Contrast structures for a system of singularly perturbed equations
A system of two singulary perturbed quasilinear equations of second order is considered in the following form \[ \varepsilon \frac{d^2y_i}{dx^2}= A_{i1} \frac{dy_1}{dx}+ A_{i2} \frac{dy_2}{dx}+ B_i, \quad i=1,2,\quad 0< x< 1, \] where \(A_{ij}\equiv A_{ij} (y_i, y_j,x)\), \(j=1,2\), \(B_i\) subject to the boundary conditions \[ y_1(0,\varepsilon)= y_1^0, \quad y_2(1,\varepsilon)= y_2^1, \quad y_1'(0,\varepsilon)= z_1^0, \quad y_2'(1,\varepsilon)= z_2^1. \] The existence of a steplike contrast structure is investigated in a system where both unknown functions in the limit have jumps in the neighborhood of a certain point when a perturbation parameter tends to zero. The main result is that there exists a function \(y(x)\) that satisfies the above problem up to a discrepancy of order \(o(\varepsilon)\).
singularly perturbed quasilinear equations
scientific article; zbMATH DE number 1440444