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Zhang, B. G.; Agarwal, R. P.

The oscillation and stability of delay partial difference equations. (English) Zbl 1062.39011

Comput. Math. Appl. 45, No. 6-9, 1253-1295 (2003).
Partial difference equations are difference equations that involve functions of two or more independent variables. Such equations occur frequently in the approximation of solutions of partial differential equations (PDEs) by finite difference methods, random walk problems, the study of molecular orbits and mathematical physics problems.
In this paper the authors consider the partial difference equation \[ A_{m+1,n}+ A_{m,n+1}- p\cdot A_{mn}+ \sum^u_{i=1} q_i A_{m-k_i,n-\ell_i}= 0,\tag{1} \] where \(p\), \(q_i\) are real numbers, \(k_i\) and \(\ell_i\in N_t\), \(i= 1,2,\dots, n\), \(N_t= t,t+1,\dots\) and \(u\) is a positive integer. Equation (1) includes the equation \[ A_{m+1,n}+ A_{m,n+1}+ A_{m-1,n}+ A_{m,n-1}- p\cdot A_{m,n}+ \sum^u_{i=1} r_i A_{m-k_i, n-\ell_i}= 0,\tag{2} \] which can be regarded as discrete analogs of the functional partial differential equation (PDE) \[ {\partial^2A\over\partial x^2}+ {\partial^2A\over\partial y^2}+ (4-p) A(x,y)+ \sum^u_{i=1} r_i A(x- \sigma_i, y-\tau_i)= 0.\tag{3} \] The oscillation of (3) has been studied by M. I. Tramov [Differ. Uravn. 20, No. 4, 721–723 (1984; Zbl 0598.35123)].
Although partial difference equation appear well before partial differential equations, the former equations have not drawn much attention as compared to their continuous counterparts.
The objective of this paper is to offer some fundamental concepts and important results in the oscillation and stability of partial difference equations.
In Section 2 a necessary and sufficient condition is introduced for all proper solutions of (1) with constant parameters to be oscillatory. In Section 3 some results for the oscillation of PDEs with variable coefficients are presented. In Section 4 the authors show some results for the oscillation of nonlinear partial difference equations. In Section 5, some linearized oscillation results are introduced. In Section 6 the stability of partial difference equations is considered.
Reviewer: Stefan Balint (Timişoara)

Cited in 11 Documents

MSC:

39A11 Stability of difference equations (MSC2000)

Keywords:

Partial difference equations; Oscillation; Stability; functional partial differential equation; variable coefficients; nonlinear partial difference equations

Citations:

Zbl 0598.35123
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\textit{B. G. Zhang} and \textit{R. P. Agarwal}, Comput. Math. Appl. 45, No. 6--9, 1253--1295 (2003; Zbl 1062.39011)
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References:

