On the existence of positive solutions and negative solutions of singular fractional differential equations via global bifurcation techniques

Abstract

We are concerned with a type of fractional differential equations attached to boundary conditions. We investigate the existence of positive solutions and negative solutions via global bifurcation techniques.

A New Approach to the Existence of Quasiperiodic Solutions for Second-Order Asymmetric p-Laplacian Differential Equations

“…Now a natural question to ask is whether the smoothness of the function ℎ( ) in (5) is further reduced; we can also obtain the same results as [13]. In this paper, we will deal with this interesting problem and answer this question in the form of Theorem 1 with more general case (1) than that of (5). Because of the presence of weak smoothness nonlinearity, the methods of seeking the existence of Aubry-Mather sets and quasiperiodic solutions for problems as [38,39] do not seem to be applicable to (1).…”

“…In recent years, all kinds of nonlinear dynamic behavior, such as the existence of positive solutions [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and signchanging solutions [17,18], the existence and uniqueness of solutions [19][20][21][22][23][24][25], the existence and multiplicity results [26][27][28][29][30], and the existence of unbounded solutions [31,32], have been widely investigated for some nonlinear ordinary differential equations and partial differential equations due to the application in many fields such as physics, mechanics, and the engineering technique fields. In the present paper, we deal with the existence of Aubry-Mather sets and quasiperiodic solutions for the second-order differential equations with a -Laplacian and an asymmetric nonlinear term…”

Positive solutions for a class of fractional infinite-point boundary value problems

“…), FDE serve as an excellent instrument for the description of memory and hereditary properties of various materials and processes. During the last few decades, much attention has been paid to the study of boundary value problems (BVP for short) of fractional differential equation, such as the nonlocal BVP [1,3,7,13,18], singular BVP [6,8,11,19,20,25], semipositone BVP [14][15][16]23], resonant BVP [2,12], and impulsive BVP [10,27]. Since only positive solutions are meaningful in most practical problems, some work has been done to study the existence of positive solutions for fractional boundary value problems by using the techniques of nonlinear analysis.…”

On the Existence of Solutions for Impulsive Fractional Differential Equations

“…In addition, since Rabinowitz established unilateral global bifurcation theorems, there have been many researches in global bifurcation theory and it has been applied to obtain the existence and multiplicity for solutions of differential equations (see, for instance, [8][9][10][11][12][13][14][15][16] and their references). However, the previous researches seldom involve both global bifurcation techniques and fractional differential equations.…”

“…However, the previous researches seldom involve both global bifurcation techniques and fractional differential equations. In [16], the following problem was studied. …”