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Positive solutions for nonlinear singular boundary value problems on the half line. (English) Zbl 1149.34011

The boundary value problem
\[ x''(t)=\mu f(t,x,x'),\;t\in(0,+\infty), \]
\[ ax(0)-b\lim_{t\to0}x'(t)=k,\;\;\lim_{t\to\infty}x'(t)=0, \]
is considered for \(\mu,a>0,\,b,k\geq0\) and \(f(t,x,z)\) singular at \(x=0\) and \(z=0\). The existence of nonnegative increasing solutions is established under the assumptions that \(f\in C([0,+\infty)\) \(\times(0,+\infty)\times(0,+\infty),R)\) and there exist functions \(\Phi\in C([0,+\infty),(0,+\infty)),\) \(g,h\in C((0,+\infty),(0,+\infty))\) and \(\beta\in C((0,+\infty),(-\infty,0))\) such that \(\|\Phi\|_\infty=\sup_{t\in[0,+\infty)}| \Phi(t)| <+\infty,\) \(\int_0^1h(s)ds<+\infty,\) for each \(c\geq1\) \(\int_0^\infty\Phi(s)\max_{{1\over c}\leq x\leq c(1+s)}h(x)ds<+\infty\) holds,
\[ | f(t,x,z)| \leq\Phi(t)h(y)g(z)\;\;\text{for}\;\;(t,x,z)\in(0,+\infty)\times(0,+\infty)\times(0,+\infty), \]
and
\[ f(t,x,z)\leq x^\gamma\beta(t)\;\;\text{for}\;\;(t,x,z)\in[0,+\infty)\times(0,+\infty)\times(0,\delta], \]
for some \(\gamma\in[0,1)\) and \(\delta>0.\) To prove the existence result the authors have constructed a sequence of nonsingular integral equation problems that approximate the problem of interest and have used a fixed point theorem for cones in Banach spaces.

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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