Relaxed submonotone mappings (Q1864350)

The authors introduce the notions of relaxed monotonicity and of (strict) relaxed submonotonicity for set-valued operators. These notions are weaker than monotonicity and (strictly) submonotonicity, and coincide with them for single-valued operators. The authors show that Clarke subdifferentials of locally Lipschitz functions defined on a separable Banach space \(X\) are relaxed submonotone on a dense \(G_{\delta}\) subset of \(X\). They also prove that if an operator \(T:X\rightrightarrows X^{\ast}\) is strictly relaxed submotonone (respectively, relaxed submonotone) and \(\text{int dom} (T)\neq\emptyset\), then \(T\) is locally bounded on \(\text{int dom}(T)\) (respectively, on some residual subset of \(\text{int dom}(T)\)). On the other hand, relaxed monotone operators are locally bounded at the set of the absorbing points of their domain.