Existence of positive solutions for a singular nonlinear fractional differential equation with integral boundary conditions involving fractional derivatives

Abstract

In this article, by using the spectral analysis of the relevant linear operator and Gelfand's formula, some properties of the first eigenvalue of a fractional differential equation are obtained. Based on these properties and through the fixed point index theory, the singular nonlinear fractional differential equations with Riemann-Stieltjes integral boundary conditions involving fractional derivatives are considered under some appropriate conditions, and the nonlinearity is allowed to be singular in regard to not only time variable but also space variable and it includes fractional derivatives. The existence of positive solutions for boundary conditions involving fractional derivatives is established. Finally, an example is given to demonstrate the validity of our main results.

Existence Results for Fractional Differential Equations with Multistrip Riemann–Stieltjes Integral Boundary Conditions

“…The details are found in [11][12][13][14][15][16] and the references therein. In [17], Sun and Zhao investigated the following fractional differential equation with integral boundary conditions:…”

Fixed-Point Theorems for Systems of Operator Equations and Their Applications to the Fractional Differential Equations

“…Boundary value problems of fractional differential equations have been investigated for many authors [17][18][19][20][21][22]. Now, there are many papers dealing with the problem for different kinds of boundary value conditions such as multipoint boundary condition [23,24], integral boundary condition [25][26][27][28][29][30][31], and many other boundary conditions [32]. From the application, we can see that the fixed-point theorems in this paper have extensive applied background.…”

Existence and Uniqueness Theorems for a Fractional Differential Equation with Impulsive Effect under Band-Like Integral Boundary Conditions

“…In addition, there have been some excellent results concerning the existence, uniqueness, and multiplicity of solutions or positive solutions to some nonlinear fractional differential equations with various nonlocal boundary conditions. As for some recent bibliographies, we refer readers to see [8][9][10][11] and the reference therein.…”