Existence And Controllability Of Multivalued Functional Differential Equations

Multivalued functional differential equations and integrodifferential equations are an important branch in the theory of nonlinear analysis, which have wide applications in many fields such as engineering, economics, optimal control and optimization theory. Therefore, it is significant to study the existence and controllability of multivalued functional differential equations and integrodifferential equations.The dissertation mainly deals with the existence of second order multivalued impulsive functional differential equation, existence and controllability of first order multivalued neutral functional differential equation and integrodifferential equation with infinite delay, controllability of Sobolev type multivalued functional differential equation and integrod-iffergntial equation with infinite delay and semilinear multivalued impulsive functional differential equation and integrodifferential equation with infinite delay. The results obtained arc presented as follows:1. Existence of second order multivalued impulsive functional differential equation with the form (p(t)y'(t))' G F(t,yt) is discussed in a finite dimensional space R. Under some growth conditions on the impulsive functions, we prove existence theorems of solutions for both the convex and nonconvex problems. Our main results arc based upon Bohnenblust-Karlin fixed point theorem, Schauder fixed point theorem combined with a selection theorem of Bressan and Colombo.2. We consider the existence of mild solutions of first order neutral multivalued functional differential equation d/dt [y (t) + g (t, y_t)} G Ay (t) + F (t, yt) and functional integrodifferential equation d/dt [y (t) + g (t, yt)} ∈ Ay (t) + ∫_o~t k (t, s) F (t, yt) with infinite delay. The controllability of the multivalued functional differential system jt [y(t) + g(t,yt)\ G Ay (t) + F (t, yt) + Bu (t) and functional integrodifferential system d/dt [y (t) + g (t. yt)\ G Ay (t) + ∫_o~t k (t, s) F (t, yt) + Bu (t) with infinite delay is also discussed. We establish our main results in the abstract phase space B by the theory of analytic semigroups and a fixed point theorem. These results improve some known ones on the corresponding equations with finite delay.3. The controllability of mild solutions of Sobolev type multivalued functional differential system (Ey (t))' - Ay (t) ∈ F(t,yt) + Bu{t) and functional mtegrodifferential system (Ey (t))' - Ay (t) ∈ ∫_o~t kv (t, s) F (s, ys) ds + Bu (l) with infinite delay is considered in the abstract phase space B. Sufficient conditions are given for the controllability of theabove systems by the integral solutions for the corresponding systems and a fixed point theorem.4. We consider the controllablity of mild solutions for multivalued semilinear impulsive functional differential system x' (t) — Ax (t) € F (t, xt) + Bu (t) with infinite delay. By introducing a suitable phase space BMh, we establish sufficient conditions for the controllablity of the above system with the theory of semigroups and a fixed point theorem. The controllability for the corresponding functional intogrodifferontial system x' (/-) - Ax (I) 6 Jo' K (t, s) F (s, xs) ds + Bu (/.) is also presented. The main theorems obtained improve some known results with finite delay in the literatures.