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Some comments on: Nonlinear extended variational inequalities without differentiability: Applications and solution methods
Nonlinear Analysis 69 (2008) 2761–2762 www.elsevier.com/locate/na
Short communication
Some comments on: Nonlinear extended variational inequalities without differentiability: Applications and solution methods Muhammad Aslam Noor Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan Received 9 June 2007; accepted 12 June 2007
In a recent paper, Konnov [1] has considered and studied a class of variational inequalities, which is called Nonlinear Extended Variational Inequalities. We would like to point out that this class was introduced and studied by Noor [2] in 1988. Since then, this class of variational inequalities has been studied extensively and is well documented in the literature. To be more precise, let K be a closed convex set in a real Hilbert space whose norm and inner product are denoted by k.k and h., .i respectively. For given nonlinear operators G, H : K → H , find x • ∈ K such that
G(x • ), H (x) − H (x • ) ≥ 0, ∀x ∈ K . (1) Inequality of type (1) was introduced by Noor [2] in 1988 and it is known as general variational inequality. For recent developments, numerical methods, sensitivity analysis, dynamical systems and its applications in various branches of pure and applied sciences, see [2–12] and the references therein. It has been shown that the general variational inequalities include a class of quasivariational inequalities, implicit complementarity problems. It is worth mentioning that the problem (17) in Konnov [1] was also introduced by Noor [11] in 1988. In the light of these historical facts, we would like to mention that problem (1) actually is not a new one as claimed by Konnov [1]. The real credit for introducing such types of variational inequalities must be given to Noor [2]. It is quite strange that in spite of these extensive research activities, the author has failed to cite the important relevant references. All these papers have appeared in leading and high quality journals of mathematical and engineering sciences. In view of these facts and comments, proper credit must be given to Noor. We would appreciate it very much if you could please take appropriate action to rectify this error and publish these up-to-date references for the sake of historical record. For further reading, see [1–5]. References [1] I.V. Konnov, Nonlinear extended variational inequalities without differentiability: Applications and solution methods, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.04.035. [2] M. Aslam Noor, General variational inequalities, Appl. Math. Lett. 1 (1988) 119–121. [3] M. Aslam Noor, General algorithms and sensitivity analysis of variational inequalities, J. Appl. Math. Stoch. Anal. 5 (1992) 29–42. [4] M. Aslam Noor, K. Inayat Noor, T.M. Rassias, Some aspects of variational inequalities, J. Comput. Appl. Math. 47 (1993) 285–312. [5] M. Aslam Noor, Wiener–Hopf equations and variational inequalities, J. Optim. Theory Appl. 79 (1993) 197–206. [6] M. Aslam Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000) 217–229.
E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.06.002
2762 [7] [8] [9] [10] [11] [12]
M.A. Noor / Nonlinear Analysis 69 (2008) 2761–2762 M. Aslam Noor, New extragradient methods for general variational inequalities, J. Math. Anal. Appl. 277 (2003) 379–395. M. Aslam Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152 (2004) 199–277. M. Aslam Noor, Merit functions for general variational inequalities, J. Math. Anal. Appl. 316 (2006) 736–752. M. Aslam Noor, A. Bnouhachem, On an iterative algorithm for general variational inequalities, Appl. Math. Comput. 185 (2007) 155–168. M. Aslam Noor, Quasi variational inequalities, Appl. Math. Lett. 1 (1988) 367–370. N. Xiu, J. Zhang, M. Aslam Noor, Tangent equations and general variational inequalities, J. Math. Anal. Appl. 258 (2001) 755–762.
Further reading [1] M. Aslam Noor, Th.M. Rassias, A class of projection methods for general variational inequalities, J. Math. Anal. Appl. 268 (2002) 334–343. [2] M. Aslam Noor, Multivalued general equilibrium problems, J. Math. Anal. Appl. 283 (2003) 140–149. [3] M. Aslam Noor, K. Inayat Noor, Self-adaptive projection methods for general variational inequalities, Appl. Math. Comput. 151 (2004) 659–670. [4] M. Aslam Noor, General variational inequalities and nonexpansive mappings, J. Math. Anal. Appl. 331 (2007) 810–822. [5] Y. Yao, M. Aslam Noor, On modified hybrid steepest-descent methods for general variational inequalities, J. Math. Anal. Appl. 334 (2007) 1276–1289.
