Acta Mathematica Scientia 2013,33B(3):712–720 http://actams.wipm.ac.cn EXISTENCE OF SOLUTIONS OF NONLINEAR FRACTIONAL PANTOGRAPH EQUATIONS ∗ K. BALACHANDRAN S. KIRUTHIKA Department of Mathematics, Bharathiar University, Coimbatore-641046, India E-mail : kbkb1956@yahoo.com; kiruthimath@gmail.com J. J. TRUJILLO Universidad de La Laguna, Departamento de An´alisis Matem´ atico, 38271 La Laguna, Tenerife, Spain E-mail : jjtrujill@ullmat.es Abstract This article deals with the existence of solutions of nonlinear fractional panto- graph equations. Such model can be considered suitable to be applied when the corresponding process occurs through strongly anomalous media. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the main result obtained in this article. Key words Fractional differential equations; pantograph equations; fixed point theorems 2010 MR Subject Classification 34A08 1 Introduction Recently fractional differential equations emerged as a new branch of applied mathematics, which has been used for many mathematical models in science and engineering [2, 13, 24, 33]. In fact, fractional differential equations is considered as an alternative model to nonlinear differential equations [9]. Theory of fractional differential equations has been extensively studied by many authors [1, 19, 20, 26]. Balachandran et al [3–5, 7, 8] studied the existence of solutions of different types of fractional differential equations whereas Hernandez et al [27] discussed the recent developments in the theory of fractional differential equations. Many models are reformulated and expressed in terms of fractional differential equations so that the physical meaning will be incorporated in the mathematical models more realistically. It is well known that in the deterministic situation there is a very special delay differential equation known as the pantograph equation y ′ (t)= ay(t)+ by(λt), 0 ≤ t ≤ T, * Received October 24, 2011; revised September 17, 2012. The second author is thankful to UGC New Delhi for providing BSR fellowship and the third author is thankful to project MTM2010-16499 from the MICINN of Spain.