The eigenvalue characterization for the constant sign Green’s functions of (k,n−k)(k,n−k) problems

This paper is devoted to the study of the sign of the Green’s function related to a general linear nth-order operator, depending on a real parameter, Tn[M]Tn[M], coupled with the (k,n−k)(k,n−k) boundary value conditions. If the operator Tn[M¯]Tn[M¯] is disconjugate for a given M̄, we describe the interval of values on the real parameter M for which the Green’s function has constant sign. One of the extremes of the interval is given by the first eigenvalue of the operator Tn[M¯]Tn[M¯] satisfying (k,n−k)(k,n−k) conditions. The other extreme is related to the minimum (maximum) of the first eigenvalues of (k−1,n−k+1)(k−1,n−k+1) and (k+1,n−k−1)(k+1,n−k−1) problems. Moreover, if n−kn−k is even (odd) the Green’s function cannot be nonpositive (nonnegative). To illustrate the applicability of the obtained results, we calculate the parameter intervals of constant sign Green’s functions for particular operators. Our method avoids the necessity of calculating the expression of the Green’s function. We finalize the paper by presenting a particular equation in which it is shown that the disconjugation hypothesis on operator Tn[M¯]Tn[M¯] for a given M̄ cannot be eliminatedAlberto Cabada was partially supported by Ministerio de Educación, Cultura y Deporte, Spain, and FEDER, project MTM2013-43014-P. Lorena Saavedra was partially supported by Ministerio de Educación, Cultura y Deporte, Spain, and FEDER, project MTM2013-43014-P, and Plan I2C scholarship, Consellería de Educación, Cultura e O.U., Xunta de Galicia, and FPU scholarship, Ministerio de Educación, Cultura y Deporte, Spain. The authors would also like to express their special thanks to the reviewer of the paper for his/her remarks, which considerably improved the content of this paperS