عملگرهای یکنوای تعمیم یافته و رویکرد قطبیِ مجموعه های یکنوای تعمیم‌یافته

ابتدا  نامساوی فنچل- موراِ  به توابع σ-محدب تعمیم داده می شود و سپس با استفاده از تابع فیتزپاتریک تعمیم یافته، تظریفی برای نامساوی فنشل- موراِ تعمیم یافته ارائه می شود. در ادامه، قطبی مجموعه های σ -یکنوا معرفی شده و نتایج مرتبط با آن مورد مطالعه قرار می گیرد.

generalized monotone operators and polarity approach to generalized monotone sets

introductionmany suppose that is a banach space with topological dual space we will denote by  the duality pairing between x and  . for , we denote by the boundary points of ω and by  the interior of ω. also we will denote by  the real nonnegative numbers. let be a set-valued map from  to . the domain and graph of  are, respectively, defined bywe recall that a set valued operator is monotone if   for all  and   for two multivalued operators  and we write  ifis an extension of , i.e., . a monotone operator is called maximal monotone if it has no monotone extension other than itself. in 1988, the fitzpatrick function of a monotone operator was introduced by fitzpatrick. the fitzpatrick function makes a bridge between the results of convex functions and results on maximal monotone operators. for a monotone operator , its fitzpatrick function is defined by                           it is a convex and norm to weak lower semicontinuous and function. let be an extended real-valued function. its effective domain is defined by  the function is called proper if . let  be a proper function. the subdifferential (in the sense of convex analysis) of  at  is defined by                      given a proper function  and a map , then  is called -convex if          for all and for all given an operator  and a map . then is called -monotone if for all  and  we have                             also is called maximal -monotone if it has no -monotone extension other than itself. we recall for a proper function  the -subdifferential of  at  is defined by        and  if .  main results  the definition we use for the fitzpatrick function is the same as for monotone operators.assume that is a proper -convex function its conjugate is defined by                                first we have the following refinement of the fenchel-moreau inequality:                     where is the indicator function and .also we have the following refinement, when is a proper, -convex and lower semicontinuous function and  is a maximal -monotone operator:                                 moreover, we approach generalized monotonicity from the point of view of the classical concept of polarity. besides, we introduce and study the notion of generalized monotone polar of a set a. moreover, we find some equivalent relations between polarity and maximal generalized monotonicity.