Cauchy problem for fractional evolution equations with Caputo derivative

Cauchy problem for nonlinear fractional differential equations with positive constant coefficient

“…For more history and basic results on fractional calculus theory, one can see monograph [1][2][3][4][5][6][7][8] and the references therein. Many researchers studied Cauchy problem and long time behavior for nonlinear fractional differential equations and obtained many interesting results by using all kinds of fixed point theorems (see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]). …”

Existence and uniqueness of mild solutions to initial value problems for fractional evolution equations

“…In 2010, Hernandez et al [8] proved that the concepts of mild solutions of fractional evolution equations considered in some previous papers were not appropriate. Based on the new definition of a mild solution obtained by employing the Laplace transform, Zhou et al [9][10][11][12][13][14] established the existence and uniqueness results for mild solution of different kinds of fractional evolution equations. Wang et al [15] revisited the nonlocal Cauchy problem for fractional evolution equations and relaxed the compactness and Lipschitz continuity on the nonlocal item given in the previous existence results.…”

Analysis of nonlinear Hadamard fractional differential equations via properties of Mittag–Leffler functions

“…The continuous time random walk theory is another example of arising of fractional derivatives in description of real physical systems (see [16]). There are several existence and stability results for all kinds of Caputo and Riemann-Liouville type nonlinear FDEs with constant coefficients (see [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]), and Hadamard type nonlinear FDEs without constant coefficient (see [1,Chapter 13] and [37][38][39][40][41][42][43]). However, the development of a related theory for Hadamard type nonlinear FDEs with constant coefficient is still in its infancy.…”