Uniqueness of monotone positive solutions for singular boundary value problems. (English) Zbl 1023.34019

Palamides, P. K.

Commun. Appl. Nonlinear Anal. 9, No. 3, 79-89 (2002).

Here, the nonlinear scalar differential equation \[ \frac 1{p(t)}(p(t)y'(t))'=q(t)f(t,y(t),p(t)y'(t)) \] is considered, where \(p\) and \(q\) are positive on \((0,1)\), “singular” at \(t=0,1\) and/or \(y=0\) and \(f\in C((0,1)\times{\mathbb{R}}^+\times{\mathbb{R}}^-)\), associated to the boundary conditions \[ \lim\limits_{t\to 0+}p(t)y'(t)=m,\quad m\leq 0,\quad\lim\limits_{t\to 1-}y(t)=0. \]
It is proved that, if \[ f(t,y,v)<0,\quad\text{for all}\quad t\in[0,1],\quad y>0,\quad v\leq 0,\tag{1} \]
\[ \lim\limits_{y\to 0+}\max_{0\leq t\leq 1}f(t,y,v)= -\infty,\quad\lim\limits_{y\to+\infty}\min_{0\leq t\leq 1}f(t,y,v)=0,\quad m_0\leq v\leq m,\tag{2} \]
\[ p\in C[0,1]\cap C^1(0,1),\quad\int\limits_0^1\frac{dt}{p(t)}<\infty,\quad q\in C(0,1),\quad\int\limits_0^1p(t)q(t)dt<\infty,\tag{3} \] and, for each \(\eta>0\), \(H>\eta\), there exists a continuous function \(\psi:[0,\infty)\to[0,\infty)\) such that \[ \int\limits_{-m}^{+\infty}\frac{ds}{\psi(s)}>\int\limits_0^1p(t)q(t)dt, \quad |f(t,y,v)|\leq\psi(|v|),\quad (t,y,v)\in[0,1]\times[\eta,H]\times{\mathbb{R}},\tag{4} \] then the boundary value problem has a positive decreasing solution.
Furthermore, there exists \(M>0\), such that for any such solution, \[ 0<y(t)\leq M,\quad 0\leq t<1. \]

Reviewer: Andrey Ivanovic Sedov (Magnitogorsk)

MSC:

34B16	Singular nonlinear boundary value problems for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B05	Linear boundary value problems for ordinary differential equations

Keywords:

boundary value problems; positive monotone solution; vector field; sublinear; superlinear; shooting method