New Boundary Value Problems for Higher Order Impulsive Fractional Differential Equations and Their Solvability

Firstly, the surveys for studies on boundary value problems for higher order ordinary differential equations and for higher order fractional differential equations are given. Secondly a simple review for studies on solvability of boundary value problems for impulsive fractional differential equations is presented. Thirdly we propose four classes of higher order linear fractional differential equations and give their exact piecewise continuous solutions. Fourthly we propose some new classes of boundary value problems for higher order fractional differential equations with impulse effects. Fifthly we establish new general methods for converting boundary value problems of impulsive fractional differential equations with the Riemann-Liouville fractional derivatives or Caputo fractional derivatives to equivalent integral equations. Sixthly by employing fixed point theorems in Banach space, we establish new existence results of solutions for these boundary value problems. Seventh, some remarks are given to show that many methods in known papers are un-suitable. Eighth, some examples are presented to illustrate the efficiency of the results obtained. Finally, possible trends of researches are given at the end of the paper.