## The existence of countably many positive solutions for some nonlinear three-point boundary problems on the half-line.(English)Zbl 1166.34304

Summary: We study the existence of countably many positive solutions for some nonlinear singular three-point boundary problems on the half-line
\begin{aligned} &(\phi(u')(t))'+a(t)f(t,u(t))=0,\quad 0<t<+\infty,\\ & u(0)-B_0(y'(\eta))=0,\quad u'(\infty)=0,\end{aligned}
where $$\phi(s):\mathbb{R}\to \mathbb{R}$$ is an increasing homeomorphism and positive homomorphism and $$\phi(0)=0$$ and $$\eta\in(0,+\infty)$$, $$a:[0,+\infty)\to[0,+\infty)$$ and has countably many singularities on $$[0,+\infty)$$. By using the fixed-point index theorem and a new fixed-point theorem in cones, the existence of countably many solutions are obtained under conditions weaker than those used by B. F. Liu and J. H. Zhang [J. Math. Anal. Appl. 309, No. 2, 505–516 (2005; Zbl 1086.34022)].

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations

Zbl 1086.34022
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### References:

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