Document Type : Research Paper


1 Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq.

2 Department of mathematics, University of Mazandaran, Babolsar, Iran.


This paper aims at establishing the existence of results for a nonstandard equilibrium problems $(EP_{N})$. The solutions of this inequality are discussed in a subset $K$ (either bounded or unbounded) of a Banach spaces $X$. Moreover, we enhance the main results by application of some differential inclusion.


Main Subjects

[1] M. Ait Mansour, Z. Chbani, and H. Riahi, Recession bifunction and solvability of noncoercive equilibrium problems, Commun. Appl. Anal., 7, (2003), pp. 369 -377.
[2] M. Alimohammady and A.E. Hashoosh, Existence Theorems For $alpha(u,v)$-monotone of nonstandard Hemivariational Inequality, Advances in Math., 10, (2015), pp. 3205-3212.
[3] Q.H. Ansari and J.C. Yao, An existence result for the generalized vector equlibrium problem, Appl. Math. Lett., 19, (1999), pp. 53-56.
[4] M.Bianchi and S. Schaible, Equilibrium problems under generalized convexity and generalized monotonicity, J. Global Optim., 30, (2004), pp. 121-134.
[5] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63, (1994), pp. 123-145.
[6] F.E. Browder, The solvability of non-linear functional equations, Duke Math. J., 30, (1963), pp. 557-566.
[7] S. Carl, V. Khoi Le, and D. Motreanu, Nonsmooth variational problems and their inequalities, Springer Monographs in Mathematics, Springer, New York, (2007).
[8] O. Chadli, Y. Chiang, and S. Huang, Topological pseudomonotonicity and vector equilibm problems, J. Math. Anal. Appl., 270, (2002), pp. 435-450.
[9] K. Fan, A generalization of Tychonoffs fixed point theorem, Math. Ann., 142, (1961), pp. 305-310.
[10] Y.P. Fang and N.J. Huang, Variational-like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory Appl., 118, (2003), pp. 327-338.
[11] N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, optimization, 59, (2010), pp. 147-160.
[12] A.E. Hashoosh and M. Alimohammady, On Well-Posedness Of Generalized Equilibrium Problems Involving α-Monotone Bifunction, J. Hyperstruct., 5, (2016), pp. 151-168.
[13] A.E. Hashoosh and M. Alimohammady, Bα,β-Operator and Fitzpatrick Functions, Jordan J. Math. Stat., 1, (2017), pp. 259-278.
[14] A.E. Hashoosh, M. Alimohammady, and M.K. Kalleji, Existence Results for Some Equilibrium Problems involving α-Monotone Bifunction, Int. J. Math. Math. Sci., 2016, (2016), pp. 1-5.
[15] U. Kamraksa and R. Wangkeeree, Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces,  J. Global Optim., 51, (2011), pp. 689 -714.
[16] B. Knaster, K. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe, Fund. Math., 14, (1929), pp. 132-137.
[17] N.K. Mahato and C. Nahak, Mixed equilibrium problems with relaxed α-monotone mapping in Banach spaces, Rend. Circ. Mat. Palermo, (2013).
[18] J.W. Peng and J.Yao, A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings., Nonlinear Anal., 71, (2009), pp. 6001-6010.
[19] A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and equilibrium problem,  J. Optim. Theory Appl., 133, (2007), pp. 359-370.
[20] R.U. Verma, On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators, J. Math. Anal. Appl. , 213, (1997), pp. 387-392.
[21] R.U. Verma, On monotone nonlinear variational inequality problems, Comment. Math. Univ. Carolin., 39, (1998), pp. 91-98.