2016/04/14 by Cabada, Alberto, Saavedra, Lorena
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1604.04245
The aim of this paper consists on the study of the following fourth-order operator: T[M] u(t)≡ u(4)(t)+p1(t) u"'(t)+p2(t) u"(t)+M u(t) , t∈ I ≡ [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=\lbrace u∈ C4(I) | u(a)=u(b)=u"(a)=u"(b)=0 \rbrace . Here p1∈ C3(I) and p2∈ C2(I). By assuming that the second order linear differential equation L2 u(t)≡ u"(t)+p1(t) u'(t)+p2(t) u(t)=0 , t∈ 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 × I. Such 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.