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

Abstract

In this paper, we consider a Riemann-Liouville type two-term fractional differential equation boundary value problem. Some positive properties of the Green's function are deduced by using techniques of analysis. As application, we obtain the existence and multiplicity of positive solutions for a fractional boundary value problem under conditions that the nonlinearity f (t, x) may change sign and may be singular at t = 0, 1 and x = 0, and we also obtain the uniqueness results of positive solution for a singular problem by means of the monotone iterative technique.

Estimation of the smallest eigenvalue of an nth‐order linear boundary value problem

“…Last, but not least, another possible area of future investigation is the extension of the results to fractional boundary value problems. The literature in this area is focused on concrete cases of the operator L with fractional derivatives, for which the properties of the associated Green function have to be explicitly determined (see, for instance, other works [20][21][22][23] ). More general results on the sign of the derivatives of the Green function of these problems would therefore be very valuable.…”

“…Other works [9][10][11][12][13][14][15][16][17][18][19] are a good account of this. It is worth remarking that much of this work in the recent years has moved into problems similar to (5) where the differential operator L contains fractional derivatives (see, for instance, the literature [20][21][22][23] ). Whereas this paper will not get into this kind of problems, it could be well extended there if results on the sign of the Green function G(x, t) like those of Almenar and Jódar 24 are proved for fractional boundary value problems.…”

Existence and Uniqueness of Positive Solutions for a Class of Nonlinear Fractional Differential Equations with Singular Boundary Value Conditions

“…e initial value problem and the boundary problem were studied in [9][10][11][12][13][14][15][16][17][18][19][20][21], respectively. e SBV problem was recently studied for NFD [22][23][24][25][26][27][28][29][30][31][32][33], mostly focused on investigating the positive solution. ey were mainly based on nonlinear analysis techniques such as Leray-Schuader theory, FP, topological theories, and mixed monotone method [34][35][36][37][38][39].…”

“…(1) To the best of our knowledge, boundary condition ( 5) is firstly considered for NFD. (2) Conditions (H2) and (H5) imposed on f are different from those in [26][27][28][29][30][31][32][33][34]. e remainder of this paper is structured as follows.…”

Explicit monotone iterative sequences for positive solutions of a fractional differential system with coupled integral boundary conditions on a half-line

“…Fractional-order differential equations is a natural generalization of the case of integer order, which has become the focus of attention involving various kinds of boundary conditions because of the wide application in mathematical models and applied sciences. Some latest results on the topic can be found in a series of papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and the references therein. In particular, a monotone iterative technique is believed to be an efficient and important method to deal with sequences of monotone solutions for initial and boundary value problems.…”