## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2013, Special Issue (2013), Article ID 102820, 44 pages.

### Hybrid and Relaxed Mann Iterations for General Systems of Variational Inequalities and Nonexpansive Mappings

L. C. Ceng, A. E. Al-Mazrooei, A. A. N. Abdou, and A. Latif

**Full-text: Open access**

#### Abstract

We introduce hybrid and relaxed Mann iteration methods for a general system of variational inequalities with solutions being also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in real smooth and uniformly convex Banach spaces. Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 102820, 44 pages.

**Dates**

First available in Project Euclid: 26 February 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1393449677

**Digital Object Identifier**

doi:10.1155/2013/102820

**Mathematical Reviews number (MathSciNet)**

MR3134168

**Zentralblatt MATH identifier**

1364.47011

#### Citation

Ceng, L. C.; Al-Mazrooei, A. E.; Abdou, A. A. N.; Latif, A. Hybrid and Relaxed Mann Iterations for General Systems of Variational Inequalities and Nonexpansive Mappings. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 102820, 44 pages. doi:10.1155/2013/102820. https://projecteuclid.org/euclid.aaa/1393449677

#### References

- F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,”
*Mathematische Zeitschrift*, vol. 100, pp. 201–225, 1967.Mathematical Reviews (MathSciNet): MR0215141

Zentralblatt MATH: 0149.36301

Digital Object Identifier: doi:10.1007/BF01109805 - L. C. Zeng, G. M. Lee, and N. C. Wong, “Ishikawa iteration with errors for approximating fixed points of strictly pseudocontractive mappings of Browder-Petryshyn type,”
*Taiwanese Journal of Mathematics*, vol. 10, no. 1, pp. 87–99, 2006. - H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,”
*Nonlinear Analysis*, vol. 61, no. 3, pp. 341–350, 2005. - W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,”
*Journal of Optimization Theory and Applications*, vol. 118, no. 2, pp. 417–428, 2003.Mathematical Reviews (MathSciNet): MR2006529

Zentralblatt MATH: 1055.47052

Digital Object Identifier: doi:10.1023/A:1025407607560 - H. Iiduka, W. Takahashi, and M. Toyoda, “Approximation of solutions of variational inequalities for monotone mappings,”
*Panamerican Mathematical Journal*, vol. 14, no. 2, pp. 49–61, 2004. - Y. Takahashi, K. Hashimoto, and M. Kato, “On sharp uniform convexity, smoothness, and strong type, cotype inequalities,”
*Journal of Nonlinear and Convex Analysis*, vol. 3, no. 2, pp. 267–281, 2002. - Y. Yao, R. Chen, and J. C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,”
*Nonlinear Analysis*, vol. 68, no. 6, pp. 1687–1693, 2008. - L. C. Ceng and J. C. Yao, “Convergence and certain control conditions for hybrid viscosity approximation methods,”
*Nonlinear Analysis*, vol. 73, no. 7, pp. 2078–2087, 2010. - J. G. O'Hara, P. Pillay, and H. K. Xu, “Iterative approaches to convex feasibility problems in Banach spaces,”
*Nonlinear Analysis*, vol. 64, no. 9, pp. 2022–2042, 2006. - L. C. Ceng and J. C. Yao, “Relaxed viscosity approximation methods for fixed point problems and variational inequality problems,”
*Nonlinear Analysis*, vol. 69, no. 10, pp. 3299–3309, 2008. - G. Cai and S. Bu, “Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces,”
*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 538–546, 2012.Mathematical Reviews (MathSciNet): MR2887397

Zentralblatt MATH: 1255.49015

Digital Object Identifier: doi:10.1016/j.mcm.2011.08.031 - R. U. Verma, “On a new system of nonlinear variational inequalities and associated iterative algorithms,”
*Mathematical Sciences Research Hot-Line*, vol. 3, no. 8, pp. 65–68, 1999. - J. L. Lions and G. Stampacchia, “Variational inequalities,”
*Communications on Pure and Applied Mathematics*, vol. 20, pp. 493–519, 1967.Mathematical Reviews (MathSciNet): MR0216344

Zentralblatt MATH: 0152.34601

Digital Object Identifier: doi:10.1002/cpa.3160200302 - L. C. Ceng, C. Y. Wang, and J. C. Yao, “Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities,”
*Mathematical Methods of Operations Research*, vol. 67, no. 3, pp. 375–390, 2008.Mathematical Reviews (MathSciNet): MR2403714