Short communication
Some comments on: Nonlinear extended variational inequalities without differentiability: Applications and solution methods Muhammad Aslam Noor Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan Received 9 June 2007; accepted 12 June 2007
In a recent paper, Konnov [1] has considered and studied a class of variational inequalities, which is called Nonlinear Extended Variational Inequalities. We would like to point out that this class was introduced and studied by Noor [2] in 1988. Since then, this class of variational inequalities has been studied extensively and is well documented in the literature. To be more precise, let K be a closed convex set in a real Hilbert space whose norm and inner product are denoted by k.k and h., .i respectively. For given nonlinear operators G, H : K → H , find x • ∈ K such that
G(x • ), H (x) − H (x • ) ≥ 0, ∀x ∈ K . (1) Inequality of type (1) was introduced by Noor [2] in 1988 and it is known as general variational inequality. For recent developments, numerical methods, sensitivity analysis, dynamical systems and its applications in various branches of pure and applied sciences, see [2–12] and the references therein. It has been shown that the general variational inequalities include a class of quasivariational inequalities, implicit complementarity problems. It is worth mentioning that the problem (17) in Konnov [1] was also introduced by Noor [11] in 1988. In the light of these historical facts, we would like to mention that problem (1) actually is not a new one as claimed by Konnov [1]. The real credit for introducing such types of variational inequalities must be given to Noor [2]. It is quite strange that in spite of these extensive research activities, the author has failed to cite the important relevant references. All these papers have appeared in leading and high quality journals of mathematical and engineering sciences. In view of these facts and comments, proper credit must be given to Noor. We would appreciate it very much if you could please take appropriate action to rectify this error and publish these up-to-date references for the sake of historical record. For further reading, see [1–5]. References [1] I.V. Konnov, Nonlinear extended variational inequalities without differentiability: Applications and solution methods, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.04.035. [2] M. Aslam Noor, General variational inequalities, Appl. Math. Lett. 1 (1988) 119–121. [3] M. Aslam Noor, General algorithms and sensitivity analysis of variational inequalities, J. Appl. Math. Stoch. Anal. 5 (1992) 29–42. [4] M. Aslam Noor, K. Inayat Noor, T.M. Rassias, Some aspects of variational inequalities, J. Comput. Appl. Math. 47 (1993) 285–312. [5] M. Aslam Noor, Wiener–Hopf equations and variational inequalities, J. Optim. Theory Appl. 79 (1993) 197–206. [6] M. Aslam Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000) 217–229.
E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.06.002
2762 [7] [8] [9] [10] [11] [12]
M.A. Noor / Nonlinear Analysis 69 (2008) 2761–2762 M. Aslam Noor, New extragradient methods for general variational inequalities, J. Math. Anal. Appl. 277 (2003) 379–395. M. Aslam Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152 (2004) 199–277. M. Aslam Noor, Merit functions for general variational inequalities, J. Math. Anal. Appl. 316 (2006) 736–752. M. Aslam Noor, A. Bnouhachem, On an iterative algorithm for general variational inequalities, Appl. Math. Comput. 185 (2007) 155–168. M. Aslam Noor, Quasi variational inequalities, Appl. Math. Lett. 1 (1988) 367–370. N. Xiu, J. Zhang, M. Aslam Noor, Tangent equations and general variational inequalities, J. Math. Anal. Appl. 258 (2001) 755–762.
Further reading [1] M. Aslam Noor, Th.M. Rassias, A class of projection methods for general variational inequalities, J. Math. Anal. Appl. 268 (2002) 334–343. [2] M. Aslam Noor, Multivalued general equilibrium problems, J. Math. Anal. Appl. 283 (2003) 140–149. [3] M. Aslam Noor, K. Inayat Noor, Self-adaptive projection methods for general variational inequalities, Appl. Math. Comput. 151 (2004) 659–670. [4] M. Aslam Noor, General variational inequalities and nonexpansive mappings, J. Math. Anal. Appl. 331 (2007) 810–822. [5] Y. Yao, M. Aslam Noor, On modified hybrid steepest-descent methods for general variational inequalities, J. Math. Anal. Appl. 334 (2007) 1276–1289.