In this paper, we deal with a class of fractional integro-differential equations involving impulsive effects and nonlocal conditions, whose principal part is of diffusion-wave type. Our aim is to establish some existence and stability results for integral solutions to the problem at hand by use of the fixed point approach.

In this paper, we investigate global existence and blow-up properties of nonnegative solutions to a nonlocal $p$-Laplacian evolution equation with weighted nonlinear nonlocal boundary condition. By using the method of upper and lower solutions, we consider some effects of weight function and nonlinear exponent on the global and blow-up solutions. In addition, we show the blow-up rate estimate, blow-up profile and blow-up set for linear diffusion case.

Revisiting the author's paper from 1995 on this topic, a fully discrete collocation method is proposed for the hypersingular integral equation arising from the double-layer approach for the solution of Neumann boundary value problems in two dimensions which is based on trigonometric differentiation to discretize the principal part of the hypersingular operator. Convergence in a Sobolev space setting is proven and the spectral convergence of the method is exhibited by numerical examples.

In the present work, we are concerned with approximations of solutions to a retarded type fractional differential equation with a deviated argument in a separable Hilbert space~$H$. We consider an integral equation associated with a given problem and then consider a sequence of approximate integral equations. We prove the existence, uniqueness and convergence to each of the approximate integral equations by using analytic semigroup theory and the fixed point method. We also prove that the limiting function satisfies the associated integral equation. Finally, we consider Faedo-Galerkin approximations of solutions and prove some convergence results.

This paper is concerned with traveling wave solutions of a nonlocal dispersal SIR epidemic model with standard incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number of the corresponding ordinary differential model and the minimal wave speed. These threshold dynamics are proved by constructing an invariant cone and applying Schauder's fixed point theorem on this cone and the Laplace transform. The main difficulties are the lack of an occurrence of a regularizing effect and the loss of the order-preserving property of this model.

We consider a nonautonomous impulsive Cau\-chy problem of parabolic type involving a nonlocal initial condition in a Banach space $X$, where the operators in linear part (possibly unbounded) depend on time $t$ and generate an evolution family. New existence theorems of mild solutions to such a problem, in the absence of compactness and Lipschitz continuity of the impulsive item and nonlocal item, are established. The non-autonomous impulsive Cauchy problem of neutral type with nonlocal initial condition is also considered. Comparisons with available literature are also given. Finally, as a sample of application, these results are applied to a system of partial differential equations with impulsive condition and nonlocal initial condition. Our results essentially extend some existing results in this area.