A class of fourth order nonlinear boundary value problem with singular perturbation
The paper considers a class of singularly perturbed nonlinear boundary value problems (SPBVP), which arises, e.g. in fluid mechanics. Unlike the traditional asymptotic analysis, the authors are concerned with the existence, asymptotic estimation and uniqueness of the solution to the following type fourth-order SPBVP without using the singular perturbation techniques, \[\varepsilon y^{(4)}(x) = f(x, y(x), y''(x), \varepsilon),\] \[y(0) = y'(0) = y''(0)= y''(1) = 0,\] where \(\varepsilon \) is a small positive parameter and \(x\in [0, 1]\). Using Laplace transform, the paper constructs Green's function for the associated fourth-order nonlinear boundary value problem without usual perturbation. Based upon some differential inequality techniques and appropriate upper-lower solutions of the model problem, the authors study the existence, uniqueness, and corresponding asymptotic estimation of the solution to the SPBVP. To construct appropriate asymptotic estimations and the existence for SPBVP, they use a reduced problem when the perturbation parameter \(\varepsilon \) is equal to \(0\). Using the upper-lower solutions theory and nonlinear analysis techniques, the existence of the auxiliary problem is obtained, and the solution to SPBVP is discussed. An example illustrates the uniqueness result for such SPBVP without using the perturbation analysis.
scientific article; zbMATH DE number 7317523