Variational Inequalities on 2-Inner Product Spaces

Document Type : Original Article

Authors
Department of Mathematics, Shahid Beheshti University, Tehran, Iran
Abstract
We introduce variational inequality problems on 2-inner product spaces and prove several existence results for variational inequalities defined on closed convex sets. Also, the relation between variational inequality problems, best approximation problems and fixed point theory is studied. 

Keywords


[1] M. Abrishami-Moghaddam, T. Sistani, Best approximation for convex subsets of 2- inner product spaces, Journal of Nonlinear Analysis and Optimization, 3(2): pp.269–278, (2012).
[2] M. Acikgoz, Some results best and 2-best approximation on 2-structures, Mathematica Morica, 13: pp. 1–11, (2009).
[3] C. Baiocchi and A. capello, variational and quasi-variational Inequalities. Applications to free boundary problems, J. wily and sons, New York, 1984.
[4] Y.J. Cho, C.S. Lin, S.S. Kim and A. Misiak, Theory of 2-Inner Product Spaces, Nova Science Publishers, Inc., New York, 2001.
[5] C. Diminnie, S. Gahler and A.White, 2-inner product spaces, Demonstratio Math. 6: 525–536, (1973).
[6] S. Elumalai, R. Vijayaragavan, Characterizations of best approximations in linear 2-normed spaces, General Mathematics Vol. 17(1): pp. 141–160, (2009).
[7] K. Fan, Some properties of convex sets to fixed point theorem, Math. Ann. 266: pp. 519–537, (1984).
[8] R.W. Freese and Y.J. Cho, Geometry of Linear 2-Normed Spaces, Nova Science Publishers, Inc., New York, 2001.
[9] S. Gahler, Lineare 2-normierte raume. Math. Nachr. 28:: 1–43, (1964).
[10] Ph. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta math. 115: pp. 271–310, (1966).
[11] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic press, New York, 1980.
[12] H. Mazaheri, R. Kazemi, Some results on 2-inner product spaces, Novi Sad J. Math., 37(2): pp. 35–40, (2007).
[13] G.J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. 29, pp.341–346, (1964).
[14] M.A. Noor, Generel variational inequalities, Appl. Math. lett. 1: pp. 119–121, (1988).
[15] M.A. Noor, Extended general variational inequalities, Appl. Math. lett. 22: pp. 182–186, (2009).
[16] Sh. Rezapour, Proximinal subspaces of 2-normed spaces, Ana. Theory Appl. 22(2): pp. 114–119, (2006).
[17] S.M. Robinson, Nonsingularity and symmetry for linear normal maps, Math. Programming, 62: pp. 415–425, (1993).
[18] T. Sistani, M. Abrishami-Moghaddam, Some results on best approximation in convex subsets of 2-normed spaces, Int.J.Math.analysis,21(3): pp. 1043–1049, (2009).