[1] Li, X. P., Partial difference equations used in the study of molecular orbits, Acta Chimica Sinica, 40, 8, 688-698 (1982), (in Chinese)
[2] Courant, R.; Fridrichs, K.; Lewye, H., On partial difference equations of mathematical physics, IBM J., 11, 215-234 (1967)
[3] Kelley, W. G.; Peterson, A. C., Difference Equations: An Introduction with Applications (1991), Academic Press: Academic Press New York · Zbl 0733.39001
[4] Agarwal, R. P., Difference Equations and Inequalities (2000), Marcel Dekker: Marcel Dekker New York · Zbl 1006.00501
[5] Cheng, S. S., Invitation to partial difference equations, (Communications in Difference Equations, Proceedings of the Fourth International Conference on Difference Equations. Communications in Difference Equations, Proceedings of the Fourth International Conference on Difference Equations, Poznan, Poland (1998), Gordon and Breach Science), 91-106 · Zbl 0988.39003
[6] Tramov, M. I., Oscillation of partial differential equations with deviating arguments, Differencial’nye Uravneniya, 20, 721-723 (1984), (in Russian) · Zbl 0598.35123
[7] Zhang, B. G.; Liu, S. T., Necessary and sufficient conditions for oscillations of partial difference equations, DCDIS, 3, 89-96 (1997) · Zbl 0874.39010
[8] Gilbert, R. P., Function Theoretic Methods in Partial Differential Equations (1969), Academic Press: Academic Press New York · Zbl 0187.35303
[9] Zhang, B. G.; Liu, S. T., Necessary and sufficient conditions for oscillations of delay partial difference equations, Differential Inclusion, 15, 1-7 (1995)
[10] Liu, S. T.; Wang, H., Necessary and sufficient conditions for oscilations of a class of delay partial difference equations, Dynamic Systems and Applications, 7, 4, 495-500 (1998) · Zbl 0922.39004
[11] Zhang, B. G.; Liu, S. T., Necessary and sufficient conditions for oscillations of hyperbolic type partial difference equations, (Proceedings of the Second International Conference of Difference Equations (1997), Gordon and Breach), 649-656 · Zbl 0892.35153
[12] Liu, S. T.; Zhang, B. G., Oscillation of a class of partial difference equations, PanAmerican Math. J., 8, 93-100 (1998) · Zbl 0958.39019
[13] Zhang, B. G.; Liu, S. T., Necessary and sufficient conditions for oscillations of linear delay partial difference equations, Discrete Dynamics in Nature and Society, 1, 265-268 (1998) · Zbl 0965.39003
[14] Györi, I.; Ladas, G., Oscillation Theory of Delay Differential Equations with Applications (1991), Clarendon Press: Clarendon Press Oxford · Zbl 0780.34048
[15] Zhang, B. G.; Liu, B. M., Necessary and sufficient conditions of partial difference equations with continuous variables, Computers Math. Applic., 38, 5/6, 163-167 (1999) · Zbl 0935.39007
[16] Zhang, B. G.; Liu, S. T., On the oscillation of two partial difference equations, J. Math. Anal. Appl., 206, 480-492 (1997) · Zbl 0877.39012
[17] Zhang, B. G.; Liu, S. T.; Cheng, S. S., Oscillation of a class of delay partial difference equations, J. Difference Equations Appl., 1, 215-226 (1995) · Zbl 0856.39015
[18] Choi, S. K.; Zhang, B. G., Oscillation of certain partial difference equations, Discrete Dynamics in Nature and Society, 2, 257-265 (1998) · Zbl 0931.39003
[19] Zhang, B. G.; Liu, S. T., Oscillations of a partial difference equation with variable coefficients, Computers Math. Applic., 36, 10-12, 235-242 (1998) · Zbl 0933.39029
[20] Zhang, B. G., Oscillation criteria of partial difference equations with continuous variables, Acta Math. Sinica, 42, 3, 487-494 (1999), (in Chinese) · Zbl 1015.39002
[21] Agarwal, R. P.; Zhou, Y., Oscillation of partial difference equations with continous variables, Mathl. Comput. Modelling, 31, 2/3, 17-29 (2000) · Zbl 1042.39511
[22] Choi, S. K.; Koo, N. J.; Zhang, B. G., The oscillation of partial difference equations with continuous arguments, Computers Math. Applic., 41, 12, 1495-1503 (2001) · Zbl 0984.39004
[23] Zhang, B. G.; Liu, S. T., Oscillation of partial difference equations, PanAmerican Math. J., 5, 61-70 (1995) · Zbl 0840.39004
