The abstract equation of the form $u=Ku\cdot L(Fu)$ is investigated in this paper. By applying a fixed point theorem for the product of operators $K$ and $A=L(F)$ defined on a Banach lattice algebra $E$, we obtain existence and uniqueness results of fixed points of the operator $T=K\cdot A$. Moreover, we state a sufficient condition on the spectral radius of a majorant linear mapping of $L$ under which the equation $u=Tu$ has the $L$-Hyers–Ulam stability. As an application, the obtained results are used to prove existence and uniqueness of solutions and Hyers–Ulam stability of a ($p_1,p_2,\dots ,p_n$)-Laplacian hybrid fractional differential system. An example is also constructed to illustrate the main results. This work contains many new ideas, and gives a unified approach applicable to several types of differential and integral equations.

We examine the solvability of fractional integral equations using the techniques of measure of noncompactness and the Petryshyn’s fixed-point theorem in Banach space concerning the Riemann-Liouville integral operator. The results obtained in this paper cover some earlier results obtained by numerous authors under weaker conditions. In the end, two applications are given to illustrate the major result.

In this study, a new numerical algorithm is proposed for solving a class of nonlinear fractional Volterra integral equations of the second kind based on our newly constructed hat functions. New functions that are called cubic hat functions (CHFs) and operational matrices of fractional order integration of these functions are applied. In a new numerical approach, the fractional order operational matrix of CHFs and the powers of weakly singular kernels of integral equations are handed down as a structure for converting the principal problem into a number of systems containing three-variable polynomial equations. Also, error analysis, convergence analysis of this method and convergence rate are investigated. In the last part, the high precision of the utilized method is shown with three examples. In addition, comparisons with Jacobi spectral Galerkin and modified hat functions methods demonstrate the improved performance of the presented approach.

By a probabilistic approach, we look at an obstacle problem with nonlinear Neumann boundary conditions for parabolic semilinear integral-partial differential equations. We prove the existence of a continuous viscosity solution of this problem. The nonlinear part of the equation and the Neumann condition satisfy the stochastic monotonicity condition on the solution variable. Furthermore, the nonlinear part is stochastic Lipschitz on the parts that depend on the gradient and the integral of the solution. It should be noted that the existence of the viscosity solution for this problem has recently been investigated using a standard monotonicity and Lipschitz conditions. We show that the solution of the related reflected generalized backward stochastic differential equations with jumps exists and is unique when the barrier is right continuous left limited (rcll) and the generators satisfy stochastic monotonicity and Lipschitz conditions. In this case, we get a comparison result.

In 2020, Chalmoukis introduced a generalization of the Volterra operator and studied its boundedness and compactness on Hardy spaces. Inspired by Chalmoukis (2020), Li, Liu and Lou (2014), and Li, Liu and Yuan (2020), we study the boundedness and compactness of the generalized Volterra operator on analytic Morrey spaces and Dirichlet type spaces.

Electromyogram (EMG) signals play a significant role in decoding muscle contraction information for robotic hand prosthesis controllers. Widely applied decoders require a large amount of EMG signals sensors, resulting in complicated calculations and unsatisfactory predictions. By the biomechanical process of single degree-of-freedom human hand movements, only several EMG signals are essential for accurate predictions. Recently, a novel predictor of hand movements adopted a multistage sequential adaptive functional estimation (SAFE) method based on the historical functional linear model (FLM) to select important EMG signals and provide precise projections.

However, SAFE repeatedly performs matrix-vector multiplications with a dense representation matrix of the integral operator for the FLM, which is computationally expensive. Noting that with a properly chosen basis, the representation of the integral operator concentrates on a few bands of the basis, the goal of this study is to develop a fast multiscale SAFE (MSAFE) method aiming at reducing computational costs while preserving (or even improving) the accuracy of the original SAFE method. Specifically, a multiscale piecewise polynomial basis is adopted to discretize the integral operator for the FLM, resulting in an approximately sparse representation matrix, and then the matrix is truncated to a sparse one. This approach not only accelerates computations but also improves robustness against noises. When applied to real hand movement data, MSAFE saves 85%$\sim$90% computing time compared with SAFE, while producing better sensor selection and comparable accuracy. In a simulation study, MSAFE shows stronger stability in sensor selection and prediction accuracy against correlated noise than SAFE.

We investigate the standing waves of fractional Schrödinger equations. Using the method of moving planes, we first prove a monotonicity of standing waves without any decay conditions at infinity. Based on the monotonicity, we obtain the nonexistence result of standing waves on the whole space.