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

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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)

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 .

Suggested Citation

  • Robert Vrabel, 2023. "( I q )–Stability and Uniform Convergence of the Solutions of Singularly Perturbed Boundary Value Problems," Mathematics, MDPI, vol. 11(12), pages 1-14, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:12:p:2717-:d:1171939
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