The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems

Abstract

In this paper, we discuss third-order full nonlinear singularly perturbed vector boundary value problems. We first present the existence of solutions for the nonlinear vector boundary value problems without perturbation by using the upper and lower solutions method and topological degree theory. Then the existence, uniqueness and asymptotic estimates of solutions for the singularly perturbed vector boundary value problems are established by constructing appropriate a lower solution-upper solution pair, as well as analysis technique. Some known results are extended.

Asymptotics of the solution to a stationary piecewise-smooth reaction-diffusion equation with a multiple root of the degenerate equation

“…For example, by using inclusion theory [13,15], the dynamics of the fishing model with the discontinuous harvesting policy, and neural networks with discontinuous neuron activation and the susceptible-infectious-recovered (SIR) epidemic model with discontinuous treatments are studied [16][17][18][19][20]. With the development of applications of mathematical models, it is necessary to introduce singular perturbation equations whose highest derivatives are multiplied by small parameters [6,7,11,[24][25][26]30,40,42]. In recent years, piecewise-smooth dynamical systems with a small parameter have been received much attention due to their wide use as mathematical models in mechanics, superconducting science, biology, etc.…”