zbMATH — the first resource for mathematics

An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. (English) Zbl 1251.47059
Let $$H$$ be a Hilbert space and let $$F$$ be a strongly monotone Lipschitz-continuous nonlinear operator $$F:H\to{H}$$. The problem under consideration is to find a point $$p^\ast\in{C}$$ such that $$\langle{F}(p^\ast),\,p=p^\ast\rangle\geq0$$ for every $$p\in{C}$$. Here, $$C\subset{H}$$ is a closed convex set of common fixed points of a finite family of nonexpansive maps $$T_i:H\to{H}$$, $$i=1,2,\dots,N$$. The authors propose an explicit iterative procedure generating a sequence which strongly converges to the unique solution of the variational inequality.

MSC:
 47J25 Iterative procedures involving nonlinear operators 47J20 Variational and other types of inequalities involving nonlinear operators (general)
Full Text:
References:
 [1] Antman, S.: The influence of elasticity in analysis: modern developments. Bull. Am. Math. Soc. 9(3), 267–291 (1983) · Zbl 0533.73001 · doi:10.1090/S0273-0979-1983-15185-6 [2] Fichera, G.: La nascita della teoria delle diseguaglianze variazionali ricordata dopo trent’anni. Accad. Naz. Lincei. 114, 47–53 (1995) [3] Stampacchia, G.: Formes bilineares coercitives sur les ensembles convexes. C. R. Hebd. Séances Acad. Sci. 258, 4413–4416 (1964) · Zbl 0124.06401 [4] Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967) · Zbl 0152.34601 · doi:10.1002/cpa.3160200302 [5] Duvaut, D., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976) · Zbl 0331.35002 [6] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980) · Zbl 0457.35001 [7] Hlavacek, I., Haslinger, J., Necas, J., Lovicek, J.: Solution of Variational Inequalities in Mechanics. Springer, New York (1982) [8] Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984) · Zbl 0536.65054 [9] Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985) · Zbl 0579.73014 [10] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Springer, New York (1985) · Zbl 0583.47051 [11] Aoyama, K., Iiduka, H., Takahashi, W.: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. (2006). doi: 10.1155/35390 · Zbl 1128.47056 · doi:10.1155/FPTA/2006/35390 [12] Yamada, Y.: The hybrid steepest-descent method for variational inequalities problems over the intersection of the fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inhently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473–504. North-Holland, Amsterdam (2001) · Zbl 1013.49005 [13] Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003) · Zbl 1043.90063 · doi:10.1023/A:1023073621589 [14] Halpern, B.: Fixed points of nonexpansive mappings. Bull. Am. Math. Soc. 73, 957–961 (1967) · Zbl 0177.19101 · doi:10.1090/S0002-9904-1967-11864-0 [15] Xu, H.K., Kim, T.H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003) · Zbl 1045.49018 · doi:10.1023/B:JOTA.0000005048.79379.b6 [16] Zeng, L.C., Ansari, Q.H., Wu, S.Y.: Strong convergence theorems of relaxed hybrid steepest-descent methods for variational inequalities. Taiwan. J. Math. 10(1), 13–29 (2006) · Zbl 1092.49013 [17] Zeng, L.C., Wong, N.C., Yao, J.Ch.: Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. J. Optim. Theory Appl. 132, 51–69 (2007) · Zbl 1137.47059 · doi:10.1007/s10957-006-9068-x [18] Wang, L., Yao, S.S.: Hybrid iteration method for fixed points of nonexpansive mappings. Thai J. Math. 5(2), 183–190 (2007) · Zbl 1167.47053 [19] Mao, Y., Li, J.: Weak and strong convergence of an iterative method for nonexpansive mappings in Hilbert spaces. Appl. Anal. Discrete Math. 2, 197–204 (2008) · Zbl 1199.47279 · doi:10.2298/AADM0802197M [20] Maingé, P.E.: Extension of the hybrid steepest descent method to a class of variational inequalities and fixed point problems with nonself-mappings. Numer. Funct. Anal. Optim. 29(7–8), 820–834 (2008) · Zbl 1159.65005 · doi:10.1080/01630560802279371 [21] Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006) · Zbl 1095.47038 · doi:10.1016/j.jmaa.2005.05.028 [22] Kang, J., Su, Y., Zhang, X.: General iterative for nonexpansive semigroups and variational inequalities in Hilbert spaces. J. Inequal. Appl. (2010). doi: 101155/2010/264052 · Zbl 1184.49014 [23] Mohamedi, I.: Iterative methods for variational inequalities over the intersection of the fixed point set of a semigroup in Banach spaces. Fixed Point Theory Appl. (2011). doi: 10.1155/620284 · Zbl 1213.65081 · doi:10.1155/2011/187439 [24] Jung, J.S.: A general iterative scheme for k-strictly pseudo-contractive mappings and optimization problems. Appl. Math. Comput. 217(2), 5581–5588 (2011) · Zbl 1213.65080 · doi:10.1016/j.amc.2010.12.034 [25] Liu, X.: Cui, Y.: The common minimal-norm fixed point of a finite family of nonexpansive mappings. Nonlinear Anal. 73, 76–83 (2010) · Zbl 1214.47050 · doi:10.1016/j.na.2010.02.041 [26] Marino, G., Xu, H.K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–346 (2007) · Zbl 1116.47053 · doi:10.1016/j.jmaa.2006.06.055 [27] Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Math., vol. 28. Cambridge University Press, Cambridge (1990) · Zbl 0708.47031 [28] Suzuki, T.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 135, 99–106 (2007) · Zbl 1117.47041 · doi:10.1090/S0002-9939-06-08435-8 [29] Zhou, H.: Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 69, 456–462 (2008) · Zbl 1220.47139 · doi:10.1016/j.na.2007.05.032 [30] Andrews, H.C., Hunt, B.R.: Digital Image Restoration. Prentice Hall, Englewood Cliffs (1977) · Zbl 0379.62098 [31] Demoment, G.: Image reconstration and restoration: Overview of common estimation structures and problems. IEEE Trans. Acoust. Speech Signal Process., 37(12), 243–253 (1985) [32] Stark, H.: Image Recovery: Theory and Applications. Academic Press, San Diego (1987) · Zbl 0627.94001 [33] Combettes, P.L.: Convex set theoretic image recovery by extrapolated iterations of parallel subgradients projections. IEEE Trans. Image Process. 6, 493–506 (1997) · doi:10.1109/83.563316
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.