### Vector variational-like inequalities with \(\eta\)-generally convex mappings

#### Abstract

Because of applications in optimization problems, mathematical programming, equilibrium problem and operations research, considerable progresses have been achieved in both theory and applications of vector variational-like inequalities. In this work, we consider vector variational-like inequalities with \(\eta\)-generally convex mappings and prove some existence results for our inequalities in the setting of Hausdorff topological vector space. The results presented in this article are more general and can be used to solve many known problems related to vector variational inequalities, variational-like inequalities and vector variational-like inequalities.

**References**- G. Y. Chen and X.Q. Yang, Vector complementarity problems and its equivalences with weak minimal element in ordered spaces,
*J. Math. Anal. Appl.***153**(1990), 136--158. - F. Giannessi,
*Theorems of Alternative, Quadratic Programs and Complementarity Problems*, Edited by R. W. Cottle, F. Giannessi and J. L. Lions, John Wiley and Sons, Chichester, 151--186, 1980. - P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations,
*Acta Math.***115**(1966), 271--310. - I. V. Konnov and J. C. Yao, On the generalized vector variational inequality problem,
*J. Math. Anal. Appl.***206**(1997), 42--58. - B. S. Lee and G. M. Lee, A vector version of Minty's lemma and application,
*Appl. Math. Lett.***12**(5)(1999), 43--50. - G. M. Lee, D. S. Kim, B. S. Lee and N. D. Yen, Vector variational inequality as a tool for studying vector optimization problems,
*Nonlinear Analysis***34**(1998), 745--765. - G. M. Lee, B. S. Lee and S. S. Chang, On vector quasi-variational inequalities,
*J. Math. Anal. Appl.***203**(1996), 626--638. - B. S. Lee, G. M. Lee and D. S. Kim, Generalized vector variational-like inequalities on locally convex Hausdorff topological vector spaces,
*Indian J. Pure Appl. Math.***28**(1997), 33--41. - S. Park, B. S. Lee and G. M. Lee, A general vector-valued variational inequality and its fuzzy extension, Internat. J. Math. and Math. Sci.
**21**(4)(1998), 637--642. - S. K. Mishra and M. A. Noor, On vector variational-like inequality problems,
*J. Math. Anal. Applics.***311**(1)(2005), 69--75. - Q. H. Ansari, On generalized vector variational-like inequalities,
*Ann. Sci. Math. Quebec***19**(1995), 131--137. - A. H. Siddiqi, Q. H. Ansari and R. Ahmad, On vector variational-like inequalities,
*Indian J. Pure Appl. Math.***28**(8)(1997), 1009--1016. - A. H. Siddiqi, Q. H. Ansari and A. Khaliq, On vector variational inequalities,
*J. Optim. Theory Appl.***84**(1)(1995), 171--180. - X. Q. Yang, Vector variational inequality and its duality,
*Nonlinear Analysis, T. M. A.***21**(1993), 869--877. - S. J. Yu and J. C. Yao, On vector variational inequalities,
*J. Optim. Theory Appl.***89**(1996), 749--769. - X. M. Yang and X. Q. Yang, Vector variational-like inequality with pseudoinvexity,
*Optimization***55**(1-2)(2006), 157--170. - Yali Zhao and Zunquan Xia, On the existence of solutions to generalized vector variational-like inequalities,
*Nonlinear Analysis***64**(9)(2006), 2075--2083. - G. Y. Chen, Existence of solution for a vector variational inequality; An extension of the Hartman-Stampacchia theorem,
*J. Optim. Theory Appl.***74**(1992), 445--456. - G. Y. Chen and G.H. Cheng,
*Vector Variational Inequality and Vector Optimization Problem*, In*Lecture Notes in Economics and Mathematical Systems*, volume 258, Springer--Verlag, (1987). - G. Y. Chen and B.D. Craven, A vector variational inequality and optimization over an efficient set,
*Zeitscrift fur Operations Research***3**(1990), 1--12.

DOI: http://dx.doi.org/10.21914/anziamj.v49i0.201

**Remember,**for most actions you have to record/upload into this online system

**and then**inform the editor/author via clicking on an email icon or Completion button.

ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.