Oscillation of Emden–Fowler-Type Neutral Delay Differential Equations

Abstract

In this work, we consider a type of second-order functional differential equations and establish qualitative properties of their solutions. These new results complement and improve a number of results reported in the literature. Finally, we provide an example that illustrates our results.

New Criteria for Sharp Oscillation of Second-Order Neutral Delay Differential Equations

“…On the other hand, in [43,44], we have presented an oscillation criterion for (1) with p(t) = 0 which is sharp for the Euler half-linear delay differential Equation (31). In the present paper, the method developed in [43,44] has been extended for neutral differential equations of the form (1), under the assumptions (H1)-(H5).…”

“…The methods mostly used in investigating the oscillatory behavior of (1) have been based on a reduction of order and comparison with oscillation of first-order delay differential equations, or on reducing (1) to a first-order Riccati inequality, based on a suitable Riccati type substitution, see e.g., [17] for more details. We note that none of the related results [3][4][5][6][7]10,[12][13][14][15][16][17][18][20][21][22]26,[28][29][30][31][32][33][34][35][36]39,42,46] involving (1) with α = 1, r(t) = 1, p(t) = 0, gives a sharp result when applied to the Euler linear delay differential equation…”

More Effective Conditions for Oscillatory Properties of Differential Equations

“…Delay differential equations contribute to many applications such as torsional oscillations which have been observed during earthquakes, see [1]. However, oscillation theory has gained particular attention due to its widespread applications in mechanical oscillations, earthquake structures, clinical applications, frequency measurements and harmonic oscillator which involves symmetrical properties; see [2,3]. In context of oscillation theory, it has been the object of many researchers who have investigated this notion for non-linear neutral differential and difference equations; the reader can refer to [4][5][6][7][8][9][10][11].…”

New Theorems for Oscillations to Differential Equations with Mixed Delays

“…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].…”