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The article presents the study of a semigroup method to semilinear partial functional differential equations with infinite delay. The study aims to understand the solution semigroup generated by a particular partial functional differential equation with infinite delay. The paper presents some basic results on existence, uniqueness and properties of integral solutions and then establishes properties of the solution operator in nonlinear case. It utilizes the Crandall and Liggett theorem in order to compute the nonlinear semigroup by an exponential formula.

Abstract: The Mackey-Glass equation, , is considered, with variable coefficients and a nonconstant delay. Under rather natural assumptions all solutions are positive and bounded. Persistence and extinction conditions are presented for this equation. In the case when there exists a constant positive equilibrium, local asymptotic stability of the constant solution and oscillation about this equilibrium are analyzed. The results are illustrated by numerical examples. In particular, it is demonstrated that with delay in both terms, a solution with positive initial conditions may become negative. [Copyright &y& Elsevier]

This paper is devoted to the study of optimization problems for dynamical systems governed by constrained delay-differential inclusions with generally nonsmooth and nonconvex data. We provide a variational analysis of the dynamic optimization problems based on their data perturbations that involve finite-difference approximations of time-derivatives matched with the corresponding perturbations of endpoint constraints. The key issue of such an analysis is the justification of an appropriate strong stability of optimal solutions under finite-dimensional discrete approximations. We establish the required pointwise convergence of optimal solutions and obtain necessary optimality conditions for delay-differential inclusions in intrinsic Euler-Lagrange and Hamiltonian forms involving nonconvex-valued subdifferentials and coderivatives of the initial data. [ABSTRACT FROM AUTHOR]