definitions and preliminary results that will be used in the sequel. Throughout this section, let C be a closed convex subset of a real Hilbert space H.
Lemma 2.1
([8])
- (i)
\(z = P_{C}x\);
- (ii)
\(\langle x - z, z - y \rangle\geq0\) for all \(y \in C\);
- (iii)
\(\|x - y\|^{2} \geq\|x - z\|^{2} + \|y - z\|^{2}\) for all \(y \in C\).
We need the following definitions about set-valued mappings for proving our main results.
Definition 2.2
([30])
Let \(B:H \rightrightarrows H\) be a set-valued mapping with the effective domain \(D(B) = \{x \in H : Bx \neq\emptyset\}\).
Also the monotone set-valued mapping B is said to be maximal if its graph \(G(B)=\{(x, y) : y \in Bx\}\) is not properly contained in the graph of any other monotone set-valued mappings.
The following property of the maximal monotone mappings is very convenient and helpful to use:
Remark 2.3
The following results play the crucial role in the next section.
Lemma 2.4
([27])
- (i)
\(z = J_{r}(I - \gamma A^{*}(I - T)A)z\);
- (ii)
\(0 \in A^{*}(I - T)Az + Bz\);
- (iii)
\(z \in B^{-1}0 \cap A^{-1}\operatorname{Fix}(T)\).
Lemma 2.5
([23])
Lemma 2.6
([35])
- (i)
\(\lim_{n \rightarrow \infty}\|x_{n}-u\|\) exists for each \(u \in C\);
- (ii)
\(\omega_{w}(x_{n}) \subset C\).
Theorem 2.7
([27])
Let \(f:C \rightarrow H\) be a monotone and k-Lipschitz continuous mapping. Assume that \(S:C \rightarrow C\) is a nonexpansive mapping such that \(\operatorname {VI}(C,f)\cap\operatorname{Fix}(S) \neq\emptyset\). Let \(\{x_{n}\}\) and \(\{ y_{n}\}\) be sequences generated by (1.5), where \(\{\lambda_{n}\}\subset[a,b]\) for some \(a,b \in(0,\frac{1}{k})\) and \(\{\alpha_{n}\} \subset[c,d]\) for some \(c,d \in(0,1)\). Then the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) converge weakly to the same point \(z \in\operatorname{VI}(C,f)\cap\operatorname{Fix}(S) \neq\emptyset\), where \(z = \lim_{n \rightarrow \infty} P_{\operatorname{VI}(C,f)\cap\operatorname{Fix}(S)}x_{n}\).
Theorem 2.8
([34])
3 Main results
Our aim in this section is to consider an iterative method by combining Nadezhkina and Takahashi’s modified extragradient method with Zhao and Yang’s algorithm for solving the split variational inequality problems and fixed point problems.
Throughout our results, unless otherwise stated, we assume that C and Q are nonempty closed convex subsets of real Hilbert spaces \(H_{1}\) and \(H_{2}\), respectively. Suppose that \(A : H_{1} \rightarrow H_{2}\) is a nonzero bounded linear operator, \(f : C\rightarrow H_{1}\) is a monotone and k-Lipschitz continuous mapping, and \(g: H_{2} \rightarrow H_{2}\) is a δ-inverse strongly monotone mapping. Suppose that \(T:H_{2} \rightarrow H_{2}\) and \(S : C \rightarrow C\) are nonexpansive. Let \(\{\mu_{n}\}, \{ \alpha_{n}\} \subset(0,1)\), \(\{\gamma_{n}\} \subset[a, b]\) for some \(a, b \in(0, \frac{1}{\|A\|^{2}})\) and \(\{\lambda_{n}\} \subset[c, d]\) for some \(c, d \in(0,\frac{1}{k})\).
Theorem 3.1
Proof
Remark 3.2
- (i)
If \(f=0\), \(T=P_{Q}\), and \(S=I\), then problem (3.1) coincides with the SFP and algorithm (3.2) reduces to algorithm (1.8) for solving the SFP.
- (ii)
If \(T=I\), then problem (3.1) coincides with the VIP and FPP and algorithm (3.2) reduces to algorithm (1.5) for solving the VIP and FPP.
- (iii)
If \(S=I\), then problem (3.1) coincides with problem 3.1 in [32] and if \(\alpha_{n}, \mu_{n} =0\), we obtain that algorithm (3.2) reduces to algorithm 3.2 in [32].
Theorem 3.3
Proof
It is clear from δ-inverse strongly monotonicity of g that it is \(\frac{1}{\delta}\)-Lipschitz continuous and so, for \(\theta\in (0,2\delta)\), we obtain that \(I-\theta g\) is nonexpansive. Since \(P_{Q}\) is firmly nonexpansive, then \(P_{Q}(I-\theta g)\) is nonexpansive. By taking \(T=P_{Q}(I-\theta g)\) in Theorem 3.1, we obtain that \(\{ x_{n}\}\) converges weakly to a point \(z\in\operatorname{VI}(C,f) \cap\operatorname {Fix}(S)\) and \(Az \in\operatorname{Fix}(P_{Q}(I-\theta g))\). It follows from \(Az =P_{Q}(I-\theta g)Az\) and Lemma 2.1 that \(Az \in\operatorname {VI}(Q,g)\). This completes the proof. □
Remark 3.4
- (i)
If \(f=0\), \(g=0\), and \(S=I\), then problem (3.19) coincides with the SFP and algorithm (3.20) reduces to algorithm (1.8) for solving the SFP.
