Singularly perturbed boundary value problems for systems of Tichonov's type in case of exchange of stabilities
Systems containing ``slow'' and ``fast'' equations are known as Tikhonov-type equations. Here, the authors deal with the ``fast'' equation: \(\varepsilon^2u''= g(u,v,x,\varepsilon)\), where \('\) denotes differentiation with respect to \(x\), and the ``slow'' equation \(v'= f(u,v,x,\varepsilon)\), or a similar equation of second-order. Let \(I_0\) denote the interval \(0<\varepsilon< \varepsilon_0\), with \(\varepsilon_0\ll 1\). \(u\) obeys the no flux across the boundary condition: \(u'(0)= u'(1)= 0\). The variable \(v\) obeys either the initial condition \(v'(0)= v^0\) or the boundary conditions \(v(0)= v^0\), \(v(1)= v^1\). The degenerate equation obtained by setting \(\varepsilon= 0\): \(g(u,v,x,0)= 0\), has two solutions: \(u=\varphi_1(v,x)\) and \(u= \varphi_2(v,x)\) such that \(\varphi_1(v,x)> \varphi_2(v,x)\) for \(v< v^0\), \(\varphi_1(v,x)= \varphi_2(v,x)\) if \(0\leq x\leq 1\), and \(\varphi_1(v,x)> \varphi_2(v,x)\) for \(v> v^0\). Thus this system differs from the original Tikhonov system. In some mathematical circles these types of equations almost automatically bring forth nonstandard analysis arguments; ``Canards'' (duck hunting), infinitely fast motions, and infinitesimal highways seem to appear from nowhere. However, it is a well-known lemma that any results derived by enlarging a space to a nonstandard model can be also derived in the old-fashioned way by taking appropriate limits. In this article, the authors take the classical approach which was pioneered by Chaplygin, Tikhonov, Levinson, O'Malley, and among others, the authors of this article. Here, the authors assume that the two linearly independent solutions may intersect each other. This makes application of most classical techniques originated by Tikhonov not applicable. It is known that such intersecting solutions may interchange their properties, including the stability of solutions. The authors study the stability of degenerate solutions, using the technique of upper and lower solutions and prove the existence of solutions in a neighborhood of an \(\varepsilon\)-neighborhood of the degenerate solutions, provided \(\varepsilon\) is sufficiently small (translate this into ``infinitesimal'' in the parallel nonstandard or ``canard'' jargon). The authors offer as a specific example the boundary value problem \[ \varepsilon^2u''= g(u,v,x,\varepsilon),\quad v''= f(u,v,x,\varepsilon), \] with existence and smoothness assumptions, to show that the solution \(U= \{u,v\}\) satisfies for every \(x\) the condition that \(U\) differs from the degenerate solution \(\widehat U\) by \(O(\varepsilon^{1/2})\). This is a well written, interesting article, which should be a challenge to the French school (M. Diener and collaborators) and other devotees of A. Robinson to demonstrate that these results can be obtained much easier and perhaps even generalized by a parallel use of nonstandard arguments.
scientific article; zbMATH DE number 1417617