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Existence and nonexistence of positive solutions for a class of nth-order three-point boundary value problems in Banach spaces. (English) Zbl 1176.34030

The authors study the existence, nonexistence, and multiplicity of positive solutions to a nonlinear three-point boundary value problem for a differential equation of order n in an ordered Banach space. The boundary value problem has the form

x (n) (t)+f(t,x(t),x ' (t),,x (n-2) (t))=θ,t[0,1],
x (i) (0)=θ,i=0,1,,n-2,
x (n-2) (1)=ρx (n-2) (η),

where ρ,η(0,1), and θ is the zero element of the Banach space. The proofs employ fixed-point theory in cones.

MSC:
34B18Positive solutions of nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
34G20Nonlinear ODE in abstract spaces
34B10Nonlocal and multipoint boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations