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

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.

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)

### 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:

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