Equilibrium of maximal monotone operator in a given set (Q2717930)

Let \(C\) be a nonempty weakly compact convex subset of a real Banach space \(E\) and \(f: E\to\overline{\mathbb{R}}\) be a lower semi-continuous convex function. The question about the equality NEWLINE\[NEWLINE\inf_E f= \inf_C fNEWLINE\]NEWLINE can be characterized by the following two different conditions: NEWLINE\[NEWLINE\forall x\not\in C\;\exists c\in C:f'(x; c-x)\leq 0\qquad (\text{that means}:\;\langle x^*,c-x\rangle\leq 0\quad \forall x^*\in\partial f(x));\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\;\exists c\in C:0\in\partial f(c)\qquad (\text{that means}:\;(\text{graph }\partial f)\cap (C\times \{0\})\neq\emptyset).\tag{2}NEWLINE\]NEWLINE Since the subdifferential mapping \(\partial f\) is monotone, we can conclude immediately that (2) implies (1). Moreover, by the maximal monotonicity theorem of Rockafellar we get even the equivalence of both conditions.NEWLINENEWLINENEWLINEIn the present paper, the author gives some generalization of these results. In the first part, the subdifferential mapping \(\partial f\) is replaced by an arbitrary maximal monotone set-valued mapping \(T: E\Rightarrow E^*\) and relations between the conditions NEWLINE\[NEWLINE\forall(x,x^*)\in \text{graph }T\quad\exists c\in C:\langle x^*,c-x\rangle\leq 0;\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\exists c\in C: 0\in T(c).\tag{4}NEWLINE\]NEWLINE are discussed. In the second part, the compactness condition of the set \(C\) is replaced by a weaker condition and it is shown that in this case condition (2) turns to NEWLINE\[NEWLINE(\text{graph\;}\partial f)\cap ((C+ B(0,\varepsilon))\times B(0,\varepsilon))\neq \emptyset\;\forall \varepsilon\in 0,NEWLINE\]NEWLINE where \(B(0,\varepsilon)\) is the ball at the origin with the radius \(\varepsilon\).