Xinguang Zhang

In this paper, we investigate the existence of positive solutions for a class of p-Laplacian tempered Riemann–Liouville fractional equations involving signed measures and sign-changing perturbations with Riemann–Stieltjes boundary conditions. By performing a spectral analysis of the associated linear operator and applying Gelfand’s formula, we establish several fundamental properties of the principal eigenvalue for the corresponding linear fractional-order differential equations. Furthermore, under suitable growth conditions on the nonlinear terms, we employ the fixed point index theory to derive new existence results for positive solutions of the proposed equations.

In this paper, we investigate the existence of positive solutions of the eigenvalue problem for a singular tempered fractional equation with a Riemann–Stieltjes integral boundary condition and signed measures. By establishing the Green function and its properties, an eigenvalue interval for the existence of positive solutions is outlined based on Schauder’s fixed-point theorem and the upper and lower solutions method. An interesting feature of this paper is that f may be singular in both the time and space variables, and the Riemann–Stieltjes integral may involve signed measures.

In this paper, we focus on the existence of solutions for a singular ‐Hessian system with augmented gradient term. By using a radial symmetric transformation and constructing a pair of suitable upper and lower solutions of the singular ‐Hessian system, the existence and the asymptotic estimate of the solution for the system are established under the case where the nonlinearities are allowed to have stronger singularity on the space variables. In particular, the nonsingular case is also considered and a sharp result is derived.

In this paper, we focus on the multiplicity of positive solutions for a singular tempered fractional initial-boundary value problem with a p-Laplacian operator and a changing-sign perturbation term. By introducing a truncation function and combing with the properties of the solution of isomorphic linear equations, we transform the changing-sign tempered fractional initial-boundary value problem into a positive problem, and then the existence results of multiple positive solutions are established by the fixed point theorem in a cone. It is worth noting that the changing-sign perturbation term only satisfies the weaker Carathèodory conditions, which implies that the perturbation term ℏ can be allowed to have an infinite number of singular points; moreover, the value of the changing-sign perturbation term can tend to negative infinity in some singular points.

In this paper, we focus on the existence of positive solutions for a class of p-Laplacian tempered fractional diffusion equations involving a lower tempered integral operator and a Riemann–Stieltjes integral boundary condition. By introducing certain new local growth conditions and establishing an a priori estimate for the Green’s function, several sufficient conditions on the existence of positive solutions for the equation are derived by using a fixed point theorem. Interesting points are that the tempered fractional diffusion equation contains a lower tempered integral operator and that the boundary condition involves the Riemann–Stieltjes integral, which can be a changing-sign measure.

In this paper, we focus on the uniqueness and iterative properties of positive solution for a coupled p-Laplacian system of singular tempered fractional equations with differential order and characteristics. Firstly, the system is converted to an integral equation, and then, a coupled iterative technique and some suitable growth conditions are proposed; furthermore, some elaborate results about the uniqueness and iterative properties of positive solutions of the system are established, which include the uniqueness, the convergence analysis, the asymptotic behavior, and error estimation, as well as the convergence rate of the positive solution. The interesting points of this paper are that the order of the system of equations is different and the nonlinear terms of the system possess the opposite monotonicity and allow for stronger singularities at space variables.

In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder’s fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables.

In this paper, we focus on the existence of positive solutions for a singular tempered sub-diffusion fractional model involving a quasi-homogeneous nonlinear operator. By using the spectrum theory and computing the fixed point index, some new sufficient conditions for the existence of positive solutions are derived. It is worth pointing out that the nonlinearity of the equation contains a tempered fractional sub-diffusion term, and is allowed to possess strong singularities in time and space variables. In particular, the quasi-homogeneous operator is a nonlinear and non-symmetrical operator.