The maximality of the sum of monotone operators in Banach space and an application to hemivariational inequalities (Q1353597)

Let \(X\) be a (not necessarily reflexive) Banach space of dual \(X^*\). Assume that \(f:X\to (-\infty,+\infty]\) is proper, convex and lower semicontinuous, and such that \(D(\partial f)\) is closed; therefore \(D(\partial f)= D(f) \). Here \(\partial f\) denotes the subdifferential of \(f\), and \(D(f)= \{x\in X: f(x)<\infty\}\). Finally, consider a single-valued monotone operator \(T: D(T)\subset X\to X^*\) with \(D(f)\subset D(T)\). The author shows that if, in addition, \(T\) is hemicontinuous on \(D(f)\), then the sum \(\partial f+T\) is maximal monotone. This extends a result by \textit{R. T. Rockafellar} [Trans. Am. Math. Soc. 149, 75-88 (1970; Zbl 0222.47017)], where \(f\) is the indicator function of a closed convex subset of \(X\). As an application, a hemivariational inequality is discussed.