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Existence results for superlinear semipositone BVP’s. (English) Zbl 0857.34032
The authors consider the existence of a positive solution to the Sturm-Liouville boundary value problem \((p(t)\cdot u')'+\lambda\cdot f(t,u)=0\), \(r<t<R\), \(a\cdot u(r)-b\cdot u'(r)=0\), \(c\cdot u(R)+d\cdot u'(R)=0\), for \(\lambda>0\) small under the condition \[ \lim_{u\to\infty} {{f(t,u)}\over u}=\infty \] uniformly on a compact subinterval of \((r,R)\). The main results of the paper are related to the cases: \(f\) is a positive possibly singular function, and \(f\) is a regular optionally positive function. The proofs of the results are based on fixed point theorems in a cone.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
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