Characterization of Approximate Monotone Operators

Results concerning local boundedness of operators have a long history. In 1994, Vesely connected the concept of approximate ´ monotonicity of an operator with local boundedness of that. It is our desire in this note to characterize an approximate monotone operator. Actually, we show that a well-known property of monotone operators, namely representing by convex functions, remains valid for the larger subclass of operators. In this general framework we establish the similar results by Fitzpatrick. Also, celebrated results of Mart´inez-Legaz and Th´era inspired us to prove that the set of maximal ε-monotone operators between a normed linear space X and its continuous dual X ∗ can be identified as some subset of convex functions on X × X ∗ .