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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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