Semilinear Neutral Differential Equations with non-instantaneous impulses, non-local conditions and infinite delay: Existence of solutions and Controllability - Lalvay Segovia, Sebastián Leonardo (2022)
In this thesis, we study the existence of solutions and controllability for retarded semilinear neutral differential equations with non-instantaneous impulses, non-local conditions, and infinite delay. First, we set the problem in a phase space satisfying the Hale-Kato axiomatic theory for retarded differential equations with infinite delay. Second, we assume that the nonlinear functions are locally Lipschitz, and Karakostas’s fixed point theorem is applied to obtain the existence of solutions. Additionally, under some additional conditions, the uniqueness is proved as well. Next, assuming that the nonlinear terms are globally Lips- chitz, we consider a more simplified system that allows us to apply the Banach contraction theorem to prove the existence of solutions. Subsequently, we study the associated control problem. On the one hand, we investigate the approximate controllability by using the technique employed by Bashirov and Ghahramanlou, which avoids the use of fixed point theorems. On the other hand, we prove the exact controllability of the same system. To this end, we transform the controllability problem into a fixed point problem. Then, under some conditions on the nonlinear terms, we use Rothe’s fixed point theorem to obtain the desired result.