Composite Generalized Variational Inequalities With Wiener-Hopf Equations

Zubair Khan, Syed Shakaib Irfan, Iqbal Ahmad, Preeti Shukla


An introduction and study of a composite generalized variational inequality problem with a composite Wiener-Hopf equation in separable real Hilbert space is performed. Projection operator technique has been employed, to establish the equivalence between the composite generalized variational inequality problem with a composite Wiener-Hopf equation. Equivalent formulation discuss the existence of solution of the problem. Under some specific conditions, the convergence analysis of the suggested iterative algorithm has been discussed. The paper also specks that the problem and results obtained are more general than the papers that are already available in the literature.


Algorithms; Composite variational inequalities; Composite Wiener-Hopf equation; Convergence analysis; Monotone operators

Full Text:



N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, (Series: Studies in Applied and Numerical Mathematics), SIAM Publishing Co., Philadelphia (1988), DOI: 10.1137/1.9781611970845.

L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, Journal of Mathematical Analysis and Applications 194 (1995), 114 – 125, DOI: 10.1006/jmaa.1995.1289.

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York (1995)..

M. A. Noor, General algorithm and sensitivity analysis for variational inequalities, Journal ofApplied Mathematics and Stochastic Analysis 5(1) (1992), 29 – 42,

M. A. Noor, General variational inequalities, Applied Mathematics Letters 1(2) (1988), 119 – 122, DOI: 10.1016/0893-9659(88)90054-7.

M. A. Noor, Some developments in general variational inequalities, Applied Mathematics and Computation 152(1) (2004), 199 – 277, DOI: 10.1016/S0096-3003(03)00558-7.

M. A. Noor, Wiener Hopf equations and variational inequalities, Journal of Optimization Theory and Applications volume 79 (1993), DOI: 10.1007/BF00941894.

M. A. Noor, Y. J.Wang and N. Xiu, Some new projection methods for variational inequalities, Applied Mathematics and Computation 137 (2003), 423 – 435, DOI: 10.1016/S0096-3003(02)00148-0.

X. Qin and M. Shang, Generalized variational inequalities involving relaxed monotone and nonexpansive mappings, Journal of Inequalities and Applications 2007 (2008), Article number 020457, DOI: 10.1155/2007/20457.

J. Shen and L. P. Pang, An approximate bundle method for solving variational inequalities, Communications in Optimization Theory 1 (2012), 1 – 18,

P. Shi, Equivalence of variational inequalities with Wiener-Hopf equations, Proceedings of the American Mathematical Society 111 (1991), 339 – 346, DOI: 10.1090/S0002-9939-1991-1037224-3.

G. Stampacchia, Formes Bilinéaires Coercitives sur les Ensembles Convexes, Comptes Rendus de l’Academie des Sciences, Paris, 258 (1964), 4413 – 4416.

R. U. Verma, Generalized variational inequalities involving multivalued relaxed monotone operators, Appl. Math. Lett. 10 (1997), 107 – 109,

C. Wu and Y. Li, Wiener Hopf equations techniques for generalized variational inequalities and fixed point problems, 4th International Congress on Image and Signal Processing IEEE 2011, 2802 – 2805, (2011), DOI: 10.1109/CISP.2011.6100758.

C. Wu, Wiener-Hopf equations methods for generalized variational equations, Journal of Nonlinear Functional Analysis 2013 (2013), Article ID 3, 1 – 10,

H. Zegeye and N. Shahzad, A hybrid approximations method for equilibrium, variational inequality and fixed point problems, Nonlinear Analysis: Hybrid Systems 4 (2010), 619 – 630, DOI: 10.1016/j.nahs.2010.03.005.

H. Zhou, Convergence theorems of fixed points for ·-strict pseudo-contraction in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications 69 (2008), 456 – 462, DOI: 10.1016/



  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905