Zentralblatt MATH: 1147.49007

Digital Object Identifier: doi:10.1007/s00186-007-0207-4 - G. M. Korpelevič, “An extragradient method for finding saddle points and for other problems,”
*Èkonomika i Matematicheskie Metody*, vol. 12, no. 4, pp. 747–756, 1976.Mathematical Reviews (MathSciNet): MR0451121 - F. Facchinei and J. S. Pang,
*Finite-Dimensional Variational Inequalities and Complementarity Problems*, vol. 1-2, Springer, New York, NY, USA, 2003.Mathematical Reviews (MathSciNet): MR1955648 - A. N. Iusem and B. F. Svaiter, “A variant of Korpelevich's method for variational inequalities with a new search strategy,”
*Optimization*, vol. 42, no. 4, pp. 309–321, 1997.Mathematical Reviews (MathSciNet): MR1609571

Zentralblatt MATH: 0891.90135

Digital Object Identifier: doi:10.1080/02331939708844365 - L. C. Ceng and J. C. Yao, “An extragradient-like approximation method for variational inequality problems and fixed point problems,”
*Applied Mathematics and Computation*, vol. 190, no. 1, pp. 205–215, 2007.Mathematical Reviews (MathSciNet): MR2335441

Zentralblatt MATH: 1124.65056

Digital Object Identifier: doi:10.1016/j.amc.2007.01.021 - M. V. Solodov and B. F. Svaiter, “A new projection method for variational inequality problems,”
*SIAM Journal on Control and Optimization*, vol. 37, no. 3, pp. 765–776, 1999.Mathematical Reviews (MathSciNet): MR1675086

Zentralblatt MATH: 0959.49007

Digital Object Identifier: doi:10.1137/S0363012997317475 - Y. Censor, A. Gibali, and S. Reich, “Two extensions of Korpelevich's extragradient method for solving the variational inequality problem in Euclidean space,” Tech. Rep., 2010.
- L. C. Zeng and J. C. Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,”
*Taiwanese Journal of Mathematics*, vol. 10, no. 5, pp. 1293–1303, 2006. - N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings,”
*Journal of Optimization Theory and Applications*, vol. 128, no. 1, pp. 191–201, 2006.Mathematical Reviews (MathSciNet): MR2201895

Zentralblatt MATH: 1130.90055

Digital Object Identifier: doi:10.1007/s10957-005-7564-z - L. C. Ceng, Q. H. Ansari, and J. C. Yao, “An extragradient method for solving split feasibility and fixed point problems,”
*Computers & Mathematics with Applications*, vol. 64, no. 4, pp. 633–642, 2012. - L. C. Ceng, M. Teboulle, and J. C. Yao, “Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems,”
*Journal of Optimization Theory and Applications*, vol. 146, no. 1, pp. 19–31, 2010.Mathematical Reviews (MathSciNet): MR2657821

Zentralblatt MATH: 1222.47091

Digital Object Identifier: doi:10.1007/s10957-010-9650-0 - L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem,”
*Nonlinear Analysis*, vol. 75, no. 4, pp. 2116–2125, 2012. - L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Relaxed extragradient iterative methods for variational inequalities,”
*Applied Mathematics and Computation*, vol. 218, no. 3, pp. 1112–1123, 2011.Mathematical Reviews (MathSciNet): MR2831366

Zentralblatt MATH: 1229.65109

Digital Object Identifier: doi:10.1016/j.amc.2011.01.061 - K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,”
*Nonlinear Analysis*, vol. 67, no. 8, pp. 2350–2360, 2007. - R. U. Verma, “Projection methods, algorithms, and a new system of nonlinear variational inequalities,”
*Computers & Mathematics with Applications*, vol. 41, no. 7-8, pp. 1025–1031, 2001. - L. C. Ceng, Q. H. Ansari, N. C. Wong, and J. C. Yao, “An extragradient-like approximation method for variational inequalities and fixed point problems,”
*Fixed Point Theory and Applications*, vol. 2011, article 22, 18 pages, 2011.Mathematical Reviews (MathSciNet): MR2823427 - L. C. Ceng, N. Hadjisavvas, and N. C. Wong, “Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems,”
*Journal of Global Optimization*, vol. 46, no. 4, pp. 635–646, 2010.Mathematical Reviews (MathSciNet): MR2601795

