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Constant sign Green’s function for simply supported beam equation. (English) Zbl 1367.34026

Summary: The aim of this paper consists on the study of the following fourth-order operator: \[ T[M]\,u(t) \equiv u^{(4)}(t)+p_1(t)u'''(t)+p_2(t)u''(t)+Mu(t),\;t \in I \equiv[a,b] , \] coupled with the two point boundary conditions: \[ u(a)=u(b)=u''(a)=u''(b)=0 . \] So, we define the following space: \[ X=\{ u \in C^4(I) : u(a)=u(b)=u''(a)=u''(b)=0 \}. \] Here, \(p_1 \in C^3(I)\) and \(p_2\in C^2(I)\). By assuming that the second order linear differential equation \[ L_2\, u(t) \equiv u''(t)+p_1(t)\,u'(t)+p_2(t)\,u(t)=0\,, t \in I, \] is disconjugate on \(I\), we characterize the parameter’s set where the Green’s function related to operator \(T[M]\) in \(X\) is of constant sign on \(I \times I\). Such a characterization is equivalent to the strongly inverse positive (negative) character of operator \(T[M]\) on \(X\) and comes from the first eigenvalues of operator \(T[0]\) on suitable spaces.

MSC:

34B27 Green’s functions for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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