Dynamical Behavior of Nonlinear Impulsive Abstract Partial Differential Equations on Networks with Multiple Time-Varying Delays and Mixed Boundary Conditions Involving Time-Varying Delays

Various real-world problems in physics, biology, neuroscience, communication and transport networks, engineering science, and so on, subject to abrupt changes at certain instants during the dynamical processes, can be well-described by impulsive partial differential equations on networks with time-varying delays. The purpose of this paper is to investigate the existence, stability, and global attractivity of a class of coupled impulsive system of nonlinear reaction-diffusion type equations on networks in which different time-varying delays appear both in the nonlinear reaction functions and in the mixed boundary conditions. The problem under consideration can be included coupled systems of reaction-diffusion and ordinary differential equations. By using the method of upper and lower solutions and its associated monotone iterations, the existence-uniqueness of the solution, the stability and attractivity analysis for quasi-monotone nondecreasing and mixed quasi-monotone reaction and impulsive functions, are considered. The results for the general system are applied to the standard PDE reaction-diffusion system without time delays and/or impulsive behavior and to the corresponding ordinary differential system. To illustrate the abstract results, problems of three-species food-chain reaction-diffusion models with time-varying delays and impulsive perturbations are considered.