Sets in the ranges of nonlinear accretive operators in Banach spaces

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 $M ∩ \overline{T(∂ G)} = ∅$. Then $M ⊂ \overline{TG}$. If, moreover, Case (i) or (ii) holds and T is of type $(S_1)$, or Case (iii) holds and T is of type $(S_2)$, then M ⊂ TG. Various results of Morales, Reich and Torrejón, and the author are improved and/or extended.