In this note we deal with a Cauchy problem governed by the following semilinear evolution differential inclusion $$ x^\prime(t)\in A(t)x(t)+F(t,x(t)) $$ and with initial data $x(0)=x_0 \in E$, where $\{A(t)\}_{t\in [0,d]}$ is a family of linear operators in the Banach space $E$ generating an evolution operator and $F$ is a Carathèodory type multifunction. We prove the existence of local and global mild solutions of the problem. Moreover, we obtain the compactness of the set of all global mild solutions. In order to obtain these results, we define a generalized Cauchy operator. Our existence theorems respectively contain the analogous results provided by Kamenskii, Obukhovskii and Zecca for inclusions with constant operator.
On the existence of mild solutions of semilinear evolution differential inclusions
CARDINALI, Tiziana;RUBBIONI, Paola
2005
Abstract
In this note we deal with a Cauchy problem governed by the following semilinear evolution differential inclusion $$ x^\prime(t)\in A(t)x(t)+F(t,x(t)) $$ and with initial data $x(0)=x_0 \in E$, where $\{A(t)\}_{t\in [0,d]}$ is a family of linear operators in the Banach space $E$ generating an evolution operator and $F$ is a Carathèodory type multifunction. We prove the existence of local and global mild solutions of the problem. Moreover, we obtain the compactness of the set of all global mild solutions. In order to obtain these results, we define a generalized Cauchy operator. Our existence theorems respectively contain the analogous results provided by Kamenskii, Obukhovskii and Zecca for inclusions with constant operator.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