[24] Tian, C.-J.; Xie, S.-L.; Cheng, S. S., Measures for oscillatory sequences, Computers Math. Applic., 36, 10-12, 149-161 (1998) · Zbl 0933.39024
[25] Tian, C. J.; Zhang, B. G., Frequent oscillation of a class of partial difference equations, Zeit. Anal. Anwend., 18, 1, 111-130 (1999) · Zbl 0923.39009
[26] Liu, S. T.; Liu, A. Q.; Guan, X. P.; Yang, J., Existence of monotone positive solutions of neutral partial difference equations, J. Math. Anal. Appl., 246, 1-14 (2000)
[27] Cheng, S. S., Sturmian theorems for hyperbolic type partial difference equations, J. of Difference Equations and Applications, 2, 375-387 (1996) · Zbl 0881.39002
[28] Cheng, S. S.; Liu, S. T.; Zhang, B. G., Positive flows of an infinite network, Communications in Applied Analysis, 1, 1, 83-90 (1997) · Zbl 0880.90039
[29] Zhang, B. G.; Liu, B. M., Oscillation criteria of certain nonlinear partial difference equations, Computers Math. Applic., 38, 11/12, 107-112 (1999) · Zbl 0972.39009
[30] Liu, S.; Guan, X.; Yang, J., Nonexistence of positive solutions of a class of nonlinear delay partial difference equations, J. Math. Anal. Appl., 234, 361-374 (1999)
[31] Wong, P. J.Y.; Agarwal, R. P., Nonexistence of unbounded nonoscillatory solutions of partial difference equations, J. Math. Anal. Appl., 214, 2, 503-605 (1997) · Zbl 0895.39005
[32] Wong, P. J.Y.; Agarwal, R. P., Oscillation criteria for nonlinear partial difference equations with delays, Computers Math. Applic., 32, 6, 57-86 (1996) · Zbl 0863.35111
[33] Wong, P. J.Y.; Agarwal, R. P., On the oscillation of partial difference equations generated by deviating arguments, Acta Math. Hungar., 79, 1/2, 1-29 (1998) · Zbl 0929.39004
[34] Agarwal, R. P.; Wong, P. J.Y., Advanced Topics in Difference Equations (1997), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0914.39005
[35] Shi, B.; Wang, Z. C.; Yu, J. S., Oscillation of nonlinear partial difference equations with delays, Computers Math. Applic., 32, 12, 29-39 (1996) · Zbl 0877.35131
[36] Shi, B.; Wang, Z. C.; Yu, J. S., Square-summable stability in parabolic Volterra difference equations, Methods and Applications of Analysis, 3, 2, 273-284 (1996) · Zbl 0868.39004
[37] Cheng, S. S.; Zhang, B. G., Qualitative theory of partial difference equations (I): Oscillation of nonlinear partial difference equations, Tamkang J. Math., 25, 3, 279-288 (1994) · Zbl 0809.39002
[38] Cheng, S. S.; Xie, S. L.; Zhang, B. G., Qualitative theory of partial difference equations (II): Oscillation criteria for direct control systems in several variables, Tamkang J. Math., 26, 1, 65-79 (1995) · Zbl 0828.39001
[39] Cheng, S. S.; Xie, S. L.; Zhang, B. G., Qualitative theory of partial difference equations (III): Forced oscillations of parabolic type partial difference equations, Tamkang J. Math., 26, 2, 177-192 (1995) · Zbl 0839.39006
[40] Cheng, S. S.; Zhang, B. G.; Xie, S. L., Qualitative theory of partial difference equations (IV): Forced oscillations of hyperbolic type partial difference equations, Tamkang J. Math., 26, 4, 337-360 (1995) · Zbl 0851.39004
[41] Cheng, S. S.; Xie, S. L.; Zhang, B. G., Qualitative theory of partial difference equations (V): Sturmian theorems for a class of partial difference equations, Tamkang J. Math., 27, 1, 89-97 (1997) · Zbl 0851.39005
[42] Ladde, G. S.; Lakshmikantham, V.; Zhang, B. G., Oscillation Theory of Differential Equations with Deviating Arguments (1987), Marcel Dekker: Marcel Dekker New York · Zbl 0832.34071
[43] Zhang, B. G.; Yu, J.-S., Comparison and linearized oscillation theorems for a nonlinear partial difference equation, ANZIAM J., 42, 552-560 (2001) · Zbl 0991.39002
[44] Zhang, B. G.; Yu, J. S., Linearized oscillation theorems for certain nonlinear delay partial difference equations, Computers Math. Applic., 35, 4, 111-116 (1998) · Zbl 0907.39017
