On a class of singular boundary value problems with singular perturbation
This paper is devoted to singular boundary value problems with singular perturbation of the form \[ \varepsilon y''-{{1}\over{{x-\lambda}}}f(x,y)y'+g(x,y)=0,\;x\in(0,\lambda)\cup(\lambda,1), \] \[ y(0)=A,\;y(\lambda)=\mu,\;y(1)=B, \] where \(\varepsilon\) is a small positive parameter, the constants \(A, B, \mu\) and \(\lambda\in[0,1]\) are such that if \(\lambda=0\) or \(\lambda=1\), then \(\mu=A\) or \(\mu=B\), respectively, and the functions \(f(x,y)\) and \(g(x,y)\) are \(C^1\)-smooth on \([0,1]\times \mathbb R\) and there is a positive constant \(\sigma_0\) such that \(f(x,y)\geq\sigma_0>0\) for \((x,y)\in[0,1]\times \mathbb R\). Using the lower and upper solutions technique and the Schauder fixed point theorem, the author studies the existence and uniqueness of solutions to the considered problem as well as their asymptotic behavior with respect to \(\varepsilon.\) Moreover, it is assumed in addition that the problem \[ f(x,y)y'=(x-\lambda)g(x,y),\;y(\lambda)=\mu, \] has a solution \(y=\phi(x)\) in \(C^2[0,1]\).
scientific article; zbMATH DE number 5998055
scientific article