Some Problems Of Generalized Convexity And Generalized Monotonicity

The generalized monotonicity have compactly connection with the generalizedconvexity and the variational inequality problems. In this paper, three problems aremainly considered. Chapter 1 discuss the generalized convexity and the generalizedmonotonicity of differentiable function. Firstly, on the base of the semistrictlyquasimonotone map, we provide the definition of semitrictly invariantquasimonotone map, relationship between semitrict prequasi-invexity and semistrictinvariant quasimonotonicity are established. Secondly, a gradient property ofsemistrictly preinvex functions is given. Chapter 2 discuss the generalized convexityand the generalized monotonicity of set-valued maps of non-differentiable functions,we mainly study the relations of strictly prequasiinvexity with clarke subdifferentiablestrictly invariant quasimonotonicity and semistrictly prequasinvexity with clarkesubdifferentiable semistrictly invariant quasimonotonicity. Chapter 3 discuss theapplications of generalized convexity and generalized monotonicity in vectorvariational-like inequality. Under the meaning of clarke subdifferentiable, we discussthe applications of pseudoinvexity and invariant pseudomonotonicity in Mintyvector variational-like inequality.