Let X be a real Banach space and G ⊂ X open and bounded. Assume that one of the following conditions is satisfied:
(i) X* is uniformly convex and T:Ḡ→ X is demicontinuous and accretive;
(ii) T:Ḡ→ X is continuous and accretive;
(iii) T:X ⊃ D(T)→ X is m-accretive and Ḡ ⊂ D(T).
Assume, further, that M ⊂ X is pathwise connected and such that M ∩ TG ≠ ∅ and M∩T(∂G)=∅. Then M⊂TG. If, moreover, Case (i) or (ii) holds and T is of type (S1), or Case (iii) holds and T is of type (S2), then M ⊂ TG. Various results of Morales, Reich and Torrejón, and the author are improved and/or extended