Positive properties of Green's function for three-point boundary value problems of nonlinear fractional differential equations and its applications

On the existence and uniqueness of the solution of a nonlinear fractional differential equation with integral boundary condition

“…The theory of fixed point to study of the solution to the boundary value problem is a consequential tool and it is played important role in both proving the existence of the solution and obtain it approximately, see [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. It can be seen a lot interest in researching on the subject of nonlocal nonlinear fractional order in both single-valued and multi-valued boundary value problems in recent years.…”

The Green’s function of a class of two-term fractional differential equation boundary value problem and its applications

“…with Dirichlet type boundary value condition. Xu and Fei [30] investigated (2) with threepoint boundary value condition. In [19], we established some new positive properties of the corresponding Green's function for (2) with multi-point boundary value condition.…”

“…Compared with the existing works, this paper has the following features. Firstly, the fractional derivative discussed in this paper is the standard Riemann-Liouville derivative, which is different from [8,9], and the linear operator of the FBVP we are considered with contains two terms, which is different from [14,19,27,28,30]; in other words, we discuss different problem which has been seldom studied before. Secondly, some meaningful properties of the Green's function for the case that 2 < α < 3 are established; this is different from [20] since Ref.…”