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This paper is devoted to the study of the sign of the Green's functionrelated to a general linear $n^{\rm th}$-order operator, depending on a realparameter, $T_n[M]$, coupled with the $(k,n-k)$ boundary value conditions.  If operator $T_n[\bar M]$ is disconjugate for a given $\bar M$, we describethe interval of values on the real parameter $M$ for which the Green's functionhas constant sign.  One of the extremes of the interval is given by the first eigenvalue ofoperator $T_n[\bar M]$ satisfying $(k,n-k)$ conditions.  The other extreme is related to the minimum (maximum) of the firsteigenvalues of $(k-1,n-k+1)$ and $(k+1,n-k-1)$ problems.  Moreover if $n-k$ is even (odd) the Green's function cannot be non-positive(non-negative).  To illustrate the applicability of the obtained results, we calculate theparameter intervals of constant sign Green's functions for particularoperators. Our method avoids the necessity of calculating the expression of theGreen's function.  We finalize the paper by presenting a particular equation in which it isshown that the disconjugation hypothesis on operator $T_n[\bar M]$ for a given$\bar M$ cannot be eliminated.

The eigenvalue Characterization for the constant Sign Green's Functions of $(k,n-k)$ problems