Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators

In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator $S(\cdot + p)+T(\cdot)$, where $p\in X$ and $S$ and $T$ are maximal monotone operators on the reflexive Banach space $X$. Then, this is used to obtain sufficient conditions for the surjectivity of $S+T$ and for the situation when $0$ belongs to the range of $S+T$. Several special cases are discussed, some of them delivering interesting byproducts.