Some classes of second order functional differential equations

This paper is dedicated to the concise presentation of some results (mainly, existence theorems and behavior of solutions) related to functional differential equations of the form d/dt[(x) over dot(t) - (Lx)(t)] = (t), (E-1) or d/dt[(x) over dot(t)] = (Vx)(t), (E-2) in which L stands for a linear operator on a certain function space, causal and continuous, while V is acting on the same function space, but it is assumed only causal and continuous (in general, nonlinear). We provide several existence results, under various assumptions. Also, we show that, under extra conditions, the solutions can present various types of behavior on a semi-axis, or on the whole real axis. Some applications are indicated, and references are provided. The results have been obtained by the author and his former students Mehran Mahdavi and Yizeng Li, during the last ten years. Some results are published here for the first time. An earlier account of these kind of problems is given in [C. Corduneanu, Abstract Volterra equations: A survey, Mathematical and Computer Modelling 32 (2000) 1503-1528; C. Corduneanu, Some existence results for functional differential equations with causal operators, Nonlinear Analysis, TMA 47 (2001) 709-716]. (C) 2008 Elsevier Ltd. All rights reserved.