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Characterization of eigenvalues for difference Equations subject to Lidstone conditions

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Abstract

This paper considers the following boundary value problem

$$\begin{gathered} ( - 1)^m \Delta ^{2m} y = \lambda F(k,y,\Delta y, \ldots ,\Delta ^{2m - 1} y), m \geqslant 1, 0 \leqslant k \leqslant N \hfill \\ \Delta ^{2i} y(0) = \Delta ^{2i} y(N + 2m - 2i) = 0, 0 \leqslant i \leqslant m - 1 \hfill \\ \end{gathered} $$

where λ > 0. We characterize the values of λ so that the boundary value problem has a positive solution. Further, explicit intervals of λ are established so that for any λ in the interval, existence of a positive solution of the boundary value problem is assured. We also present several examples to dwell upon the importance of the results obtained.

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Authors and Affiliations

  1. Division of Mathematics, Nanyang Technological University, 469, Bukit Timah Road, 259756, Singapore, Singapore

    Patricia J. Y. Wong

  2. Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, 119260, Singapore, Singapore

    Ravi P. Agarwal

Authors
  1. Patricia J. Y. Wong
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  2. Ravi P. Agarwal
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Correspondence to Patricia J. Y. Wong.

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Wong, P.J.Y., Agarwal, R.P. Characterization of eigenvalues for difference Equations subject to Lidstone conditions. Japan J. Indust. Appl. Math. 19, 1–18 (2002). https://doi.org/10.1007/BF03167445

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  • DOI: https://doi.org/10.1007/BF03167445

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