The analysis of the dependence of integral operators on perturbations plays an important role in the study of inverse problems and of perturbed boundary value problems. In this paper, we focus on the mapping properties of the volume potentials with weakly singular periodic kernels. Our main result is to prove that the map which takes a density function and a periodic kernel to a (suitable restriction of the) volume potential is bilinear and continuous with values in a Roumieu class of analytic functions. This result extends to the periodic case of some previous results obtained by the authors for nonperiodic potentials, and it is motivated by the study of perturbation problems for the solutions of boundary value problems for elliptic differential equations in periodic domains.

We derive some sufficient conditions for the existence of a square-mean almost automorphic solution for a stochastic cellular neural network on time scales by using Krasnoselskii’s fixed point theorem. The Banach contraction principle is also used to show the uniqueness of a solution with some additional restrictions. Exponential stability of the obtained solution is also discussed by taking a suitable Lyapunov functional on time scale. At the end, we give a numerical example to illustrate the effectiveness of the obtained theoretical results. Our results are valid for general time scale, and the cases $\mathbb{𝕋}=ℝ$ or $\mathbb{Z}$ are particular cases.

We construct and study a new polyconvolution $\ast \left(f,g,h\right)\left(x\right)$ of functions $f$, $g$ and $h$ for the Hartley integral transform $H_2$. We will show that polyconvolution satisfies the following factorization equality:

$$H_2\left(f\underset{2}{\ast }g\right)\left(y\right)=\left(H_{2}f\right)\left(y\right)\left(H_{2}g\right)\left(y\right)\left(H_{2}h\right)\left(y\right),\forall y\in ℝ.$$

We prove the existence of this polyconvolution in the space $L\left(ℝ\right)$. As examples, applications to solve an integral equation of polyconvolution type and a system of integral equations of polyconvolution type are presented.

An integral equation approach is adopted to price American-style non-recourse stock loans with finite maturities. In particular, we use Fourier transform to reduce the partial differential equation governing the price of an American-style non-recourse stock loan into an ordinary differential equation, the solution of which can be easily found (in the Fourier space) and analytically inverted into the original space. As a result, we can decompose the price of an American non-recourse stock loan into two main components: the price of its European counterpart and the early exit premium. This decomposition allows us to break the pricing problem of American stock loans into two simple steps: 1) finding the optimal exit prices; 2) calculating the American non-recourse stock loans. Our obtained numerical results appear to fit very well with those obtained from literature.

The purpose of this paper is twofold. We first carry out an analysis of a class of noncompact weakly singular Volterra integral equations whose kernels possess both an end-point and diagonal singularities. A numerical method based on piecewise collocation scheme on uniform meshes including its convergence analysis are presented. We will then indicate the usefulness of our main concerned equation in the numerical study of an initial-value problem for fractional differential equations with variable coefficient. As a result, we reformulate the linear fractional differential equations with Erdélyi–Kober derivative to a particular type of the underlying Volterra equations with weakly singular kernels. Under certain verifiable conditions on the coefficient, the existence and uniqueness results as well as the numerical solution of the resulting equation by spline collocation method on piecewise polynomial space are analyzed. The reliability and efficiency of this approach are finally demonstrated by some numerical experiments.

This paper is devoted to the existence and nonexistence of traveling wave solutions for an SEIR model in combination with random dispersal and nonlocal dispersal, which can be seen as a continuity work of Tian and Yuan (16). The main difficulties lie in the fact that the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity. We use a proper iteration technique to construct a pair of upper and lower solutions, find a new nonmonotone operator and then apply Schauder fixed point theorem to obtain the threshold dynamics for this model. Our results also show that the diffusion ability of the exposed individuals and the infected individuals can accelerate the speed of the spread of the disease while the nonlocal interaction between the infective and the susceptible individuals can speed up the spread of the disease.

We study the existence and regularity of solutions for neutral integro-differential equations with nonlocal condition by applying the theory of resolvent operators established recently in the literature. Since the nonlinear terms of the systems involve spacial derivatives, we make full use of the theory of fractional power, the $\alpha$-norm, and Schauder’s fixed point theorem to discuss the problems. An example is given to illustrate the applications of the obtained results.