K. BALACHANDRAN , S. KIRUTHIKA , J. J. TRUJILLO . EXISTENCE OF SOLUTIONS OF NONLINEAR FRACTIONAL PANTOGRAPH EQUATIONS[J]. Acta mathematica scientia, Series B, 2013 , 33(3) : 712 -720 . DOI: 10.1016/S0252-9602(13)60032-6

[1] Agarwal R P, Benchohra M, Hamani S. A survey on the existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl Math, 2010, 109: 973–1033

[2] Ahmed E, Elgazzar A S. On fractional order differential equations model for nonlocal epidemics. Phys A, 2007, 379: 607–614

[3] Balachandran K, Park J Y. Nonlocal Cauchy problem for abstract fractional semilinear evolution equations. Nonlinear Anal Theory Methods and Appl, 2009, 71: 4471–4475

[4] Balachandran K, Kiruthika S. Existence of solutions of abstract fractional impulsive semilinear evolution equations. Electron J Qual Theory Differ Equ, 2010, (4): 1–12

[5] Balachandran K, Kiruthika S. Existence results for fractional integrodifferential equations with nonlocal condition via resolvent operators. Comput Math Appl, 2011, 62: 1350–1358

[6] Balachandran K, Trujillo J J. The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Anal Theory Methods and Appl, 2010, 72: 4587–4593

[7] Balachandran K, Kiruthika S, Trujillo J J. Existence results for fractional impulsive integrodifferential equations in Banach spaces. Commun in Nonlinear Sci and Numer Simul, 2011, 16: 1970–1977

[8] Balachandran K, Kiruthika S, Trujillo J J. On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces. Comput Math Appl, 2011, 62: 1157–1165

[9] Bonilla B, RiveroM, Rodr´?guez-Germ´a L, Trujillo J J. Fractional differential equations as alternative models to nonlinear differential equations. Appl Math Comput, 2007, 187: 79–88

[10] Byszewski L. Theorems about the existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problem. J Math Anal Appl, 1991, 162: 494–505

[11] Caputo M. Linear models of dissipation whose Q is almost frequency independent. Part II. Geophys J R Astron Soc, 1967, 13: 529–539

[12] Derfel G A, Iserles A. The pantograph equation in the complex plane. J Math Anal Appl, 1997, 213: 117–132

[13] El-Sayed A M A. Fractional-order diffusion-wave equation. Int J Theor Phys, 1996, 35: 311–322

[14] Iserles A. On the generalized pantograph functional-differential equation. Eur J Appl Math, 1993, 4: 1–38

[15] Iserles A. Exact and discretized stability of the pantograph equation. Appl Numer Math, 1997, 24: 295–308

[16] Iserles A, Liu Y. On pantograph integro-differential equations. J Integral Equations Appl, 1994, 6: 213–237

[17] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006

[18] Langley J K. A certain functional-differential equation. J Math Anal Appl, 2000, 244 : 564–567

[19] Lakshmikantham V. Theory of fractional functional differential equations. Nonlinear Anal Theory Methods and Appl, 2008, 69: 3337–3343

[20] Lakshmikantham V, Vatsala A S. General uniqueness and monotone iterative technique for fractional differential equations. Appl Math Lett, 2008, 21: 828–834

[21] Liu M Z. Asymptotic behaviour of functional-differential equations with proportional time delays. Eur J Appl Math, 1996, 7: 11–30

[22] Li D, Liu M Z. Runge-Kutta methods for the multi-pantograph delay equation. Appl Math Comput, 2005, 163: 383–395

[23] Liu M Z, Li D. Properties of analytic solution and numerical solution of multi-pantograph equation. Appl Math Comput, 2004, 155: 853–871

[24] Metzler R, Klafter J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep, 2000, 339: 1–77

[25] Miller K S, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley, 1993

[26] N’Gu´er´ekata G M. A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Analysis: Theory, Methods and Applications, 2009, 70: 1873–1876

[27] Hern´andez E, O’Regan D, Balachandran K. On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal Theory Methods and Appl, 2010, 73: 3462–3471

[28] Ockendon J R, Taylor A B. The dynamics of a current collection system for an electric locomotive. Proc R Soc London, Ser.A, 1971, 322: 447–468

[29] Podlubny I. Fractional Differential Equations. New York: Academic Press, 1998

[30] Samko S G, Kilbas A A, Marichev O I. Fractional Integrals and Derivatives; Theory and Applications. Amsterdam: Gordon and Breach, 1993

[31] Sezer M, Yalcinbas S, Sahin N. Approximate solution of multi-pantograph equation with variable coeffi-cients. J Comput Appl Math, 2008, 214: 406–416

[32] Smart D R. Fixed Point Theorems. Cambridge: Cambridge University Press, 1980

[33] Xu H. Analytical approximations for population growth model with fractional order. Commun in Nonlinear Sci and Numer Simul, 2009, 14: 1978–1983

[34] Yu Z H. Variational iteration method for solving the multi-pantograph delay equation. Phys Lett A, 2008, 372: 6475–6479