An upper-lower solution method for the eigenvalue problem of Hadamard-type singular fractional differential equation

Abstract

In this paper, we are concerned with the eigenvalue problem of Hadamard-type singular fractional differential equations with multi-point boundary conditions. By constructing the upper and lower solutions of the eigenvalue problem and using the properties of the Green function, the eigenvalue interval of the problem is established via Schauder’s fixed point theorem. The main contribution of this work is on tackling the nonlinearity which possesses singularity on some 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].…”

“…Due to the widespread application of differential equations in practice, in recent decades, many theories and methods of nonlinear analysis, such as the spaces theories [26][27][28][29][30][31], smoothness theories [32][33][34][35], operator theories [36][37][38], fixed-point theorems [18,21,24,25,[39][40][41], subsuper solution methods [17,[42][43][44][45], monotone iterative techniques [12,[46][47][48][49][50][51][52][53] and the variational method [54][55][56][57][58], have been developed to study various differential equations. For example, by adopting the fixed point theorem of the mixed monotone operator, Zhou et.…”

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].…”