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The existence of positive solutions and oscillation of solutions of higher-order difference equations with forcing terms. (English) Zbl 0933.39026
Summary: Sufficient conditions are established for the existence of positive solutions, and oscillation of all bounded solutions of the neutral difference equation $$\Delta^p[x_n- cx_{n-l}]+ q_nf(x_{n-k})=h_n, \quad n\ge n_0,$$ where $\Delta$ is the forward difference operator $\Delta x_n= x_{n+1}-x_n$, $l$ and $k$ are integers, $c\ne\pm 1$ is a real number, and $\{q_n\}$ and $\{h_n\}$ are real sequences. It is also shown that some of these sufficient conditions are necessary.

##### MSC:
 39A11 Stability of difference equations (MSC2000)
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##### References:
 [1] Cheng, S.S.; Yan, T.C.; Li, H.J.: Oscillation criteria for second order difference equation. Funkcial. ekvac. 34, 223-239 (1991) · Zbl 0773.39001 [2] Graef, J.R.; Spikes, P.W.: Asymptotic decay of oscillatory solutions of forced nonlinear difference equations. Dynamical systems and applications 3, 95-102 (1994) · Zbl 0790.39002 [3] He, X.-Z.: Oscillatory and asymptotic behavior of second order nonlinear difference equations. J. math. Anal. appl. 175, 482-498 (1993) · Zbl 0780.39001 [4] J. Jaros and I.P. Stavroulakis, Necessary and sufficient conditions for oscillation of difference equations with several delays, Utilitas Math. (to appear). [5] Lalli, B.S.; Zhang, B.G.; Li, J.Z.: On the oscillation of solutions and existence of positive solutions of neutral difference equations. J. math. Anal. appl. 158, 213-233 (1991) · Zbl 0732.39002 [6] Lalli, B.S.: Oscillation theorems for neutral difference equations. Computers math. Applic. 28, No. 1--3, 191-202 (1994) · Zbl 0807.39004 [7] Popenda, J.: The oscillation of solutions of difference equations. Computers math. Applic. 28, No. 1--3, 271-279 (1994) · Zbl 0807.39006 [8] Zhicheng, W.; Jianshe, Y.: Oscillation of second order nonlinear difference equations. Funkcial. ekvac. 34, 313-319 (1991) · Zbl 0742.39003 [9] Zhang, B.G.; Yu, J.S.: On the existence of asymptotically decaying positive solutions of second order neutral differential equations. J. math. Anal. appl. 166, 1-11 (1992) · Zbl 0754.34075 [10] Agarwal, R.P.: Difference calculus with applications to difference equations. Int. ser. Num. math. 71, 95-110 (1984) · Zbl 0592.39001 [11] Agarwal, R.P.: Properties of solutions of higher order nonlinear difference equations II. An. sti. Univ. ”al. I. cuza” iasi, 85-96 (1985) [12] Zafer, A.; Dahiya, R.S.: Oscillation of a neutral difference equation. Appl. math. Lett. 6, No. 2, 71-74 (1993) · Zbl 0772.39001 [13] Zafer, A.: Oscillatory and asymptotic behavior of higher order difference equations. Mathl. comput. Modelling 21, No. 4, 43-50 (1995) · Zbl 0820.39001 [14] Agarwal, R.P.: Difference equations and inequalities. (1992) · Zbl 0925.39001