Mixed-Type Functional Differential Equations: A $C_{0}$-Semigroup Approach

Abstract

In this paper we study certain systems of mixed-type functional differential equations, from the point of view of the C 0 -semigroup theory. In general, this type of equations are not well-posed as initial value problems. But there are also cases where a unique differentiable solution exists. For these cases and in order to achieve our goal, we first rewrite the system as a classical Cauchy problem in a suitable Banach space. Second, we introduce the associated semigroup and its infinitesimal generator and prove important properties of these operators. As an application, we use the results to characterize the null controllability for those systems, where the control u is constrained to lie in a non-empty compact convex subset Ω of R n .

On the stability and behavior of solutions in mixed differential equations with delays and advances

“…On this subject, we call the attention of the reader to previous works 12,13 . However, it is possible that this initial value problem can be applied (see previous studies 1,2,5,14–20 ). A similar application was made for mixed type difference equations in Yeniçerioglu et al 21 …”

“…The linear autonomous mixed type differential equation is a special version of Equation () where and are nonzero real numbers and each and are positive real numbers ( and ). Such equations can arise in the study of traveling waves in regions with nonlocal interactions initiated in previous works 18,19 . As is well known, this equation can be obtained from (), under the assumption that and are step functions with numbers and of jump points.…”

“…Hupkes and Augeraud‐Véron 16 reviewed the well‐posedness initial value problems for scalar functional algebraic and differential functional equations of mixed type. Similarly, it is shown how the system can be written as a classical Cauchy problem in a suitable Banach space, provided that the initial value problem related to the linear autonomous mixed‐type differential equation given in Mármol and Vanegas 19 is well presented. Then the semigroup associated with the ordinary differential equation and its infinitesimal generator are given and some important properties of these operators are proved.…”