(<i>I</i>_q)–Stability and Uniform Convergence of the Solutions of Singularly Perturbed Boundary Value Problems

Abstract

In this paper, using the notion of (Iq)–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [a,b] have been established for four different types of boundary conditions for a singularly perturbed differential equation εy″=f(x,y,y′), a≤x≤b. In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f(x,y,y′)=0.

Keywords

• second-order ordinary differential equation
• boundary value problem
• Neumann conditions
• periodic conditions
• three- and four-point conditions
• singular perturbation