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Positive solutions for singular third-order boundary value problem with dependence on the first order derivative on the half-line. (English) Zbl 1203.34038
Using a fixed point theorem in a cone, the authors give sufficient conditions for the existence of countably many positive solutions of the boundary value problem
\[ \bigl(\varphi(-u'')(t)\bigr)'+a(t)f(t,u,u')=0,\;\;0<t<+\infty, \]
\[ u(0)-\beta u'(0)=0,\;u'(\infty)=0,\;u''(0)=0, \] where \(a:[0,+\infty)\to[0,+\infty)\) has countably many singularities, \(f\in C([0,+\infty)^3,[0,+\infty))\) is such that \(f(t,(1+t)u,u')\) is bounded on \([0,+\infty)\) if \(u\) is bounded, \(f(t,0,0)\not\equiv0\) on any subinterval of \((0,+\infty)\), and \(\varphi:R\to R,\) with \(\varphi(0)=0\), is an increasing homeomorphism and positive homomorphism, i.e. \(\varphi\) is a continuous bijection such that \(\varphi^{-1}\) is also continuous, \(\varphi(x)\leq \varphi(y)\) for all \(x,y\in R\) such that \(x\leq y\) and \(\varphi(xy)=\varphi(x)\varphi(y)\) for all \(x,y\in[0,+\infty)\). Also, \(a\) and \(\varphi^{-1}\) satisfy the conditions
\[ 0<\int_0^{+\infty}a(t)dt<+\infty,\;\int_0^{+\infty}\varphi^{-1}\Bigl(\int_0^sa(\tau)d\tau\Bigr)ds<+\infty \] and
\[ \int_0^{+\infty}s\varphi^{-1}\Bigl(\int_0^sa(\tau)d\tau\Bigr)ds<+\infty \]
and there is a sequence \(\{t_i\}_{i=1}^{\infty}\) such that \(\lim_{t\to t_i}a(t)=\infty,\) \(\lim_{i\to\infty}t_i=t_o<+\infty\) and either \(1\leq t_{i+1}<t_i\), \(t_0>1\) or \(0<t_{i+1}<t_i<1,\) \(0<t_0<1\).

MSC:
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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