Impulsive fractional partial differential equations

This paper deals with Cauchy problem for a class of impulsive partial hyperbolic differential equations involving the Caputo derivative. Our first purpose is to show that the formula of solutions in cited papers are incorrect. Next, we reconsider a class of impulsive fractional partial hyperbolic differential equations and introduce a correct formula of solutions for Cauchy problem in R n . Further, some sufficient conditions for existence of the solutions are established by applying fixed point method. At last, we consider the Cauchy problem in a Banach space via the technique of measures of noncompactness and Monch's fixed point theorem. Some examples are given to illustrate our results.

[1]  A. Vityuk,et al.  Existence of solutions of systems of partial differential equations of fractional order , 2004 .

[2]  Francesco Mainardi,et al.  Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics , 2012, 1201.0863.

[3]  Yong Zhou,et al.  EXISTENCE AND UNIQUENESS FOR FRACTIONAL NEUTRAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY , 2009 .

[4]  Zhanbing Bai,et al.  On positive solutions of a nonlocal fractional boundary value problem , 2010 .

[5]  R. Agarwal,et al.  A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions , 2010 .

[6]  R. Metzler,et al.  Relaxation in filled polymers: A fractional calculus approach , 1995 .

[7]  Mouffak Benchohra,et al.  Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order , 2010 .

[8]  Michal Fečkan,et al.  On the new concept of solutions and existence results for impulsive fractional evolution equations , 2011 .

[9]  Gaston M. N'Guérékata,et al.  Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay , 2010, Appl. Math. Comput..

[10]  I. Podlubny.  Fractional differential equations , 1998 .

[11]  T F Nonnenmacher,et al.  A fractional calculus approach to self-similar protein dynamics. , 1995, Biophysical journal.

[12]  Yong Zhou.  Basic Theory of Fractional Differential Equations , 2014 .

[13]  H. Mönch,et al.  Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces , 1980 .

[14]  S. Abbas,et al.  EXISTENCE THEORY FOR IMPULSIVE PARTIAL HYPERBOLIC FUNCTIONAL DIFFERENTIAL EQUATIONS INVOLVING THE CAPUTO FRACTIONAL DERIVATIVE , 2010 .

[15]  Yong Zhou,et al.  Existence and controllability results for fractional semilinear differential inclusions , 2011 .

[16]  Mouffak Benchohra,et al.  Topics in Fractional Differential Equations , 2012 .

[17]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[18]  Bashir Ahmad,et al.  Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory , 2010 .

[19]  Yong Zhou,et al.  On the concept and existence of solution for impulsive fractional differential equations , 2012 .

[20]  L. Gaul,et al.  Damping description involving fractional operators , 1991 .

[21]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[22]  Yong Zhou,et al.  Analysis of nonlinear fractional control systems in Banach spaces , 2011 .

[23]  Shuqin Zhang,et al.  Existence of positive solution for some class of nonlinear fractional differential equations , 2003 .

[24]  Ravi P. Agarwal,et al.  Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay , 2010 .