A convex-analytical approach to extension results for \(n\)-cyclically monotone operators (Q2460927)

Let \((X, \langle \cdot, \cdot \rangle)\) be a real Hilbert space. A subset \(G\) of \(X\times X\) is said to be \(n\)-monotone if \((a_1, a_1^*),\dots, (a_{n},a_{n}^*) \in G\Rightarrow \sum_{i=1}^{n} \langle a_{i+1}-a_i,a_i^* \rangle \leq 0\), where \(a_{n+1}=a_1\). A multivalued operator \(A:X\rightarrow 2^X\) is called \(n\)-monotone if its graph is \(n\)-monotone. The function \(F_{A,n}:X\times X\rightarrow [-\infty, +\infty]\), defined by \(F_{A,n}(x,x^*)= \sup\{\sum_{i=1}^{n-2} \langle a_{i+1}-a_i,a_i^* \rangle +\langle x-a_{n-1}, a_{n-1}^*\rangle+\langle a_1,x^*\rangle: (a_i,a_i^*)\in \operatorname{Gr} A\), \(1\leq i\leq n-1 \}\), is called the Fitzpatrick function of order \(n\) associated to the multivalued operator \(A\). For \(f:X\times X\rightarrow [-\infty, +\infty]\), let \(f^*\) be the Fenchel conjugate of \(f\). The authors prove the following result: If \(A:X\rightarrow 2^X\) is an \(n\)-monotone multivalued operator, with nonempty graph, satisfying \(\langle a, a^*\rangle \leq F_{A,n}^* (a, a^*)\) for all \((a, a^*)\in X\times X\), then for each \(w\in X\) there exists \(x\in \overline{\operatorname{conv}} \operatorname{dom} A\) such that \(\{(x, w-x)\}\cup \operatorname{Gr} A\) is \(n\)-monotone. This result provides a new proof, more elementary and rooted in convex analysis, for Theorem~3.1 in [\textit{M. D. Voisei}, Stud. Cercet. Ştiinţ., Ser. Mat., Univ. Bacău 9, 235--242 (1999; Zbl 1055.47510)].