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

Abstract

<abstract><p>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 <sup>[<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b20">20</xref>]</sup>. 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 <sup>[<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b20">20</xref>]</sup> 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.</p></abstract>