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Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems. (English) Zbl 0970.34019
The author considers the boundary value problem (BVP) $Lu=f(t, u),\;0<t<1,\tag{1}$ $\alpha u(0)-\beta u'(0)=0,\;\gamma u(1)+\delta u'(1)= 0, \tag{2}$ with $$f\in C(I\times\mathbb{R}_+, \mathbb{R}_+)$$, $$I=[0,1]$$, $$\mathbb{R}_+= [0, \infty)$$, $$\alpha,\beta, \gamma,\delta\geq 0$$ with $$\gamma\beta+ \alpha\gamma+ \alpha\delta >0$$ and $$Lu=-(p(t)u')' +q(t)u$$. Here, $$p\in C^1(I, (0, \infty))$$, $$q\in C(I,\mathbb{R}_+)$$. The author also assumes that $$\lambda=0$$ is not an eigenvalue of $$Lu=\lambda u$$ subject to conditions (2). He establishes the existence of positive solutions to BVP(1), (2). Conditions are given in terms of the relative behavior of the quotient $$(f(t,u))/u$$ for $$u$$ near 0 and $$\infty$$ with respect to the smallest positive eigenvalue of $$Lu=\lambda u$$ satisfying the boundary conditions (2).

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B24 Sturm-Liouville theory
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##### References:
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