Zentralblatt MATH: 1198.47081

Digital Object Identifier: doi:10.1007/s10898-009-9454-7 - L. C. Ceng, S. M. Guu, and J. C. Yao, “Finding common solutions of a variational inequality, a general system of variational inequalities, and a fixed-point problem via a hybrid extragradient method,”
*Fixed Point Theory and Applications*, vol. 2011, Article ID 626159, 22 pages, 2011. - L. C. Ceng and A. Petruşel, “Krasnoselski-Mann iterations for hierarchical fixed point problems for a finite family of nonself mappings in Banach spaces,”
*Journal of Optimization Theory and Applications*, vol. 146, no. 3, pp. 617–639, 2010.Mathematical Reviews (MathSciNet): MR2720605

Zentralblatt MATH: 1210.47094

Digital Object Identifier: doi:10.1007/s10957-010-9679-0 - Y. Yao, Y. C. Liou, S. M. Kang, and Y. Yu, “Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces,”
*Nonlinear Analysis*, vol. 74, no. 17, pp. 6024–6034, 2011.Mathematical Reviews (MathSciNet): MR2833373 - K. Aoyama, H. Iiduka, and W. Takahashi, “Weak convergence of an iterative sequence for accretive operators in Banach spaces,”
*Fixed Point Theory and Applications*, vol. 2006, Article ID 35390, 13 pages, 2006.Mathematical Reviews (MathSciNet): MR2235489

Zentralblatt MATH: 1128.47056

Digital Object Identifier: doi:10.1155/FPTA/2006/35390 - J. S. Jung, “Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 302, no. 2, pp. 509–520, 2005.Mathematical Reviews (MathSciNet): MR2107851

Zentralblatt MATH: 1062.47069

Digital Object Identifier: doi:10.1016/j.jmaa.2004.08.022 - L. C. Ceng, H. K. Xu, and J. C. Yao, “Strong convergence of an iterative method with perturbed mappings for nonexpansive and accretive operators,”
*Numerical Functional Analysis and Optimization*, vol. 29, no. 3-4, pp. 324–345, 2008.Mathematical Reviews (MathSciNet): MR2412411

Zentralblatt MATH: 1140.47050

Digital Object Identifier: doi:10.1080/01630560801998203 - S. Kamimura and W. Takahashi, “Weak and strong convergence of solutions to accretive operator inclusions and applications,”
*Set-Valued Analysis Devoted to the Theory of Multifunctions and Its Applications*, vol. 8, no. 4, pp. 361–374, 2000.Mathematical Reviews (MathSciNet): MR1802240

Zentralblatt MATH: 0981.47036

Digital Object Identifier: doi:10.1023/A:1026592623460 - A. Kangtunyakarn, “Iterative scheme for a nonexpansive mapping, an $\eta $-strictly pseudo-contractive mapping and variational inequality problems in a uniformly convex and 2-uniformly smooth Banach space,”
*Fixed Point Theory and Applications*, vol. 2013, article 23, 21 pages, 2013.Mathematical Reviews (MathSciNet): MR3023868 - H. K. Xu, “Inequalities in Banach spaces with applications,”
*Nonlinear Analysis*, vol. 16, no. 12, pp. 1127–1138, 1991. - S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,”
*SIAM Journal on Optimization*, vol. 13, no. 3, pp. 938–945, 2002.Mathematical Reviews (MathSciNet): MR1972223

Digital Object Identifier: doi:10.1137/S105262340139611X - H. K. Xu, “Iterative algorithms for nonlinear operators,”
*Journal of the London Mathematical Society*, vol. 66, no. 1, pp. 240–256, 2002.Mathematical Reviews (MathSciNet): MR1911872

Zentralblatt MATH: 1013.47032

Digital Object Identifier: doi:10.1112/S0024610702003332 - S. S. Chang, “Some problems and results in the study of nonlinear analysis,”
*Nonlinear Analysis*, vol. 30, no. 7, pp. 4197–4208.Mathematical Reviews (MathSciNet): MR1603564 - S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 67, no. 2, pp. 274–276, 1979.Mathematical Reviews (MathSciNet): MR528688

Zentralblatt MATH: 0423.47026

Digital Object Identifier: doi:10.1016/0022-247X(79)90024-6 - Y. J. Cho, H. Zhou, and G. Guo, “Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings,”
*Computers & Mathematics with Applications*, vol. 47, no. 4-5, pp. 707–717, 2004. - K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,”
*Nonlinear Analysis*, vol. 67, no. 8, pp. 2350–2360, 2007. - H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 298, no. 1, pp. 279–291, 2004.Mathematical Reviews (MathSciNet): MR2086546

