Asymptotic expansion method for some nonlinear two point boundary value problems with rapidly oscillating coefficients (Q2470122)

Consider the two point boundary value problem \[ (a(x/\varepsilon)(u^{\varepsilon})')' + b_1(x/\varepsilon)\psi_1((u^{\varepsilon})') + b_2(x/\varepsilon)\psi_2(u^{\varepsilon})= f(x),\quad 0 < x < m\varepsilon; \] \[ u^{\varepsilon}(0)= u_0,\qquad u^{\varepsilon}(m\varepsilon)=u_1, \] \(m\) is a positive integer, \(f\in C^{\infty}[0,m\varepsilon]\), \(b_1(\cdot)\), \(b_1(\cdot)\) are \(m\varepsilon\)-periodic and \(\psi_1(\cdot)\), \(\psi_2(\cdot)\) some functions. The problem is studied by using asymptotic expansions of the solutions with respect to \(\varepsilon\) and by comparison with the solutions of the two point boundary value problem \[ (a_0U')' + b_0\psi_1(U') + \psi_2(U) = F(x),\quad 0 < x < m\varepsilon; \qquad U(0)=u_0,\quad U(m\varepsilon)=u_1, \] where \(F(x)\) is a function of \(f\) and its derivatives.