scholarly journals A Systematic Approach to Higher-Order Necessary Conditions in Optimization Theory

1984 ◽  
Vol 22 (2) ◽  
pp. 211-238 ◽  
Author(s):  
Dennis S. Bernstein
Keyword(s):  
Systematic Approach ◽  
Necessary Conditions ◽  
Optimization Theory ◽  
Higher Order

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.

scholarly journals Minimization of Vectors of Curvilinear Functionals on Second-Order Jet Bundle: Dual Program Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Mihai Postolache
Keyword(s):  
Variational Problem ◽  
Necessary Conditions ◽  
Optimization Theory ◽  
General Setting ◽  
Second Order ◽  
Mechanical Work ◽  
Program Theory ◽  
Jet Bundle ◽  
Dual Program ◽  
Suitable Framework

A previous paper (2011), Pitea and Postolache, considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problem, subject to PDE and/or PDI constraints. They have chosen the suitable framework offered by the second-order jet bundle, and initiated an optimization theory for this class of problems by introducing necessary conditions. As natural continuation of these results, the present work introduces a dual program theory, the general setting, and the theory which is new as a whole, containing our results.

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