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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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