A survey on impulsive fractional differential equations

Abstract Recently, in series of papers we have proposed different concepts of solutions of impulsive fractional differential equations (IFDE). This paper is a survey of our main results about IFDE. We present several types of such equations with various boundary value conditions as well. Concept of solutions, existence results and examples are presented. Proofs are only sketched.

[1]  S. Sivasundaram,et al.  Existence of solutions for impulsive integral boundary value problems of fractional order , 2010 .

[2]  Yong Zhou,et al.  Existence of mild solutions for fractional neutral evolution equations , 2010, Comput. Math. Appl..

[3]  H. Redwane EXISTENCE RESULTS FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS IN ORLICZ SPACES , 2010 .

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

[5]  X. Xiang,et al.  Nonlinear impulsive integro-differential equations of mixed type and optimal controls , 2006 .

[6]  Colin Atkinson,et al.  Rational Solutions for the Time-Fractional Diffusion Equation , 2011, SIAM J. Appl. Math..

[7]  Gisèle M. Mophou,et al.  Existence and uniqueness of mild solutions to impulsive fractional differential equations , 2010 .

[8]  N. Nyamoradi Multiplicity of Nontrivial Solutions for Boundary Value Problem for Impulsive Fractional Differential Inclusions Via Nonsmooth Critical Point Theory , 2015 .

[9]  Donal O’Regan,et al.  A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations , 2016 .

[10]  M. Benchohra,et al.  IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES , 2009 .

[11]  Guotao Wang,et al.  Systems of first order impulsive functional differential equations with deviating arguments and nonl , 2011 .

[12]  JinRong Wang,et al.  On a new class of impulsive fractional differential equations , 2014, Appl. Math. Comput..

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

[14]  S. Kiruthika,et al.  EXISTENCE OF SOLUTIONS OF ABSTRACT FRACTIONAL IMPULSIVE SEMILINEAR EVOLUTION EQUATIONS , 2010 .

[15]  JinRong Wang,et al.  On recent developments in the theory of boundary value problems for impulsive fractional differential equations , 2012, Comput. Math. Appl..

[16]  Yuanji Cheng A generalized Fucik type eigenvalue problem for p-Laplacian , 2009 .

[17]  Y. Zhou,et al.  Presentation of solutions of impulsive fractional Langevin equations and existence results , 2013 .

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

[19]  Stepan Tersian,et al.  Existence of solutions to boundary value problem for impulsive fractional differential equations , 2014 .

[20]  Yong-sheng Ding,et al.  A generalized Gronwall inequality and its application to a fractional differential equation , 2007 .

[21]  S. Sivasundaram,et al.  Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations , 2009 .

[22]  J. B. Díaz,et al.  A fixed point theorem of the alternative, for contractions on a generalized complete metric space , 1968 .

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

[24]  Michal Fečkan,et al.  Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions , 2017 .