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Coexistence of positive solutions of nonlinear three-point boundary value and its conjugate problem. (English) Zbl 1125.34017

This paper deals with a three-point boundary value problem of the form

-u '' =f(t,u(t)),u ' (0)=0,u(1)=αu(η),

together with its conjugate BVP

-v '' =f(s,v(s)),v ' (0)=0,v + ' (η)-v - ' (η)=αv ' (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 λ. Assuming

lim sup x0 + max 0t1 f(t,x)/x<λ<lim inf x+ min 0t1 f(t,x)/x


lim sup x+ max 0t1 f(t,x)/x<λ<lim inf x0 min 0t1 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.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
47H10Fixed point theorems for nonlinear operators on topological linear spaces