In this paper, under some super- and sub-linear growth conditions, we study the existence of positive solutions for a high-order Riemann–Liouville type fractional integral boundary value problem involving fractional derivatives. Our analysis methods are based on the fixed point index and nonsymmetric property of the Green function. Additionally, we provide some valid examples to illustrate our main results.

The major goal of this manuscript is to investigate the existence, uniqueness, and stability of a q-fractional Langevin differential equation with q-fractional integral conditions. We demonstrate the existence and uniqueness of the solution to the proposed q-fractional Langevin differential equation using the Banach contraction principle and Schaefer’s fixed-point theorem. We also elaborate on different kinds of Ulam stability. The theoretical outcomes are verified by examples.

Hyperspectral images (HSIs) restoration is an important preprocessing step. The spectral vectors in HSI can be separated into different classification based on the land‐covers, which means the spectral space can be regarded as the union of several low‐rank subspaces. Subspace low‐rank representation (SLRR) is powerful in exploring the inner low‐rank structure and has been applied for HSI restoration. However, the traditional SLRR only seek for the rank‐minimum representation under a given dictionary, which may treat the structured sparse noise as inherent low‐rank components. In addition, the SLRR framework cannot make full use of the spatial information. In this study, a framework named subspace representation with low‐rank constraint and spatial‐spectral total variation is proposed for HSI restoration. In which, an artificial rank constraint is introduced to control the rank of the representation result, which can improve the removal of the structured sparse noise and exploit the intrinsic structure of spectral space more effectively. Meanwhile, the spatial‐spectral total variation regularisation is applied to enhance the spatial and spectral smoothness. Several experiments conducted in simulated and real HSI datasets demonstrate that the proposed method can achieve a state‐of‐the‐art performance both in visual quality and quantitative assessments.