On the oscillation of two partial difference equations. (English) Zbl 0877.39012

The authors consider two partial difference equations: \[ A(m,n)+ P_{m,n}A_{m-k,n-\ell}=0\tag{1} \] and \[ A(m,n)+ \sum^u_{i=1} P_i(m,n)f_i(A_{n-k_i,n-\ell_i})=0,\tag{2} \] with \(A(m,n):= A_{m-1,n}+ A_{m,n-1}- A_{m,n}\), where \(P_{m,n}\), \(P_i(m,n)\) are positive functions defined on \(\mathbb{N}^2_0\), here \(\mathbb{N}_0=\{0,1,2,\dots\}\); \(k,\ell,k_i,\ell_i\in \mathbb{N}_0\); \(f_i:\mathbb{N}_0\to\mathbb{R}\) are nondecreasing and continuous, \(xf_i(x)>0\) for \(x\neq 0\). Clearly (1) is a special case of (2) when \(u=1\) and \(f(x)=x\).
The main sufficient condition ensuring oscillation of all solutions of (1) is proved by contradiction in Theorem 2.1, and some variants and corollaries are also obtained. By using a discrete analogy of Green’s formula established by the authors in an earlier paper, two sufficient conditions for the oscillation of equation (2) are also proved. Some illustrating examples are given.
We note that, in the conduction (iv) of Theorem 3.1, the number \(\mu_i\) should read \(\eta_i\).


39A12 Discrete version of topics in analysis
39A10 Additive difference equations
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