Weighted pseudo almost automorphic mild solutions for two-term fractional order differential equations

Using the notion of Weyl fractional derivative, we study the existence and uniqueness of weighted pseudo almost automorphic mild solutions for a class of two-term fractional order abstract differential equation in the form D t α u ' ( t ) + µ D t β u ( t ) = A u ( t ) + D t α f ( t , u ( t ) ) , t ? 0 , 0 < α ? β ? 1 , µ ? 0 where A is a closed linear operator defined in a Banach space X and the forcing term f is Sp-weighted pseudo almost automorphic.

[1]  C. Lizama,et al.  Compact almost automorphic solutions to integral equations with infinite delay , 2009 .

[2]  Mark M. Meerschaert,et al.  Brownian subordinators and fractional Cauchy problems , 2007, 0705.0168.

[3]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[4]  Deepmala,et al.  A Study on Some Problems on Existence of Solutions for Nonlinear Functional-Integral Equations , 2013 .

[5]  H. Ding,et al.  Weighted pseudo almost automorphic functions and WPAA solutions to semilinear evolution equations , 2014 .

[6]  Jin Liang,et al.  Decomposition of weighted pseudo-almost periodic functions , 2010 .

[7]  L. Mishra,et al.  Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar , 2015 .

[8]  Alexander Pankov,et al.  Stepanov-like almost automorphic functions and monotone evolution equations , 2008 .

[9]  C. Lizama,et al.  On a connection between powers of operators and fractional Cauchy problems , 2012 .

[10]  Hui-Sheng Ding,et al.  A composition theorem for weighted pseudo-almost automorphic functions and applications , 2010 .

[11]  Erkan Nane.  Higher order PDE’s and iterated processes , 2005, math/0508262.

[12]  Anatoly N. Kochubei,et al.  Cauchy problem for fractional diffusion equations , 2003 .

[13]  M. Zitane,et al.  Weighted Pseudo-Almost Automorphic Solutions to a Neutral Delay Integral Equation of Advanced Type , 2012 .

[14]  Rico Zacher,et al.  Maximal regularity of type Lp for abstract parabolic Volterra equations , 2005 .

[15]  C. Lizama,et al.  Asymptotic behavior of fractional order semilinear evolution equations , 2013, Differential and Integral Equations.

[16]  W. Arendt.  Vector-valued laplace transforms and cauchy problems , 2002 .

[17]  Indira Mishra,et al.  Weighted pseudo almost automorphic solution of an integro-differential equation, with weighted Stepanov-like pseudo almost automorphic forcing term , 2013, Appl. Math. Comput..

[18]  Toka Diagana,et al.  Existence of weighted pseudo-almost periodic solutions to some classes of differential equations with Sp-weighted pseudo-almost periodic coefficients , 2010 .

[19]  L. Mishra,et al.  Dirichlet Average of Generalized Miller-Ross Function and Fractional Derivative , 2015 .

[20]  Gisèle Mophou,et al.  Weighted pseudo almost automorphic functions and applications to abstract differential equations , 2009 .

[21]  On the Existence of Locally Attractive Solutions of a Nonlinear Quadratic Volterra Integral Equation of Fractional Order , 2010 .

[22]  Yongkun Li,et al.  Weighted pseudo almost automorphic functions with applications to abstract dynamic equations on time scales , 2013 .

[23]  Jing-huai Liu,et al.  Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations , 2010 .

[24]  D. Benson,et al.  Radial fractional‐order dispersion through fractured rock , 2004 .

[25]  Gisèle M. Mophou,et al.  Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations , 2011, Appl. Math. Comput..

[26]  Juan J. Nieto,et al.  Pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to semi-linear differential equations in Hilbert spaces , 2011 .

[27]  Juan J. Nietoc,et al.  Pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to a partial functional differential equation in Banach spaces , 2012 .

[28]  Rong-Nian Wang,et al.  Abstract fractional Cauchy problems with almost sectorial operators , 2012 .

[29]  Yong-Kui Chang,et al.  New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations , 2012 .

[30]  Wei Lin,et al.  Square-mean weighted pseudo almost automorphic solutions for non-autonomous stochastic evolution equations , 2013 .

[31]  Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay , 2008 .

[32]  Rudolf Gorenflo,et al.  Nonlinear two-term time fractional diffusion-wave problem , 2010 .

[33]  Syed Abbas.  WEIGHTED PSEUDO ALMOST AUTOMORPHIC SEQUENCES AND THEIR APPLICATIONS , 2010 .

[34]  Francesco Mainardi,et al.  Subordination pathways to fractional diffusion , 2011, 1104.4041.

[35]  Khalil Ezzinbi,et al.  Weighted pseudo-almost automorphic solutions for some partial functional differential equations , 2011 .

[36]  Howard H. Chang,et al.  Minimum energy as the general form of critical flow and maximum flow efficiency and for explaining variations in river channel pattern , 2004 .

[37]  Carlos Lizama,et al.  An operator theoretical approach to a class of fractional order differential equations , 2011, Appl. Math. Lett..