Solutions for a Singular Hadamard-Type Fractional Differential Equation by the Spectral Construct Analysis

Abstract

In this paper, we consider the existence of positive solutions for a Hadamard-type fractional differential equation with singular nonlinearity. By using the spectral construct analysis for the corresponding linear operator and calculating the fixed point index of the nonlinear operator, the criteria of the existence of positive solutions for equation considered are established. The interesting point is that the nonlinear term possesses singularity at the time and space variables.

Upper and Lower Solution Method for a Singular Tempered Fractional Equation with a p-Laplacian Operator

“…In fact, singularity may occur in the transmission process of a turbulent flow in highly heterogeneous porous media, as some unpredictable factors force the transmission process from a phase into another different phase or state. In past decades, many works have been completed for various singular nonlinear equations; for more details, we refer the reader to [17][18][19][20][21][22][23][24][25].…”

The extreme solutions for a σ‐Hessian equation with a nonlinear operator

“…Various nonlinear analysis theories and methods may be used to study the 𝜎-Hessian equation, such as the spaces theories [12][13][14][15][16][17][18][19][20][21], smoothness theories [22][23][24][25][26][27], operator theories [28][29][30][31], fixed point theorems [32][33][34][35][36], sub-super solution methods [37][38][39], monotone iterative techniques [40,41], and the variational method [42][43][44]. For example, by adopting the sub-super solution method, Zhang et al [37] recently established the interval of the eigenvalue in which the existence of solutions for the following singular augmented 𝜎-Hessian equation is guaranteed…”

“…Various nonlinear analysis theories and methods may be used to study the $$$ \sigma $$$‐Hessian equation, such as the spaces theories [12–21], smoothness theories [22–27], operator theories [28–31], fixed point theorems [32–36], sub‐super solution methods [37–39], monotone iterative techniques [40, 41], and the variational method [42–44]. For example, by adopting the sub‐super solution method, Zhang et al [37] recently established the interval of the eigenvalue in which the existence of solutions for the following singular augmented $$$ \sigma $$$‐Hessian equation is guaranteed where $$$ {B}_1=\left\{x\in {\mathrm{\mathbb{R}}}^M:|x|<1\right\} $$$, …”

Existence of positive solutions for third-order semipositone boundary value problems on time scales

“…where λ > 0 is a parameter, 1/2 < p < q < 1 are constants, f : (0, 1) × [0, ∞) → R, e : (0, 1) → R and ω : [q, 1] → [0, ∞) are continuous functions, and e ∈ L(0, 1). For more details about multiple point boundary value problems and integral boundary value problems, we refer the reader to the survey of [22,36,38] and [11,23,24,26,30,31].…”