[45] Zhang, B. G., Oscillation theorems for certain nonlinear delay partial difference equations, Nonlinear Analysis, 41, 447-454 (2000) · Zbl 0951.39004
[46] Liu, B. M.; Zhang, B. G., Linearized oscillation theorems for certain nonlinear delay partial difference equations, (Proceeding of International Conference on Dynamical Systems and Differential Equations. Proceeding of International Conference on Dynamical Systems and Differential Equations, Shanghai, June 9-11, 1998 (1998), Shanghai Jiaotong University Press: Shanghai Jiaotong University Press Shanghai), 89-96 · Zbl 1043.39502
[47] Liu, S. T.; Cheng, S. S., Existence of positive solutions for partial difference equations, Far East Journal Math. Sci., 5, 3, 381-392 (1997)
[48] Liu, S. T.; Cheng, S. S., Nonexistence of positive solutions of a nonlinear partial difference equation, Tamkang J. Math., 28, 51-58 (1997) · Zbl 0877.39003
[49] Domshlak, Y.; Cheng, S. S., Sturmian theorems for a partial difference equation, Functional Differential Equations, 3, 83-97 (1996) · Zbl 0859.39003
[50] Cheng, S. S.; Zhang, B. G., Nonexistence criteria for positive solutions of discrete elliptic equations, Fasciculi Mathematici, 28, 19-30 (1998) · Zbl 0907.39001
[51] Cheng, S. S.; Liu, S. T.; Zhang, G., A multivariate oscillation theorem, Fasciculi Mathematici, 30, 15-22 (1999) · Zbl 0949.39004
[52] Wong, P. J.Y., Eventually positive and monotonely decreasing solutions of partial difference equations, Computers Math. Applic., 35, 4, 35-45 (1998) · Zbl 0914.39006
[53] Erbe, L. H.; Xia, H.; Yu, J. S., Global stability of a linear nonautonomous delay difference equation, J. of Difference Equations and Appl., 1, 2, 151-161 (1995) · Zbl 0855.39007
[54] Zhang, B. G.; Tian, C. J.; Wong, P. J.Y., Global attractivity of difference equations with variable delay, DCDIS, 6, 307-319 (1999) · Zbl 0938.39016
[55] Tian, C. J.; Zhang, B. G., On the global attractivity of partial difference equations, J. of Ocean University of Qingdao, 28, 4, 688-694 (1998) · Zbl 0973.39004
[56] Zhang, B.; Deng, X., The stability of certain partial difference equations, Computers Math. Applic., 42, 3-5, 419-425 (2001) · Zbl 1002.39023
[57] Zhang, B. G.; Tian, C. J., Stability criteria for a class of linear delay partial difference equations, Computers Math. Applic., 38, 2, 37-43 (1999) · Zbl 0943.39003
[58] Lin, Y. Z.; Cheng, S. S., Stability criteria for two partial difference equations, Computers Math. Applic., 32, 7, 87-103 (1996) · Zbl 0867.39003
[59] Lin, Y. Z.; Cheng, S. S., Exponential stability of a partial difference equation with nonlinear perturbation, Acta. Math. Appl. Sinica, 15, 1, 98-108 (1999) · Zbl 1125.39302
[60] Cheng, S. S., Stability of partial difference equations, (New Developments in Difference Equations and Applications, Proceedings of the Third International Conference on Difference Equations (1999), Gordon and Breach Science), 93-135 · Zbl 0939.39010
[61] Lin, Y. Z.; Cheng, S. S., Bounds for solutions of a three-point partial difference equation, Acta Mathematica Scientia, 18, 1, 107-112 (1998) · Zbl 0909.39002
[63] Xie, S. L.; Cheng, S. S., Stability criteria for parabolic type partial difference equations, J. Comput. Math. Appl., 75, 57-66 (1996) · Zbl 0866.65056
[64] Shi, B., Stability of the positive steady-state solutions of systems of nonlinear Volterra difference equations of population models with diffusion and infinite delay, Internat. J. Math. and Math. Sci., 23, 4, 261-270 (2000) · Zbl 0958.39007
[65] Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), Academic Press: Academic Press New York · Zbl 0484.15016
[66] Shi, B.; Wang, Z. C.; Yu, J. S., Asymptotic constancy of solutions of linear parabolic Volterra difference equations, Computers Math. Applic., 32, 8, 65-77 (1996) · Zbl 0873.39007
[67] Zhang, Z. Q.; Yu, J. S., Global attractivity of linear partial difference equations, Differential Equations and Dynamical Systems, 7, 2, 147-157 (1999) · Zbl 0978.39009
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