( I q )–Stability and Uniform Convergence of the Solutions of Singularly Perturbed Boundary Value Problems

Robert Vrabel ()
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Robert Vrabel: Institute of Applied Informatics, Automation and Mechatronics, Slovak University of Technology in Bratislava, Bottova 25, 917 01 Trnava, Slovakia

Mathematics, 2023, vol. 11, issue 12, 1-14

Abstract: In this paper, using the notion of ( I q )–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 ε y ″ = 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; (Iq)–stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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