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Zbl 1125.34017
Wang, Shuli; Liu, Jinsheng
Coexistence of positive solutions of nonlinear three-point boundary value and its conjugate problem.
(English)
[J] J. Math. Anal. Appl. 330, No. 1, 334-351 (2007). ISSN 0022-247X

This paper deals with a three-point boundary value problem of the form $$-u''= f(t,u(t)), \quad u'(0)=0, \ u(1)=\alpha u(\eta),$$ together with its conjugate BVP $$-v''= f(s,v(s)), \; v'(0)=0, \; v'_+(\eta)-v'_-(\eta)=\alpha v'(1), \; v(1)=0.$$ It is interesting to note that the linear problems associated with the two different boundary conditions have the same first eigenvalue $\lambda$. Assuming $$\limsup_{x\to 0^+} \max_{0\leq t \leq 1} f(t,x)/x < \lambda < \liminf_{x\to +\infty} \min_{0\leq t \leq 1} f(t,x)/x$$ or $$\limsup_{x\to +\infty} \max_{0\leq t \leq 1} f(t,x)/x < \lambda < \liminf_{x\to 0} \min_{0\leq t \leq 1} f(t,x)/x,$$ the existence of at least one positive solution is proved. The other main result provides (technical) sufficient conditions for the existence of at least two positive solutions. The proof is developed in the framework of fixed point index theory in cones.
[Anna Capietto (Torino)]
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34B10 Multipoint boundary value problems
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: three-point boundary value problem; positive solutions; fixed point index

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