Operator inclusions and operator-differential inclusions
In Chapter 2, we first introduce a generalized inverse differentiability for set-valued mappings and consider some of its properties. Then, we use this differentiability, Ekeland's Variational Principle and some fixed point theorems to consider constrained implicit function and open mapping theorems and surjectivity problems of set-valued mappings. The mapping considered is of the form F(x, u) + G (x, u). The inverse derivative condition is only imposed on the mapping x F(x, u), and the mapping x G(x, u) is supposed to be Lipschitz. The constraint made to the variable x is a closed convex cone if x F(x, u) is only a closed mapping, and in case x F(x, u) is also Lipschitz, the constraint needs only to be a closed subset. We obtain some constrained implicit function theorems and open mapping theorems. Pseudo-Lipschitz property and surjectivity of the implicit functions are also obtained. As applications of the obtained results, we also consider both local constrained controllability of nonlinear systems and constrained global controllability of semilinear systems. The constraint made to the control is a time-dependent closed convex cone with possibly empty interior. Our results show that the controllability will be realized if some suitable associated linear systems are constrained controllable. In Chapter 3, without defining topological degree for set-valued mappings of monotone type, we consider the solvability of the operator inclusion y0 N1(x) + N2 (x) on bounded subsets in Banach spaces with N1 a demicontinuous set-valued mapping which is either of class (S+) or pseudo-monotone or quasi-monotone, and N2 is a set-valued quasi-monotone mapping. Conclusions similar to the invariance under admissible homotopy of topological degree are obtained. Some concrete existence results and applications to some boundary value problems, integral inclusions and controllability of a nonlinear system are also given. In Chapter 4, we will suppose u A (t,u) is a set-valued pseudo-monotone mapping and consider the evolution inclusions x' (t) + A(t,x((t)) f (t) a.e. and (d)/(dt) (Bx(t)) + A (t,x((t)) f(t) a.e. in an evolution triple (V,H,V*), as well as perturbation problems of those two inclusions.