Geodesic
Strong singularities in mixed boundary value problems
Mathematica Bohemica, Tome 131 (2006) no. 4, pp. 393-409

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We study singular boundary value problems with mixed boundary conditions of the form \[ (p(t)u^{\prime })^{\prime }+ p(t)f(t,u,p(t)u^{\prime })=0, \quad \lim _{t\rightarrow 0+}p(t)u^{\prime }(t)=0, \quad u(T)=0, \] where $[0,T]\subset {\mathbb{R}}.$ We assume that ${\mathbb{R}}^2,$ $f$ satisfies the Carathéodory conditions on $(0,T)\times $ $p\in C[0,T]$ and ${1/p}$ need not be integrable on $[0,T].$ Here $f$ can have time singularities at $t=0$ and/or $t=T$ and a space singularity at $x=0$. Moreover, $f$ can change its sign. Provided $f$ is nonnegative it can have even a space singularity at $y=0.$ We present conditions for the existence of solutions positive on $[0,T).$
We study singular boundary value problems with mixed boundary conditions of the form \[ (p(t)u^{\prime })^{\prime }+ p(t)f(t,u,p(t)u^{\prime })=0, \quad \lim _{t\rightarrow 0+}p(t)u^{\prime }(t)=0, \quad u(T)=0, \] where $[0,T]\subset {\mathbb{R}}.$ We assume that ${\mathbb{R}}^2,$ $f$ satisfies the Carathéodory conditions on $(0,T)\times $ $p\in C[0,T]$ and ${1/p}$ need not be integrable on $[0,T].$ Here $f$ can have time singularities at $t=0$ and/or $t=T$ and a space singularity at $x=0$. Moreover, $f$ can change its sign. Provided $f$ is nonnegative it can have even a space singularity at $y=0.$ We present conditions for the existence of solutions positive on $[0,T).$
DOI : 10.21136/MB.2006.133975
Classification : 34B15, 34B16, 34B18
Keywords: singular mixed boundary value problem; positive solution; lower function; upper function; convergence of approximate regular problems 
Rachůnková, Irena. Strong singularities in mixed boundary value problems. Mathematica Bohemica, Tome 131 (2006) no. 4, pp. 393-409. doi: 10.21136/MB.2006.133975
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