A note on one-dimensional time fractional ODEs

We prove or re-prove in this note several important results regarding one dimensional time fractional ODEs following our previous work. Here we use the definition of Caputo derivative proposed in (arXiv:1612.05103,arXiv:1708.08384) based on a convolution group. In particular, we establish generalized comparison principles consistent with the new definition of Caputo derivatives. In addition, we establish the full asymptotic behaviors of the solutions for $D_c^{\gamma}u=Au^p$. Lastly, we provide a simplified proof for the strict monotonicity and stability in initial values for the time fractional differential equations with weak assumptions.

[1]  Lei Li,et al.  Cauchy problems for Keller–Segel type time–space fractional diffusion equation , 2017, Journal of Differential Equations.

[2]  Vicente Vergara,et al.  Optimal Decay Estimates for Time-Fractional and Other NonLocal Subdiffusion Equations via Energy Methods , 2013, SIAM J. Math. Anal..

[3]  Alexis Vasseur,et al.  A Parabolic Problem with a Fractional Time Derivative , 2015, 1501.07211.

[4]  Jian-Guo Liu,et al.  A note on deconvolution with completely monotone sequences and discrete fractional calculus , 2017 .

[5]  Jian-Guo Liu,et al.  Continuous and discrete one dimensional autonomous fractional ode , 2017, 1703.02636.

[6]  Jian-Guo Liu,et al.  Some Compactness Criteria for Weak Solutions of Time Fractional PDEs , 2017, SIAM J. Math. Anal..

[7]  Josimar Ramirez,et al.  Generalized Monotone Iterative Technique for Caputo Fractional Differential Equation with Periodic Boundary Condition via Initial Value Problem , 2012 .

[8]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[9]  Arak M. Mathai,et al.  Mittag-Leffler Functions and Their Applications , 2009, J. Appl. Math..

[10]  Francesco Mainardi,et al.  On Mittag-Leffler-type functions in fractional evolution processes , 2000 .

[11]  Dennis G. Weis.  Asymptotic behavior of some nonlinear Volterra integral equations , 1975 .

[12]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

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

[14]  Yury Luchko,et al.  Maximum principle for the generalized time-fractional diffusion equation , 2009 .

[15]  Lei Li,et al.  A Generalized Definition of Caputo Derivatives and Its Application to Fractional ODEs , 2016, SIAM J. Math. Anal..