Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 9 (2014), 1 – 54 EXPLICIT STABILITY CONDITIONS FOR NEUTRAL TYPE VECTOR FUNCTIONAL DIFFERENTIAL EQUATIONS. A SURVEY Michael I. Gil’ Abstract. This paper is a survey of the recent results of the author on the stability of linear and nonlinear neutral type functional differential equations. Mainly, vector equations are considered. In particular, equations whose nonlinearities are causal mappings are investigated. These equations include neutral type, ordinary differential, differential-delay, integro-differential and other traditional equations. Explicit conditions for the Lyapunov, exponential, input-to-state and absolute stabilities are derived. Moreover, solution estimates for the considered equations are established. They provide us the bounds for the regions of attraction of steady states. A part of the paper is devoted to the Aizerman type problem from the the absolute stability theory. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the generalized Bohl - Perron principle, positivity conditions for fundamental solutions of scalar equations and properties of the so called generalized norm 1 Introduction 1. This paper is a survey of the recent results of the author on the stability of the neutral type linear and nonlinear vector functional differential equations. Functional differential equations naturally arise in various applications, such as control systems, mechanics, nuclear reactors, distributed networks, heat flows, neural networks, combustion, interaction of species, microbiology, learning models, epide- miology, physiology, and many others. The theory of functional differential equations has been developed in the works of V. Volterra, A.D. Myshkis, N.N. Krasovskii, B. Razumikhin, N. Minorsky, R. Bellman, A. Halanay, J. Hale and other mathematicians. The problem of the stability analysis of neutral type equations continues to attract the attention of many specialists despite its long history. It is still one of the 2010 Mathematics Subject Classification: 34K20; 34K99; 93D05; 93D25. Keywords: functional differential equations; neutral type equations; linear and nonlinear equations; exponential stability; absolute stability; L 2 -stability, input-to-state stability, causal mappings; Bohl - Perron principle; Aizerman problem. ****************************************************************************** http://www.utgjiu.ro/math/sma