# American Institute of Mathematical Sciences

October  2008, 4(4): 727-738. doi: 10.3934/jimo.2008.4.727

## Some characterizations and applications on strongly $\alpha$-preinvex and strongly $\alpha$-invex functions

 1 College of Mathematics Science, Inner Mongolia University, Hohhot, 010021, China

Received  October 2007 Revised  July 2008 Published  November 2008

In this paper, a new class of generalized convex functions is introduced, which is called the strongly quasi $\alpha$-preinvex functions. Some properties of strongly $\alpha$-preinvex functions are studied. We establish the relationships among the strongly quasi $\alpha$-preinvex functions, strongly quasi $\alpha$-invex functions and strongly quasi $\alpha\eta$-monotonicity under some suitable conditions. As applications, a class of perturbed variational-like inequality problems is introduced, some relationships between the perturbed variational-like inequality and optimization problems are established under the assumptions of strongly $\alpha$-invex functions.
Citation: Cai-Ping Liu. Some characterizations and applications on strongly $\alpha$-preinvex and strongly $\alpha$-invex functions. Journal of Industrial & Management Optimization, 2008, 4 (4) : 727-738. doi: 10.3934/jimo.2008.4.727
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