In this article, we are concerned with the oscillation of the fractional differential equation r(t) D α − y η (t) − q(t)f ⎛ ⎝ ∞ t (v − t) −α y(v)dv ⎞ ⎠ = 0 for t > 0, where D α − y is the Liouville right-sided fractional derivative of order a (0,1) of y and h >0 is a quotient of odd positive integers. We establish some oscillation criteria for the equation by using a generalized Riccati transformation technique and an inequality. Examples are shown to illustrate our main results. To the best of author's knowledge, nothing is known regarding the oscillatory behavior of the equation, so this article initiates the study. MSC (2010): 34A08; 34C10.
The paper deals with the forced oscillation of the fractional differential equationm-q a x is the Riemann-Liouville fractional integral of order m -q of x, and b k (k = 1, 2, . . . , m) are/is constants/constant. We obtain some oscillation theorems for the equation by reducing the fractional differential equation to the equivalent Volterra fractional integral equation and by applying Young's inequality. We also establish some new oscillation criteria for the equation when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator. The results obtained here improve and extend some existing results. An example is given to illustrate our theoretical results. MSC: 34A08; 34C10
In this paper, we derive some sufficient conditions for the oscillation and asymptotic behavior of the nth-order nonlinear neutral delay dynamic equationson time scales, where α > 0 is a constant, γ > 0 is a quotient of odd positive integers and λ = ±1. Our results in this paper not only extend and improve some known results but also present a valuable unified approach for the investigation of oscillation and asymptotic behavior of nth-order nonlinear neutral delay differential equations and nth-order nonlinear neutral delay difference equations. Examples are provided to show the importance of our main results.
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