Study On The Solutions Of Boundary Value Problems For Some Classes Of Generalized Fractional Differential Equations

Compared with integral calculus,fractional calculus is more clear and accurate in describing many practical problems.At present,there are more researches on Riemann-Liouville and Caputo type fractional calculus in the existing achievements.In the later appearing of the fractional Hilfer derivative of orderα and type β,if β=0,it becomes Riemann-Liouville fractional derivative;If β=1,it is the Caputo type fractional derivative;If 0＜β＜1,the Hilfer fractional derivative is a derivative between the above two special derivatives,it can be seen that the Hilfer fractional derivative is reasonably extended on the basis of the above two commonly used fractional derivative operator definitions,and is a generalized fractional derivative.The results show that the generalized fractional differential equation can better describe the operation process of some complex systems,so as to become an effective tool for mathematical modeling in signal processing,rheology,optics and other fields.Because of the generality of the definition of Hilfer fractional derivative,it is challenging to study the generalized fractional differential equation with Hilfer fractional derivative.At present,the boundary value problem of generalized fractional differential equation is just starting,and there is still a lot of research space.Therefore,by proving and using several appropriate fixed point theorems,the following four kinds of boundary value problems for generalized fractional differential equations are studied in this paper:1)Existence of unique solutions to multi-point boundary value problems of Hilfer fractional differential equations;2)Existence of solutions to boundary value problems of left and right Hilfer mixed fractional differential equations;3)Existence of solutions to boundary value problems of Hilfer fractional differential equations with pulses;4)Existence of solutions to boundary value problems of Caputo and Hilfer mixed fractional differential equations with p-Laplacian operators.On the basis of proving the new fixed point theorem and selecting the appropriate fixed point theorem,this paper not only studies the existence of the solutions of the above four kinds of new boundary value problems,but also applies the conclusions to concrete examples,and then illustrates the applicability of the conclusions obtained.