- (ii)
If \(g=0\) and \(Q=H_{2}\), then problem (3.19) coincides with the VIP and FPP and algorithm (3.20) reduces to algorithm (1.5) for solving the VIP and FPP.
- (iii)
If \(S=I\), then problem (3.19) coincides with problem 3.1 in [32] and if \(\alpha_{n}, \mu_{n} =0\), then algorithm (3.20) reduces to algorithm (1.10).
4 Applications
In this section, by using the main results, we give some applications to the weak convergence of the produced algorithms for the equilibrium problem, zero point problem and convex minimization problem.
- (A1)
\(F(x,x)=0\) for all \(x \in C\);
- (A2)
F is monotone, that is, \(F(x,y) + F(y,x) \leq0\) for all \(x, y \in C\);
- (A3)
for each \(x, y, z \in C\), \(\limsup_{t\downarrow0} F(tz + (1 - t)x, y) \leq F(x, y)\);
- (A4)
for each fixed \(x \in C\), \(y \mapsto F(x,y)\) is lower semicontinuous and convex,
- (i)
\(T_{r}\) is single-valued and firmly nonexpansive;
- (ii)
\(\operatorname{Fix}(T_{r}) = \operatorname{EP}(C,F)\);
- (iii)
\(\operatorname{EP}(C,F)\) is closed and convex.
The following result is related to the equilibrium problems by applying Theorem 3.1.
Theorem 4.1
Proof
Since \(T_{r}\) is nonexpansive, the proof follows from Theorem 3.1 by taking \(T_{r}=T\). □
The following results are the application of Theorem 3.1 to the zero point problem.
Theorem 4.2
Proof
Since \(J_{r}\) is firmly nonexpansive and \(\operatorname{Fix}(J_{r})=B^{-1}0\), the proof follows from Theorem 3.1 by taking \(J_{r}=T\). □
Theorem 4.3
Proof
Since F is δ-inverse strongly monotone, then \(I-rF\) is nonexpansive. By the nonexpansiveness of \(J_{r}\), we obtain that \(J_{r}(I-rF)\) is also nonexpansive. We know that \(z \in(B+F)^{-1}0\) if and only if \(z=J_{r}(I-rF)z\). Thus the proof follows from Theorem 3.1 by taking \(J_{r}(I-rF)=T\). □
By applying Theorem 3.3, we get the following result.
Theorem 4.4
Proof
We obtain the following result for solving the split minimization problems and fixed point problems by applying Theorem 3.3.
Theorem 4.5
Declarations
Acknowledgements
The first author is thankful to the Science Achievement Scholarship of Thailand. We would like express our deep thanks to the Department of Mathematics, Faculty of Science, Naresuan University for the support.
Funding
The research was supported by the Science Achievement Scholarship of Thailand and Naresuan University.
Authors’ contributions
All authors contributed equally to the work. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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
- Alghamdi, M.A., Shahzad, N., Zegeye, H.: On solutions of variational inequality problems via iterative methods. Abstr. Appl. Anal. 2014, Article ID 424875 (2014) MathSciNetGoogle Scholar
- Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011) View ArticleGoogle Scholar
- Billups, S.C., Murty, K.G.: Complementarity problems. J. Comput. Appl. Math. 124, 303–318 (2000) MathSciNetView ArticleGoogle Scholar
- Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994) MathSciNetMATHGoogle Scholar
- Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002) MathSciNetView ArticleGoogle Scholar
- Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004) MathSciNetView ArticleGoogle Scholar
- Byrne, C., Censor, Y., Gibali, A., Reich, S.: Weak and strong convergence of algorithms for the split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012) MathSciNetMATHGoogle Scholar
- Ceng, L.C., Ansari, Q.H., Yao, J.C.: Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Anal. 74, 5286–5302 (2011) MathSciNetView ArticleGoogle Scholar
- Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006) View ArticleGoogle Scholar
- Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994) MathSciNetView ArticleGoogle Scholar
- Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012) MathSciNetView ArticleGoogle Scholar
- Cho, S.Y., Qin, X., Yao, J.C., Yao, Y.: Viscosity approximation splitting methods for monotone and nonexpansive operators in Hilbert spaces. J. Nonlinear Convex Anal. 19, 251–264 (2018) MathSciNetMATHGoogle Scholar
- Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005) MathSciNetMATHGoogle Scholar
- Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992) MathSciNetView ArticleGoogle Scholar
- Fang, N.N., Gong, Y.P.: Viscosity iterative methods for split variational inclusion problems and fixed point problems of a nonexpansive mapping. Commun. Optim. Theory 2016, Article ID 11 (2016) Google Scholar
- Gibali, A.: Two simple relaxed perturbed extragradient methods for solving variational inequalities in Euclidean spaces. J. Nonlinear Var. Anal. 2, 49–61 (2018) View ArticleGoogle Scholar
