Abstract
In this paper, we establish the qualitative behavior of the even-order advanced differential equation a υ y κ − 1 υ β ′ + ∑ i = 1 j q i υ g y η i υ = 0 , υ ≥ υ 0 . The results obtained are based on the Riccati transformation and the theory of comparison with first- and second-order equations. This new theorem complements and improves a number of results reported in the literature. Two examples are presented to demonstrate the main results.
Highlights
Advanced differential equations are of practical importance, which model a phenomenon in which the rate of change of a quantity depends on present and future values of the quantity
Population genetics [3], the study of wavelets [4], population growth [5], the field of time symmetric electrodynamics [6], neural networks [7], optimal control problems with delay [8], economics [8], dynamical systems, mathematics of networks, optimization, electrical power systems, materials, energy j ≥ 1, etc. [9] have been studied using advanced differential equations and many approaches discussed in [10,11,12,13,14,15,16,17,18,19,20,21,22] can be presented for solution of such equations
In 1980, Shah et al [23] discussed the uniqueness and existence of the solution to nonlinear and linear such types equations, while the oscillation properties of the solution were investigated by Ladas and Stavroulakis [24], and after that, in the last decade, Further refinements and improvements in the theory of advanced differential equations have been made by different researchers and it is still an active of research in engineering and applied sciences
Summary
Advanced differential equations are of practical importance, which model a phenomenon in which the rate of change of a quantity depends on present and future values of the quantity. [9] have been studied using advanced differential equations and many approaches discussed in [10,11,12,13,14,15,16,17,18,19,20,21,22] can be presented for solution of such equations. In 1980, Shah et al [23] discussed the uniqueness and existence of the solution to nonlinear and linear such types equations, while the oscillation properties of the solution were investigated by Ladas and Stavroulakis [24], and after that, in the last decade, Further refinements and improvements in the theory of advanced differential equations have been made by different researchers and it is still an active of research in engineering and applied sciences. Equation (1) is said to be oscillatory if all of its solutions are oscillatory
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