Universal bounds for eigenvalues of Schrödinger operators on Riemannian manifolds.

Wang, Qiaoling; Xia, Changyu

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This paper is concerned with a class of nonlinear difference inequalities which include many different classes of difference inequalities and equations as special cases. By means of a Riccati type transformation, necessary and sufficient conditions for the existence of eventually positive solutions and positive nonincreasing solutions are obtained. Various type of comparison theorems are also derived as applications, which extends many theorems in the literature.