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Behavior of solutions for a Robin problem. (English) Zbl 0704.34033
The paper studies the behavior as \(k\to 0+\) of the solutions of the Robin problem \(y''=\theta^ 2g_ k(y)\), \(-y'(-1)+cy(-1)=a\), \(y'(1)+cy(1)=a\) relative to the solutions of \(y''=\theta^ 2g_ 0(y)\) with the same boundary conditions. It is assumed that \(g_ k\in C^ 1([0,a/c])\) \((k>0)\), \(g_ 0\in C^ 1((0,a/c])\) (notice that \(g_ 0\) may be singular at \(y=0)\), \(g_ k(y)>0\) for \(y\in (0,a/c]\), \(g_ 0(0)=0\), \(g_ k(y)\sim y^ b\) as \(y\to 0\) for some \(b\geq 1\) independent of k, \(g_ k(y)\uparrow g_ 0(y)\) uniformly on each [\(\epsilon\),a/c] \((\epsilon >0)\) as \(k\to 0+\) and \(g_ 0(y)\sim y^{-d}\) as \(y\to 0\) for some \(d>0\). Such models can be applied to describe the steady-state of certain reaction-diffusion chemical kinetics. Conditions for existence and for uniqueness or multiplicity of solutions and convergence to the solutions of \(y''=\theta^ 2g_ 0(y)\) are obtained.
Reviewer: C.A.Braumann

MSC:
34B99 Boundary value problems for ordinary differential equations
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