Abstract
The purpose of this paper is to suggest and analyze a number of iterative algorithms for solving the generalized set-valued variational inequalities in the sense of Noor in Hilbert spaces. Moreover, we show some relationships between the generalized set-valued variational inequality problem in the sense of Noor and the generalized set-valued Wiener-Hopf equations involving continuous operator. Consequently, by using the equivalence, we also establish some methods for finding the solutions of generalized set-valued Wiener-Hopf equations involving continuous operator. Our results can be viewed as a refinement and improvement of the previously known results for variational inequality theory.
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He, J.H.: Variational iteration method—Some recent results and new interpretations. J. Comput. Appl. Math. 207(1), 3–17 (2007)
Fang, S.C., Peterson, E.L.: Generalized variational inequalities. J. Optim. Theory Appl. 38, 363–383 (1982)
Hemeda, A.A.: Variational iteration method for solving wave equation. Comput. Math. Appl. 56(8), 1948–1953 (2008)
Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Dekker, New York (1995)
Noor, M.A.: General variational inequalities. Appl. Math. Lett. 1, 119–121 (1988)
Noor, M.A.: Wiener-Hopf equations and variational inequalities. J. Optim. Theory Appl. 79, 197–206 (1993)
Noor, M.A.: Generalized set-valued variational inequalities. Le Matematiche (Catania) 52, 3–24 (1997)
Noor, M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)
Noor, M.A., Al-Said, E.A.: Iterative methods for generalized nonlinear variational inequalities. Comput. Math. Appl. 33(8), 1–11 (1997)
Noor, M.A., Al-Said, E.A.: Wiener-Hopf equations technique for quasimonotone variational inequalities. J. Optim. Theory Appl. 103(3), 705–714 (1999)
Pitonyak, A., Shi, P., Shiller, M.: On an iterative method for variational inequalities. Numer. Math. 58, 231–242 (1990)
Qin, X., Noor, M.A.: General Wiener-Hopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces. Appl. Math. Comput. 201(1–2), 716–722 (2008)
Robinson, S.M.: Normal maps induced by linear transformations. Math. Oper. Res. 17, 691–714 (1992)
Robinson, S.M.: Sensitivity analysis of variational inequalities by normal maps. In: Giannessi, F., Maugeri, A. (eds.) Variational Inequalities and Network Equilibrium Problems. Plenum, New York (1995)
Rus, I.A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001)
Shi, P.: Equivalence of variational inequalities with Wiener-Hopf equations. Proc. Am. Math. Soc. 111, 339–346 (1991)
Speck, F.O.: General Wiener-Hopf Factorization Methods. Pitman Advanced Publishing Program, London (1985)
Stampacchia, G.: Formes bilineaires coercitives surles ensembles conveys. C. A. Acad. Sci. Paris 258, 4413–4416 (1964)
Wazwaz, A.M.: The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations. Comput. Math. Appl. 54(7–8), 933–939 (2007)
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This work was supported by The Thailand Research Fund (Project No. MRG5180178) and Faculty of Science, Naresuan University, Thailand.
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Petrot, N. Existence and algorithm of solutions for general set-valued Noor variational inequalities with relaxed (μ,ν)-cocoercive operators in Hilbert spaces. J. Appl. Math. Comput. 32, 393–404 (2010). https://doi.org/10.1007/s12190-009-0258-1
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DOI: https://doi.org/10.1007/s12190-009-0258-1
Keywords
- General set-valued Noor variational inequality problem
- The generalized set-valued Wiener-Hopf equations involving continuous operator
- Relaxed (μ,ν)-cocoercive set-valued operator