scholarly journals HIGHER-ORDER VARIATIONAL SETS, VARIATINAL DERIVATIVES AND HIGHER-ORDER NECESSARY CONDITIONS IN ABSTRACT MATHEMATICAL PROGRAMMING

10.5109/13392 ◽  
1988 ◽  
Vol 23 (1/2) ◽  
pp. 9-40 ◽  
Author(s):  
Nagata Furukawa ◽  
Yuji Yoshinaga
Keyword(s):  
Mathematical Programming ◽  
Necessary Conditions ◽  
Higher Order ◽  
Variational Sets

Oscillation and asymptotic behavior of a higher-order neutral delay difference equation with variable delays under Δ m

2021 ◽  
Vol 71 (1) ◽  
pp. 129-146
Author(s):  
Chittaranjan Behera ◽  
Radhanath Rath ◽  
Prayag Prasad Mishra
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Variable Delays ◽  
All Solutions ◽  
Delay Difference

Abstract In this article we obtain sufficient conditions for the oscillation of all solutions of the higher-order delay difference equation Δ m ( y n − ∑ j = 1 k p n j y n − m j ) + v n G ( y σ ( n ) ) − u n H ( y α ( n ) ) = f n , $$\begin{array}{} \displaystyle \Delta^{m}\big(y_n-\sum_{j=1}^k p_n^j y_{n-m_j}\big) + v_nG(y_{\sigma(n)})-u_nH(y_{\alpha(n)})=f_n\,, \end{array}$$ where m is a positive integer and Δ xn = x n+1 − xn . Also we obtain necessary conditions for a particular case of the above equation. We illustrate our results with examples for which it seems no result in the literature can be applied.

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