## Existence of positive solutions for second-order semipositone differential equations on the half-line.(English)Zbl 1117.34033

The authors are concerned with the existence of positive solutions of a semi-positone Sturm-Liouville boundary value problem on the half-line of the form
$(p(t)x'(t))'+f(t,x)+q(t)=0,$
$\alpha_1 x(0)-\beta_1\lim_{t\rightarrow 0^+}p(t)x'(t)=0,$
$\alpha_2 \lim_{t\rightarrow +\infty}x(t)-\beta_2\lim_{t\rightarrow +\infty}p(t)x'(t)=0,$
where $$\alpha_i\geq 0,\; \beta_i>0\; (i=1,2)$$ with $\alpha_2\beta_1+\alpha_1\beta_2 +\alpha_1\alpha_2\int^{+\infty}_0 \frac{ds}{p(s)}>0;\quad q:(0,\infty)\to R$
is Lebesgue integrable. Their main results generalize those obtained in Yansheng Liu [Appl. Math. Comput. 144, 543–556 (2003; Zbl 1036.34027)].
Reviewer: Ruyun Ma (Lanzhou)

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations

### Keywords:

semipositone; half line; positive solutions; cone

Zbl 1036.34027
Full Text:

### References:

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