Generalized (f,λ)-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces

Abstract

In this paper, we expanded from the convex case to the nonconvex case in the setting of reflexive smooth Banach spaces, the concept of the $ f $-generalized projection $ \pi^{f}_S:X^*\to S $ initially introduced for convex sets and convex functions in [19,20]. Indeed, we defined the $ (f, \lambda) $-generalized projection operator $ \pi^{f, \lambda}_S:X^*\to S $ from $ X^* $ onto a nonempty closed set $ S $. We proved many properties of $ \pi^{f, \lambda}_S $ for any closed (not necessarily convex) set $ S $ and for any lower semicontinuous function $ f $. Our principal results broaden the scope of numerous theorems established in [19,20] from the convex setting to the nonconvex setting. An application of our main results to solutions of nonconvex variational problems is stated at the end of the paper.

Keywords

• $ f $-generalized projection
• $ (f， \lambda) $-generalized projection
• $ p $-uniformly convex banach spaces
• $ q $-uniformly smooth banach spaces
• nonconvex variational problem
• uniformly generalized prox-regular