\(L_p\)-Continuous selections of fixed points of~multifunctions with~ decomposable values. ~III. Applications
The article under review refers to the author's series [ibid. 40, No. 3, 695-709 (1999; Zbl 0924.54027); 40, No. 5, 1167-1181 (1999; Zbl 0930.54020), see the review above] which contains an abstract background for semilinear differential inclusions of evolution type. The author gives an extension of the theory of \(L_p\)-continuous selections for fixed points of multifunctions to studying semilinear evolution inclusions of the form \[ \dot x(t)\in A(\xi)x(t) + F(t,x(t),\xi), \] where \(A(\xi)\), \(\xi\in M\), is a family of densely defined linear operators such that, for every \(\xi\in M\), the operator \(A(\xi)\) generates a \(C_0\)-semigroup \(S(\xi)(t)\), \(t\in T\), of linear continuous operators acting in a separable Banach space \(X\), and \(F\) is a nonconvex multivalued map. The author introduces into consideration the concept of parametric solution and establishes relations between the concepts of parametric and usual solutions. He proves existence results not only for parametric solutions but also for continuous selections whose values are parametric solutions. Moreover, the proof of the so-called relaxation theorem is given for these selections and topological properties of the set of parametric selections are exposed such as the total retractness property, arcwise connectedness, and others. Similar properties are also discussed for usual solutions.
scientific article; zbMATH DE number 1396714
scientific article