On solvability of derivative dependent doubly singular boundary value problems. (English) Zbl 1198.34028
Summary: We establish the existence of solutions of the singular boundary value problem
\[ -(p(x)y ^{\prime}(x))^{\prime}=q(x)f(x,y,py^{\prime})\text{ for }0<x\leq b, \]
and
\[ \lim_{x\rightarrow0^{+}}p(x)y^{\prime}(x)=0 , \alpha _{1} y(b)+\beta_{1} p(b)y ^{\prime}(b)=\gamma_{1}\text{ with }p(0)=0 \]
and \(q(x)\) is allowed to have an integrable discontinuity at \(x=0\). So, the problem may be doubly singular. Here, we consider \(\lim_{x\rightarrow 0^{+}}\frac{q(x)}{p'(x)}\neq0\), therefore, \(\lim_{x\rightarrow0^{+}}p(x)y'(x)=0\) does not imply \(y^{\prime}(0)=0\) unless \(\lim_{x\rightarrow0^{+}}f(x,y(x),p(x)y'(x))=0\).
\[ -(p(x)y ^{\prime}(x))^{\prime}=q(x)f(x,y,py^{\prime})\text{ for }0<x\leq b, \]
and
\[ \lim_{x\rightarrow0^{+}}p(x)y^{\prime}(x)=0 , \alpha _{1} y(b)+\beta_{1} p(b)y ^{\prime}(b)=\gamma_{1}\text{ with }p(0)=0 \]
and \(q(x)\) is allowed to have an integrable discontinuity at \(x=0\). So, the problem may be doubly singular. Here, we consider \(\lim_{x\rightarrow 0^{+}}\frac{q(x)}{p'(x)}\neq0\), therefore, \(\lim_{x\rightarrow0^{+}}p(x)y'(x)=0\) does not imply \(y^{\prime}(0)=0\) unless \(\lim_{x\rightarrow0^{+}}f(x,y(x),p(x)y'(x))=0\).
MSC:
| 34B16 | Singular nonlinear boundary value problems for ordinary differential equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
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