Oscillation criteria for second-order neutral equations with distributed deviating argument☆
Introduction
In this paper, we are concerned with the oscillation problem for the second-order neutral delay differential equations with distributed deviating argumentwhere . In this paper, we assume that
and for ;
for ;
for ;
and is not eventually zero on any half linear ;
for has a continuous and positive partial derivative on with respect to the first variable t and nondecreasing with respect to the second variable , respectively, and for ;
is nondecreasing, and the integral of Eq. (1.1) is in the sense of Riemann–Stieltijes.
We restrict our attention to those solutions of Eq. (1.1) which exist on some half linear and satisfy for any . As usual, such a solution of Eq. (1.1) is called oscillatory if the set of its zeros is unbounded from above, otherwise, it is said to be nonoscillatory. Eq. (1.1) is called oscillatory if all its solutions are oscillatory.
In the last decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of solutions for different classes of second-order differential equations [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].
The oscillation problem for nonlinear delay equation such asas well as for the linear ordinary differential equationand the neutral delay differential equationhave been studied by many authors with different methods. Some results can be found in [1], [2], [3], [4], [5], [6], [7], [8], [10], [14] and references therein. Recently, In [13], by using Riccati technique and averaging functions method, Ruan established some general oscillation criteria for second-order neutral delay differential equationSahiner [11] obtained some general oscillation criteria for neutral delay differential equationsis oscillatory. In [9], by using Riccati technique and averaging functions method, Wang established some general oscillation criteria for second-order neutral delay differential equation with distributed deviating argumentVery recently, Xu and Weng [12] also have established oscillation criteria for study Eq. (1.1), by using the generalized Riccati transformation.
In this paper, by using a generalized Riccati technique and the integral averaging technique and following the results of Wong [15] and Philos [16], we establish some oscillation criteria for Eq. (1.1), which complement and extend the results in [11], [12]. We will use the function class Y to study the oscillatory of Eq. (1.1), we say that a function belongs to the function class Y denoted by , if , where , which satisfies for , and has the partial derivative on E such that is locally integrable with respect to .
We define the operator byand the function is defined byIt is easy to verify that is a linear operator and satisfiesIn addition, we will make use of the following conditions:
There exists a positive real number M such that for ;
for .
Lemma 1.1
If , then
Section snippets
When is monotone
In this section, we shall deal with the oscillation for Eq. (1.1) under the assumptions and the following assumption.
exists, and for . Theorem 2.1 Let hold. Eq. (1.1) is oscillatory provided that for each , there exist a function and such thatwhereand the operator A is defined by (1.8) and the function
When is not monotone
In this section, we shall deal with the oscillation for Eq. (1.1) under the assumptions and the following assumption.
and for . Theorem 3.1 Let and hold. Eq. (1.1) is oscillatory provided that for each , there exists functions and such thatwhereand the operator T is defined by (1.8) and the function
Interval oscillation criteria
In this section, we will establish several interval oscillation criteria for Eq. (1.1). Theorem 4.1 Let hold. Eq. (1.1) is oscillatory provided that for each , there exist a function , and two constants such thatwhere the operator A is defined by (1.8) and the function is defined by (1.9). Proof With the proof of Theorem 2.1, when t and l are replaced by d and c, respectively, we easily see that every solution of Eq. (1.1) has
Examples
Example 5.1 Consider the nonlinear differential equationwhereIf we take ,Hence, by Corollary (2.1) we have Eq. (5.1) is oscillatory for . Example 5.2 Consider the nonlinear differential equationwhere
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This research was partial supported by the NNSF of China (10771118).