AMO - Advanced Modeling and Optimization, Volume 15, Number 2, 2013 Existence and controllability result for fractional neutral stochastic integro-differential equations with infinite delay Toufik Guendouzi 1 , Iqbal Hamada 2 1,2 Laboratory of Stochastic Models, Statistical and Applications Tahar Moulay University PO.Box 138 En-Nasr, 20000 Saida, Algeria Abstract This paper is concerned with the existence of mild solutions and the approximate controllability for a class of fractional neutral stochastic integro-differential equations with infinite delay in Hilbert spaces. Firstly, a sufficient condition for the existence is obtained under non-Lipschitz conditions by means of Sadovskii’s fixed point theorem. Secondly, the approximate controllability of nonlinear fractional stochastic system is discussed, under the assumption that the corresponding linear system is approximately controllable. Finally, an example is given to illustrate the theory. Key words and phrases: Existence result, approximate controllability, stochastic fractional differential equations, fixed point technique, infinite delay. Mathematics Subject Classification: 34K30, 34K50, 26A33. 1 Introduction The subject of fractional calculus and its applications has gained a lot of importance during the past three decades, mainly because it has become a powerful tool in modeling several complex phenomena in numerous seemingly diverse and widespread fields of science and engineering [7, 10, 15]. Recently, there has been a significant development in the existence and uniqueness of solutions of initial and boundary value problem for fractional differential equations [24]. Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention in the last decades. But the literature related to neutral fractional differential equations is very limited and we refer the reader to [23]. On the other hand, the study of controllability plays an important role in the control theory and engineering [2, 11]. In recent years, various controllability problems for different kinds of dynamical systems have been studied in many publications [1, 3, 5, 6]. From the mathematical point of view, the problems of exact and approximate controllability are to be distinguished. However, the concept of exact controllability is usually too strong and has limited applicability. 1 Corresponding author. E-mail addresses: tf.guendouzi@gmail.com (T. Guendouzi), iqbalhamada@gmail.com (I. Hamada). The work described in this paper is financially supported by The National Agency of Development of University Research (ANDRU), Algeria (SMA 2011-2014). *AMO-Advanced Modeling and Optimization. ISSN: 1841-4311. 281