Some Kinds Of Generalized Invex Functions And Their Properties

In this thesis,the new concepts of generalized invex functions,generalized preinvex functions and generalized invariant monotone operators are introduced.The relationships among generalized convexity,generalized invexity and generalized invariant monotonicity are discussed.The applications in mathematical programming and variational inequality on these new generalized convexities and generalized monotonicities are studied.The thesis consists of five parts as following:In part 1,a new class of generalized convex functions in Banach space,termed semistrictly invex functions,is introduced.Under the condition of locally Lipschitz,the characterizations of semistrictly invex functions corresponding Clarke's subdifferential are given.The relationships are investigated between semistrictly invex functions and invex functions,between semistrictly invex functions and semistrictly preinvex functions.In part 2,the concept of semistrictly quasi-invex functions is introduced in Banaeh space.We give examples to illustrate that semistrictly quasi-invex functions are more wider than quasi-invex functions and pseudo-invex functions.The properties of nonsmooth semistrictly quasi-invex functions for corresponding Clarke's subdifferential are discussed.The relationships are investigated between semistrictly quasi-invex functions and quasi-invex functions.In part 3,two new classes of generalized convex functions,termed strongly prequasi -invex functions and strongly quasi-invex functions,are introduced.The interesting relationships are investigated between strong prequasi-invex functions and strongly quasiinvex functions,between strong quasi-invex functions and strongly pseudoinvex functions. Finally,Strongly prequasi-invex functions are used in the study of multiobjective optimization problems.In part 4,the new concepts of generalized(ρ,θ)-ηinvex functions and generalized (ρ,θ)-ηinvariant monotone operators are introduced.The relationships between generalized (ρ,θ)-ηinvexity of functions and generalized(ρ,θ)-ηinvariant monotonicity of corresponding Clarke's subdifferential are studied.We also introduce the concept of strongly(ρ,θ)-ηinvariant pseudomonotone operators and give its necessary conditions.The results obtained in this part can be viewed as a refinement and improvement of the results in[30].In part 5,the concepts of strongly quasiα-preinvex functions and strongly quasi aη-monotone operators are introduced.We establish the relationships among the strongly quasiα-preinvex functions,strongly quasiα-invex functions and strongly quasiαη-monotone operators under some suitable conditions.Some properties of stronglyα-preinvex functions are obtained.In particular,a class of perturbed variational-like inequality problems is introduced.Some relationships are established between the perturbed variational-like inequality and optimization problems under the assumptions of stronglyα-invex functions.