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Existence results for semipositone boundary value problems. (English) Zbl 1021.34017
The authors consider the boundary value problems $(p(t) u')'+\lambda f(t,u)+ e(t, u)= 0,\quad r< t< R,$ $au(r)- bp(r) u'(r)= 0,\quad cu(R)+ dp(R) u'(R)= 0,$ where $$f$$ and $$e: [r,R]\times [0,\infty)\to \mathbb{R}$$ are two continuous functions satisfying $$f\geq 0$$ and $$|e|\leq M$$ for some $$M> 0$$. Using fixed-point theorems in cones, they show the existence of at least one positive solution if either $$f$$ is superlinear at infinity and $$\lambda> 0$$ be small enough or $$f$$ is sublinear at infinity and $$\lambda> 0$$ is large enough. Their results extend some known results in the particular case where $$e\equiv 0$$, cf. [V. Anuradha, D. D. Hai and R. Shivaji, Proc. Am. Math. Soc. 124, No. 3, 757-763 (1996; Zbl 0857.34032)].

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
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