Singular perturbation of boundary value problem for a vector fourth order nonlinear differential equation

Abstract

We study the vector boundary value problem with boundary perturbations:

$$\begin{gathered} \varepsilon ^2 y^{(4)} = f(x,y,y'',\varepsilon ,\mu ) (\mu< x< 1 - \mu ) \hfill \\ y(x,\varepsilon ,\mu )\left| {_{x = \mu } = A_1 (\varepsilon ,\mu ), y(x,\varepsilon ,\mu )} \right|_{x = 1 - \mu } = B_1 (\varepsilon ,\mu ) \hfill \\ y''(x,\varepsilon ,\mu )\left| {_{x = \mu } = A_2 (\varepsilon ,\mu ), y''(x,\varepsilon ,\mu )} \right|_{x = 1 - \mu } = B_2 (\varepsilon ,\mu ) \hfill \\ \end{gathered}$$

where yf, A_j andB _j (j=1,2) are n-dimensional vector functions and ε, μ are two small positive parameters. This vector boundary value problem does not appear to have been studied, although the scalar boundary value problem has been treated. Under appropriate assumptions, using the method of differential inequalities we find a solution of the vector boundary value problem and obtain the uniformly valid asymptotic expansions.