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Existence and multiplicity of positive solutions for multi-parameter three-point differential equations system. (English) Zbl 1113.34017

The author studies, via fixed point index theory, the existence and multiplicity of positive solutions of a system of second order differential equations of the type

u '' (t)+λa 1 (t)f 1 (u(t),v(t))=0,t(0,1),v '' (t)+μa 2 (t)f 2 (u(t),v(t))=0,t(0,1),

subject to the boundary conditions

u(0)=0,u(1)=α 1 u(η 1 ),v(0)=0,v(1)=α 2 u(η 2 )·

Here, λ and μ are positive parameters, 0<η 2 η 1 <1, α 1 ,α 2 (0,1), and a 1 ,a 2 ,f 1 ,f 2 are continuous functions.

The author of this paper remarks that, unlike what usually happens in previous results for systems, here the two nonlinearities possess a somewhat different behavior. In fact, the function a 1 is positive and f 1 is bounded below by a negative constant, whereas f 2 is positive and the function a 2 is allowed to change sign.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE