New Oscillation Criteria for Second-Order Forced Quasilinear Functional Differential Equations

Abstract

We establish some new interval oscillation criteria for a general class of second-order forced quasilinear functional differential equations with -Laplacian operator and mixed nonlinearities. It especially includes the linear, the one-dimensional p-Laplacian, and the prescribed mean curvature quasilinear differential operators. It continues some recently published results on the oscillations of the second-order functional differential equations including functional arguments of delay, advanced, or delay-advanced types. The nonlinear terms are of superlinear or supersublinear (mixed) types. Consequences and examples are shown to illustrate the novelty and simplicity of our oscillation criteria. ( ) satisfies a usual growth condition and the coefficients ( ), and ( ) are positive only on some intervals where ( ) changes the sign. Recently, Bai and Liu [1] have studied the oscillation of second-order delay differential equation: ( ( ) ( )) + ∑ =1 ( ) ( − ) + ∑ =1 ( ) ( − ) sgn ( − ) = ( ) , ≥ 0 ,

Parametrically Excited Oscillations of Second-Order Functional Differential Equations and Application to Duffing Equations with Time Delay Feedback

“…Obviously assumption (11) is a particular case of assumption (9). Hence, this proof is very similar to the proof of Lemma 2 and so it is left to the reader.…”

“…In the following slightly simpler version of Lemma 2, inequality (9) is replaced with an asymptotic condition and, at the same time, the limit in (10) is relaxed with the limit inferior. Thus, conditions (9) and (10) are replaced with more practical ones.…”

“…Since (V) is supposed to be odd and increasing function, just before (3), and ( ) satisfies (14), the proof of Lemma 25 in the first case, that is, ( ) ( ( )) ≤ 0 for all ∈ ( ( ), ( )), is the same as the proof of [9,Corollaries 17 and 18]. But in the second case, that is, ( ) ( ( )) ≥ 0 for all ∈ ( ( ), ( )), the proof is as follows: if previous inequality holds, then ( ) (− ( )) ≤ 0 for all ∈ ( ( ), ( )) and, therefore, to the function − ( ) one can apply the first case of this lemma and consequently one obtains…”

“…It is shortly called the parametrically excited oscillations of (1). In Section 4 we discuss some known oscillation criteria published in [1][2][3][4][5][6][7][8][9] which allow the parametrically excited oscillations but only in the form of examples. On various problems concerning the functional differential equations we refer the reader to [10][11][12][13][14][15] and the references therein.…”