A General Comparison Result for Higher Order Nonlinear Difference Equations With Deviating Arguments |
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Authors: | John R Graef Agnes Miciano-Cariño Chuanxi Qian |
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Institution: | 1. Department of Mathematics , University of Tennessee at Chattanooga , Chattanooga , TN , 37403 , USA;2. Institute of Computer Science , University of the Philippines , Los Ba?os, Laguna , Philippines;3. Department of Mathematics and Statistics , Mississippi State University , Mississippi State , MS , 39762 , USA |
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Abstract: | The authors consider m -th order nonlinear difference equations of the form D m p x n + i h j ( n , x s j ( n ) )=0, j =1,2,( E j ) where m S 1, n ] N 0 ={0,1,2,…}, D 0 p x n = x n , D i p x n = p n i j ( D i m 1 p x n ), i =1,2,…, m , j x n = x n +1 m x n , { p n 1 },…,{ p n m } are real sequences, p n i >0, and p n m L 1. In Eq. ( E 1 ) , p = a and p n i = a n i , and in Eq. ( E 2 ) , p = A and p n i = A n i , i =1,2,…, m . Here, { s j ( n )} are sequences of nonnegative integers with s j ( n ) M X as n M X , and h j : N 0 2 R M R is continuous with uh j ( n , u )>0 for u p 0. They prove a comparison result on the oscillation of solutions and the asymptotic behavior of nonoscillatory solutions of Eq. ( E j ) for j =1,2. Examples illustrating the results are also included. |
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Keywords: | Comparison Theorem Difference Equations With Deviating Arguments Nonlinear Difference Equations Oscillation |
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