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.
We study in the space of continuous functions defined on [0,T] with values in a real Banach space E the periodic boundary value problem for abstract inclusions of the form ⎧ $x ∈ S(x(0), sel_{F}(x))$ ⎨ ⎩ x (T) = x(0), where, $F:[0,T] × 𝓚 → 2^E \∅$ is a multivalued map with convex compact values, 𝓚 ⊂ E, $sel_{F}$ is the superposition operator generated by F, and S: 𝓚 × L¹([0,T];E) → C([0,T]; 𝓚 ) an abstract operator. As an application, some results are given to the periodic boundary value problem for nonlinear differential inclusions governed by m-accretive operators generating not necessarily a compact semigroups.
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