Zentralblatt MATH: 1061.47060

Digital Object Identifier: doi:10.1016/j.jmaa.2004.04.059 - R. E. Bruck, Jr., “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,”
*Transactions of the American Mathematical Society*, vol. 179, pp. 251–262, 1973.Mathematical Reviews (MathSciNet): MR0324491

Zentralblatt MATH: 0265.47043

Digital Object Identifier: doi:10.1090/S0002-9947-1973-0324491-8 - K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,”
*Taiwanese Journal of Mathematics*, vol. 5, no. 2, pp. 387–404, 2001. - H. Zhou, L. Wei, and Y. J. Cho, “Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces,”
*Applied Mathematics and Computation*, vol. 173, no. 1, pp. 196–212, 2006.Mathematical Reviews (MathSciNet): MR2203381

Zentralblatt MATH: 1100.65049

Digital Object Identifier: doi:10.1016/j.amc.2005.02.049 - N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 125, no. 12, pp. 3641–3645, 1997.Mathematical Reviews (MathSciNet): MR1415370

Zentralblatt MATH: 0888.47034

Digital Object Identifier: doi:10.1090/S0002-9939-97-04033-1 - T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,”
*Journal of Mathematical Analysis and Applications*, vol. 305, no. 1, pp. 227–239, 2005. \endinputMathematical Reviews (MathSciNet): MR2128124

Zentralblatt MATH: 1068.47085

Digital Object Identifier: doi:10.1016/j.jmaa.2004.11.017

### More like this

- Mann-Type Extragradient Methods for General Systems of Variational Inequalities with Multivalued Variational Inclusion Constraints in Banach Spaces

Ceng, Lu-Chuan, Latif, Abdul, and Al-Mezel, Saleh A., Abstract and Applied Analysis, 2013 - Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces

Ceng, Lu-Chuan and Wen, Ching-Feng, Journal of Applied Mathematics, 2013 - Hybrid Viscosity Approaches to General Systems of Variational Inequalities with Hierarchical Fixed Point Problem Constraints in Banach Spaces

Ceng, Lu-Chuan, Al-Mezel, Saleh A., and Latif, Abdul, Abstract and Applied Analysis, 2013

- Mann-Type Extragradient Methods for General Systems of Variational Inequalities with Multivalued Variational Inclusion Constraints in Banach Spaces

Ceng, Lu-Chuan, Latif, Abdul, and Al-Mezel, Saleh A., Abstract and Applied Analysis, 2013 - Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces

Ceng, Lu-Chuan and Wen, Ching-Feng, Journal of Applied Mathematics, 2013 - Hybrid Viscosity Approaches to General Systems of Variational Inequalities with Hierarchical Fixed Point Problem Constraints in Banach Spaces

Ceng, Lu-Chuan, Al-Mezel, Saleh A., and Latif, Abdul, Abstract and Applied Analysis, 2013 - Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions
with Finitely Many Variational Inequality Constraints in Banach Spaces

Ceng, Lu-Chuan, Latif, Abdul, and Al-Mazrooei, Abdullah E., Abstract and Applied Analysis, 2013 - Hybrid Extragradient Methods for Finding Zeros of Accretive Operators and Solving Variational Inequality and Fixed Point Problems in Banach Spaces

Ceng, Lu-Chuan and Wen, Ching-Feng, Abstract and Applied Analysis, 2013 - Modified Hybrid Steepest-Descent Methods for General Systems of Variational
Inequalities with Solutions to Zeros of
m
-Accretive Operators in Banach Spaces

Ceng, Lu-Chuan and Wen, Ching-Feng, Abstract and Applied Analysis, 2013 - Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

Ceng, Lu-Chuan, Liao, Cheng-Wen, Pang, Chin-Tzong, and Wen, Ching-Feng, Journal of Applied Mathematics, 2014 - A New Hybrid Projection Algorithm for System of Equilibrium Problems and
Variational Inequality Problems and Two Finite Families of Quasi-
ϕ
-Nonexpansive Mappings

Phuangphoo, Pongrus and Kumam, Poom, Abstract and Applied Analysis, 2013 - A Viscosity Hybrid Steepest Descent Method for Equilibrium Problems,
Variational Inequality Problems, and Fixed Point Problems of Infinite Family
of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

Che, Haitao and Pan, Xintian, Abstract and Applied Analysis, 2013 - Hybrid Iterative Scheme by a Relaxed Extragradient Method for Equilibrium
Problems, a General System of Variational Inequalities and Fixed-Point Problems of a
Countable Family of Nonexpansive Mappings

Dong, Qiao-Li, Guo, Yan-Ni, and Su, Fang, Journal of Applied Mathematics, 2012