On solutions of differential inclusions in homogeneous spaces of functions
This paper is concerned with the following class of semilinear differential inclusions \[ x'(t)\in Ax(t)+f(t),\;t\in\mathbb R, \] where the multivalued linear operator \(A\in ML(E)\) (\(ML\) is the collection of all closed linear multivalued maps in \(E\)) and \(f\) belong to the homogeneous space \({\mathcal U}(\mathbb R,E)\). The theory of multivalued linear operators [see \textit{R. Cross}, Multivalued linear operators. Pure and Applied Mathematics, 213. New York, NY: Marcel Dekker (1998; Zbl 0911.47002)]. The fundamental ideas for the first result are based on the spectrum properties of \(U(1)\subset \mathbb C\backslash S\) (\(U(.)\) is a generalized \(C_0\)-semigroup and \(S\) is the unit circle), for the second result the authors prove that the conditions of Theorem 16 are necessary for the existence and uniqueness of mild solutions of the above problem in each specially homogeneous space, \(C(\mathbb R,E),\;L^p(\mathbb R,E), S_p(\mathbb R,E), 1\leq p<\infty, AP(\mathbb R,E), APS_p(\mathbb R,E).\) Also, the authors prove the existence and uniqueness of \(T\)-periodic mild solutions, where \(A\in ML(E)\) is a sectorial operator. An example of applications to feedback control systems governed by differential equations of Sobolev type in a Banach space is presented.
scientific article; zbMATH DE number 5073025
scientific article