- Hartman, P., Stampacchia, G.: On some non-linear elliptic differential-functional equations. Acta Math. 115, 271–310 (1966) MathSciNetView ArticleGoogle Scholar
- Kim, J.K., Salahuddin: A system of nonconvex variational inequalities in Banach spaces. Commun. Optim. Theory 2016, Article ID 20 (2016) Google Scholar
- Korpelevich, G.M.: An extragradient method for finding saddle points and for other problems. Èkon. Mat. Metody 12, 747–756 (1976) MathSciNetMATHGoogle Scholar
- Mancino, O.G., Stampacchia, G.: Convex programming and variational inequalities. J. Optim. Theory Appl. 9(1), 3–23 (1972) MathSciNetView ArticleGoogle Scholar
- Qin, X., Cho, S.Y., Wang, L.: Strong convergence of an iterative algorithm involving nonlinear mappings of nonexpansive and accretive type. Optimization 67, 1377–1388 (2018). https://doi.org/10.1080/02331934.2018.1491973 MathSciNetView ArticleMATHGoogle Scholar
- Qin, X., Yao, J.C.: Projection splitting algorithms for nonself operators. J. Nonlinear Convex Anal. 18, 925–935 (2017) MathSciNetMATHGoogle Scholar
- Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43, 153–159 (1991) MathSciNetView ArticleGoogle Scholar
- Stampacchia, G.: Formes bilineaires coercivites sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964) MathSciNetMATHGoogle Scholar
- Suwannaprapa, M., Petrot, N., Suantai, S.: Weak convergence theorems for split feasibility problems on zeros of the sum of monotone operators and fixed point sets in Hilbert spaces. Fixed Point Theory Appl. 2017, 6 (2017) MathSciNetView ArticleGoogle Scholar
- Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2009) MATHGoogle Scholar
- Takahashi, W., Nadezhkina, N.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2006) MathSciNetView ArticleGoogle Scholar
- Takahashi, W., Toyoda, M.: Weak convergence theorem for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003) MathSciNetView ArticleGoogle Scholar
- Takahashi, W., Wen, C.F., Yao, J.C.: An implicit algorithm for the split common fixed point problem in Hilbert spaces and applications. Appl. Anal. Optim. 1, 423–439 (2017) MathSciNetGoogle Scholar
- Takahashi, W., Xu, H.K., Yao, J.C.: Iterative methods for generalized split feasibility problems in Hilbert spaces. Set-Valued Var. Anal. 23(2), 205–221 (2015) MathSciNetView ArticleGoogle Scholar
- Tian, M., Jiang, B.N.: Weak convergence theorem for variational inequality problems with monotone mapping in Hilbert spaces. J. Inequal. Appl. 2016, 286 (2016) MathSciNetView ArticleGoogle Scholar
- Tian, M., Jiang, B.N.: Weak convergence theorem for a class of split variational inequality problems and applications in a Hilbert space. J. Inequal. Appl. 2017, 123 (2017) MathSciNetView ArticleGoogle Scholar
- Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004) MathSciNetView ArticleGoogle Scholar
- Xu, H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010) MathSciNetView ArticleGoogle Scholar
- Xu, H.K.: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 150, 360–378 (2011) MathSciNetView ArticleGoogle Scholar
- Yao, Y., Marino, G., Liou, Y.C.: A hybrid method for monotone variational inequalities involving pseudocontractions. Fixed Point Theory Appl. 2011, 180534 (2011) MathSciNetMATHGoogle Scholar
- Yao, Y.H., Agarwal, R.P., Postolache, M., Liou, Y.C.: Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. Fixed Point Theory Appl. 2014, 183 (2014) MathSciNetView ArticleGoogle Scholar
- Yao, Y.H., Liou, Y.C., Yao, J.C.: Split common fixed point problem for two quasi-pseudocontractive operators and its algorithm construction. Fixed Point Theory Appl. 2015, 127 (2015) View ArticleGoogle Scholar
- Yao, Y.H., Liou, Y.C., Yao, J.C.: Iterative algorithms for the split variational inequality and fixed point problems under nonlinear transformations. J. Nonlinear Sci. Appl. 10, 843–854 (2017) MathSciNetView ArticleGoogle Scholar
- Yuan, H.: A splitting algorithm in a uniformly convex and 2-uniformly smooth Banach space. J. Nonlinear Funct. Anal. 2018, Article ID 26 (2018) Google Scholar
- Zegeye, H., Shahzad, N., Yao, Y.H.: Minimum-norm solution of variational inequality and fixed point problem in Banach spaces. Optimization 64, 453–471 (2015) MathSciNetView ArticleGoogle Scholar
- Zeng, L.C., Yao, J.C.: Strong convergence theorems for fixed point problems and variational inequality problems. Taiwan. J. Math. 10(5), 1293–1303 (2006) View ArticleGoogle Scholar
- Zhou, H., Zhou, Y., Feng, G.: Iterative methods for solving a class of monotone variational inequality problems with applications. J. Inequal. Appl. 2015, 68 (2015) MathSciNetView ArticleGoogle Scholar