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

In this work, we prove some new oscillation theorems for second-order neutral delay differential equations of the form (a(ξ)((v(ξ)+ b(ξ)v(θ(ξ)))′))′+ c(ξ)G1(v(κ(ξ)))+ d(ξ)G2(v(ς(ξ))) = 0 under canonical and non-canonical operators, that is, ∫ ∞ ξ0 dξ a(ξ) = ∞ and ∫ ∞ ξ0 dξ a(ξ) < ∞. We use the Riccati transformation to prove our main results. Furthermore, some examples are provided to show the effectiveness and feasibility of the main results.


Introduction
It is well known that differential equations have many applications in research, for example, population growth, decay, Newton's law of cooling, glucose absorption by the body, spread of epidemics, Newton's second law of motion, and interacting species competition. They appear in the study of many real-world problems (see [1][2][3]).
Here, we mention some recent developments of oscillation theory to neutral differential equations.
In [4], Santra et al. have studied explicit criteria for the oscillation of second-order differential equations with several sub-linear positive neutral coefficients of the form and obtained some new sufficient conditions for the oscillation of (1). Santra et al. [5] have studied asymptotic behavior of a class of second-order nonlinear neutral differential equations with multiple delays of the form and obtained some new sufficient conditions for the oscillation of solution of (2) under a non-canonical operator with various ranges of the neutral coefficient b. In another paper [6], Santra et al. have established some new oscillation theorems to neutral differential equations with mixed delays under a canonical operator with 0 ≤ b < 1. 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. In particular, by using different methods, the following papers were concerned with the oscillation of various classes of half-linear/Emden-Fowler differential equations and half-linear/Emden-Fowler differential equations with different neutral coefficients: the paper [24] was concerned with neutral differential equations assuming that 0 ≤ b(ξ) < 1 and b(ξ) > 1, where b is the neutral coefficient; the paper [25] was concerned with neutral differential equations assuming that 0 ≤ b(ξ) < 1; the paper [27] was concerned with neutral differential equations assuming that b(ξ) is nonpositive; the papers [28,32] were concerned with neutral differential equations in the case where b(ξ) > 1; the paper [31] was concerned with neutral differential equations assuming that 0 ≤ b(ξ) ≤ b 0 < ∞ and b(ξ) > 1; the paper [33] was concerned with neutral differential equations in the case where 0 ≤ b(ξ) ≤ b 0 < ∞; the paper [34] was concerned with neutral differential equations in the case when 0 ≤ b(ξ) = b 0 = 1; whereas the paper [30] was concerned with differential equations with a nonlinear neutral term assuming that 0 ≤ b(ξ) ≤ b < 1. These examples have the same research topic as that of this paper.

Proof.
Proof is similar to that in Theorem 1, only we have to choose for the current theorem.

Oscillation Results under (C2)
In this section, we will prove some new oscillation results for (3) under the assumption (C2).

Examples
In this section, we will give two examples to illustrate our main results.

Example 2.
Let us consider the following equation Therefore, by Corollary 8, all solutions of (21) oscillate.

Conclusions
In this paper, we defined some new general Riccati transformations to study the oscillation of second-order differential equations of neutral type with two nonlinear functions G 1 and G 2 and proved new oscillation theorems under canonical and non-canonical operators with the help of the general Riccati transformation. It would be of attentiveness to analyze the oscillation of (3) for sub-linear, super-linear and integral neutral coefficients; for more details, we refer the reader to the papers [24,27,28,[30][31][32][33][34].