Multiple positive solutions to singular positone and semipositone m-point boundary value problems of nonlinear fractional differential equations

Abstract

In this paper, we consider the properties of Green's function for the nonlinear fractional differential equation boundary value problemis the standard Riemann-Liouville derivative. Here our nonlinearity f may be singular at u = 0. As an application of Green's function, we give some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leray-Schauder nonlinear alternative and a fixed point theorem on cones. MSC: 34B15

A generalization of the compression cone method for integral operators with changing sign kernel functions

“…To construct the cone in a space such as C[0, T], usually a positive Green's function for the BVP is required to ensure a positive kernel for the integral operator. Consequently, it leads to existence of positive solutions for the original BVP [12,17,21,23,24,26]. Changing sign solutions have drawn relatively less attention.…”

Positive solutions for eigenvalue problems of fractional q-difference equation with ϕ-Laplacian

“…As a branch of q-calculus, fractional q-calculus was first proposed by Al-Salam and Agarwal in the 1960s; see [2,3]. Fractional q-calculus has a wide range of applications in many fields, such as cosmic strings and black holes, quantum theory, aerospace dynamics, biology, economics, control theory, medicine, electricity, signal processing, image processing, biophysics, blood flow phenomenon, and so on; see [4][5][6][7][8][9][10] and the references therein. The fractional q-difference equations are very important, and their basic theory has been continuously developed.…”

Existence and Nonexistence of Positive Solutions for Fractional Three-Point Boundary Value Problems with a Parameter

“…Fractional calculus has played an important role in the modeling of different physical and natural phenomena, such as fluid mechanics, control system, and many other branches of engineering. In recent years, there are many papers concerning the existence of positive solutions for nonlinear fractional differential equations; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references cited therein.…”

“…Xu et al [2] showed the existence of multiple positive solutions to singular positone and semipositone m-point boundary value problems of nonlinear fractional differential equations 0 + ( ) + ( , ( )) = 0, 0 < < 1, (0) = 0,…”