Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 502839, 9 pages http://dx.doi.org/10.1155/2013/502839 Research Article Approximate Controllability of Fractional Sobolev-Type Evolution Equations in Banach Spaces N. I. Mahmudov Eastern Mediterranean University, Gazimagusa, T.R. North Cyprus, Mersin 10, Turkey Correspondence should be addressed to N. I. Mahmudov; nazim.mahmudov@emu.edu.tr Received 3 January 2013; Accepted 1 February 2013 Academic Editor: Jen-Chih Yao Copyright © 2013 N. I. Mahmudov. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We discuss the approximate controllability of semilinear fractional Sobolev-type diferential system under the assumption that the corresponding linear system is approximately controllable. Using Schauder fxed point theorem, fractional calculus and methods of controllability theory, a new set of sufcient conditions for approximate controllability of fractional Sobolev-type diferential equations, are formulated and proved. We show that our result has no analogue for the concept of complete controllability. Te results of the paper are generalization and continuation of the recent results on this issue. 1. Introduction Many social, physical, biological, and engineering problems can be described by fractional partial diferential equations. In fact, fractional diferential equations are considered as an alternative model to nonlinear diferential equations. In the last two decades, fractional diferential equations (see Samko et al. [1] and the references therein) have attracted many scientists, and notable contributions have been made to both theory and applications of fractional diferential equations. Recently, the existence of mild solutions and stability and (approximate) controllability of (fractional) semilinear evolution system in Banach spaces have been reported by many researchers; see [2–36]. We refer the reader to El- Borai [3, 4], Balachandran and Park [5], Zhou and Jiao [6, 7] Hern´ andez et al. [8], Wang and Zhou [9], Sakthivel et al. [12, 13], Debbouche and Baleanu [14], Wang et al. [15–21], Kumar and Sukavanam [22], Li and Yong [37], Dauer and Mahmu- dov [28], Mahmudov [27, 29], and the references therein. Complete controllability of evolution systems of Sobolev type in Banach spaces has been studied by Balachandran and Dauer [23], Ahmed [24], and Feckan et al. [2]. However, the approximate controllability of fractional evolution equations of Sobolev type has not been studied. Motivated by the above-mentioned papers, we study the approximate controllability of a class of fractional evolution equations of Sobolev type:     ( ()) =  () +  () +  (,  ()) ,  ∈ [0, ] ,  (0) =  0 , (1) where  : () ⊂  →  and  : () ⊂  →  are linear operators from a Banach space  to . Te control function  takes values in a Hilbert space  and ∈  2 ([0, ], ). :→ is a linear bounded operator. Te function  ∈  ([0, ] × , ) will be specifed in the sequel. Te fractional derivative     , 0<<1, is understood in the Caputo sense. Our aim in this paper is to provide a sufcient condition for the approximate controllability for a class of fractional evolution equations of Sobolev type. It is assumed that  −1 is compact, and, consequently, the associated linear control system (35) is not exactly controllable. Terefore, our approximate controllability results have no analogue for the concept of complete controllability. In Section 5, we give an