SURVEY AND NEW RESULTS ON BOUNDARY-VALUE PROBLEMS OF SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS

Firstly we prove existence and uniqueness of solutions of Cauchy problems of linear fractional differential equations (LFDEs) with two variable coefficients involving Caputo fractional derivative, Riemann-Liouville derivative, Caputo type Hadamard derivative and Riemann-Liouville type Hadamard fractional derivatives with order q is an element of [n - 1, n) by using the iterative method. Secondly we obtain exact expressions for piecewise continuous solutions of the linear fractional differential equations with a constant coefficient and a variable one. These results provide new methods to transform an impulsive fractional differential equation (IFDE) to a fractional integral equation (FIE). Thirdly, we propose four classes of boundary value problems of singular fractional differential equations with impulse effects. Sufficient conditions are given for the existence of solutions of these problems. We allow the nonlinearity p(t)f(t, x) in fractional differential equations to be singular at t = 0, 1. Finally, we point out some incorrect formulas of solutions in cited papers. A new Banach space and the compact properties of subsets are proved. By establishing a new framework to find the solutions for impulsive fractional boundary value problems, the existence of solutions of three classes boundary value problems of impulsive fractional differential equations with multi-term fractional derivatives are established.