Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients
© L. H. Xu and J. Yang. 2010
Received: 12 November 2009
Accepted: 11 February 2010
Published: 8 March 2010
This paper is concerned with a class of nonlinear delay partial difference equations with positive and negative coefficients, which also contains forcing terms. By making use of frequency measures, some new oscillatory criteria are established.
Partial difference equations are difference equations that involve functions with two or more independent integer variables. Such equations arise from considerations of random walk problems, molecular structure problems, and numerical difference approximation problems. Recently, there have been a large number of papers devoted to partial difference equations, and the problem of oscillatory of solutions and frequent oscillatory solutions for partial difference equations is receiving much attention.
In , authors considered oscillatory behavior of the partial difference equations with positive and negative coefficients of the form
but they have not discussed frequent oscillations of this equation.
In , authors considered oscillatory behavior for nonlinear partial difference equations with positive and negative coefficients of the form
In , authors considered frequent oscillation in the nonlinear partial difference equation
In , authors considered oscillations of the partial difference equations with several nonlinear terms of the form,
and in  authors considered frequent oscillations of these equations.
In , authors considered unsaturated solutions for partial difference equations with forcing terms
In this paper, we will consider the equation of the following form:
The usual concepts of oscillation or stability of steady state solutions do not catch all their fine details, and it is necessary to use the concept of frequency measures introduced in  to provide better descriptions. In this paper, by employing frequency measures, some new oscillatory criteria of (1.6) are established.
To the best of our knowledge, nothing is known regarding the qualitative behaviour of the solutions of (1.6), because these equations contain positive and negative coefficients, and also contain forcing terms.
Our plan is as follows. In the next section, we will recall some of the basic results related to frequency measures. Then we obtain several criteria for all solutions of (1.6) to be frequently oscillatory and unsaturated. In the final section, we give one example to illustrate our results.
The following results are concerned with the frequency measures and their proofs are similar to those in .
For any real double sequence defined on a subset of the level set is denoted by The notations and are similarly defined. Let be a real double sequence. If then is said to be frequently positive, and if , then is said to be frequently negative.
is said to be frequently oscillatory if it is neither frequently positive nor frequently negative. If then is said to have unsaturated upper positive part, and if then is said to have unsaturated lower positive part. is said to have unsaturated positive part if .
The concepts of frequently oscillatory and unsaturated double sequences were introduced in [5–11]. It was also observed that if a double sequence is frequently oscillatory or has unsaturated positive part, then it is oscillatory; that is, is not positive for all large and nor negative for all large and Thus if we can show that every solution of (1.6) is frequently oscillatory or has unsaturated positive part, then every solution of (1.6) is oscillatory.
3. Frequently Oscillatory Solutions
which is a contradiction.
is infinite. Again we may arrive at a contradiction as above. The proof is complete.
is infinite. This can lead to a contradiction again. The proof is complete.
4. Unsaturated Solutions
The methods used in the above proofs can be modified to obtain the following results for unsaturated solutions.
is infinite and a subsequent contradiction.
is infinite. Then by discussions similar to these in the proof of Theorem 3.2 again, we may arrive at a contradiction. The proof is complete.
Combining Theorems 3.3 and 4.1, we have the following Theorem 4.2 and the proof of this theorem is omitted.
is infinite. Then, we can also lead to a contradiction. The proof is complete.
We remark that every nontrivial solution of (1.6) has an unsaturated lower positive part under the same conditions as Theorems 4.1, 4.2, or 4.3. So we can obtain that every nontrivial solution of (1.6) has an unsaturated positive part.
We give one example to illustrate our previous results.
Then according to Theorems 3.2 or 3.3, we know that every nontrivial solution of (5.1) is frequently oscillatory. If , we see that all conditions in Theorems 4.1, 4.2, or 4.3 are satisfied. Thus, every nontrivial solution of (5.1) has an unsaturated upper positive part.
This project was supported by the NNSF of P.R.China (60604004) and by Natural Science Foundation of Hebei province (no. 07M005).
- Liu ST, Zhang BG: Oscillatory behavior of delay partial difference equations with positive and negative coefficients. Computers & Mathematics with Applications 2002,43(8-9):951-964. 10.1016/S0898-1221(02)80005-0MathSciNetView ArticleMATHGoogle Scholar
- Liu ST, Liu YQ, Deng FQ: Oscillation for nonlinear delay partial difference equations with positive and negative coefficients. Computers & Mathematics with Applications 2002,43(10-11):1219-1230. 10.1016/S0898-1221(02)00093-7MathSciNetView ArticleMATHGoogle Scholar
- Yang J, Zhang YJ, Cheng SS: Frequent oscillation in a nonlinear partial difference equation. Central European Journal of Mathematics 2007,5(3):607-618. 10.2478/s11533-007-0017-1MathSciNetView ArticleMATHGoogle Scholar
- Zhang BG, Xing Q: Oscillation of certain partial difference equations. Journal of Mathematical Analysis and Applications 2007,329(1):567-580. 10.1016/j.jmaa.2006.07.002MathSciNetView ArticleMATHGoogle Scholar
- Yang J, Zhang YJ: Frequent oscillatory solutions of a nonlinear partial difference equation. Journal of Computational and Applied Mathematics 2009,224(2):492-499. 10.1016/j.cam.2008.05.035MathSciNetView ArticleMATHGoogle Scholar
- Zhu ZQ, Cheng SS: Unsaturated solutions for partial difference equations with forcing terms. Central European Journal of Mathematics 2006,4(4):656-668. 10.2478/s11533-006-0030-9MathSciNetView ArticleMATHGoogle Scholar
- Tian CJ, Xie S-L, Cheng SS: Measures for oscillatory sequences. Computers & Mathematics with Applications 1998,36(10–12):149-161. 10.1016/S0898-1221(98)80017-5MathSciNetView ArticleMATHGoogle Scholar
- Cheng SS: Partial Difference Equations, Advances in Discrete Mathematics and Applications. Volume 3. Taylor & Francis, London, UK; 2003:xii+267.Google Scholar
- Zhu ZQ, Cheng SS: Frequently oscillatory solutions of neutral difference equations. Southeast Asian Bulletin of Mathematics 2005,29(3):627-634.MathSciNetMATHGoogle Scholar
- Zhu ZQ, Cheng SS: Frequently oscillatory solutions for multi-level partial difference equations. International Mathematical Forum 2006,1(29–32):1497-1509.MathSciNetMATHGoogle Scholar
- Zhang B, Zhou Y: Qualitative Analysis of Delay Partial Difference Equations, Contemporary Mathematics and Its Applications. Volume 4. Hindawi Publishing Corporation, New York, NY, USA; 2007:viii+374.View ArticleGoogle Scholar
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