On an eigenvalue problem for some nonlinear transformations of multi-dimensional arrays

Date of Completion

January 2007






In this dissertation it is shown that certain transformations of multi-dimensional arrays posses unique positive solutions. These transformations are composed of linear components defined in terms of Stieltjes matrices, and semi linear components similar to uku 3. ^ In particular, the analysis of the linear components extends some results of the Perron-Frobenius theory [11] to multi-dimensional arrays. ^ It is shown that any number greater than the smallest positive eigenvalue of the linear part is an eigenvalue of the transformation and that the corresponding positive eigenvector is unique. Moreover, such positive eigenvectors form a monotone increasing and continuous function of the corresponding eigenvalues. The connection with discrete time independent Gross-Pitaevskii equation (GPE) [4] is also shown. This equation plays a key role in modelling Bose-Einstein condensate [4] at near absolute zero temperatures. The Bose-Einstein Condensation, and the corresponding mathematical models are subject of current theoretical and numerical research. ^ We also study sign patterns of solutions of semi linear partial differential equation discretized by finite difference methods. It is shown that in somewhat more general case of one dimensional semi linear equations with arbitrary totally positive matrices, the number of sign changes in the solution does not increase with each iteration. ^