Second-Order Differential Equation: Oscillation Theorems and Applications

Abstract

Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the form ς y u ′ y a ′ + p y u c ϑ y = 0 , for y ≥ y 0 , under the assumption ∫ ∞ ς η − 1 / a = ∞ . Two cases are considered for a < c and a > c , where a and c are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.

New Oscillation Theorems for Second-Order Differential Equations with Canonical and Non-Canonical Operator via Riccati Transformation

“…In [7], Bazighifan et al have studied oscillatory properties of even-order ordinary differential equations with variable coefficients. For more details on the oscillation theory of neutral delay differential equations, we refer the reader to the papers [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In particular, the study of oscillation of halflinear/Emden-Fowler (neutral) differential equations with deviating arguments (delayed or advanced arguments or mixed arguments) has numerous applications in physics and engineering (e.g., half-linear/Emden-Fowler differential equations arise in a variety of realworld problems, such as in the study of p-Laplace equations and chemotaxis models); see, e.g., the papers [23][24][25][26][27][28][29][30][31][32][33][34] for more details.…”

New Theorems for Oscillations to Differential Equations with Mixed Delays

“…It is interesting to notice that, in the aforementioned works, the authors obtained only sufficient conditions that ensure the oscillation of the solutions of the considered equations. A problem worthy of investigations is the study of necessary and sufficient conditions for oscillation, and some satisfactory answers were given in [11][12][13][14][15][16][17][18]. Finally, the interested readers are referred to the following papers and to the references therein for some recent results on the oscillation theory for ordinary differential equations of several orders [19][20][21][22][23][24][25][26][27].…”

Novel Schemes for Cauchy-Riemann System of Equations with Cauchy Conditions

“…Researchers have proposed several numerical and analytical methods to obtain the solutions of complex systems. For instance, Santra et al [1] presented the oscillation theorem and also discussed the consistency analysis of second-order dierential equations in their exploration. They proved several theorems for the validity of the solutions of this class of dierential system.…”