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Unsaturated solutions for partial difference equations with forcing terms. (English) Zbl 1118.39006

The paper is concerned with the partial difference equation \[ \triangle_1u(i-1,j)+\triangle_2u(i,j-1)+P_1(i,j)u(i-1,j)+P_2(i,j)u(i,j-1)+P_3(i,j)u(i,j)=f(i,j) \] where \((i,j)\in Z_+^2=\{1,2,3,\ldots\}\times\{1,2,3,\ldots\}\), \(f\), \(P_i\), \(i=1,2,3\) are real functions of two integer arguments, \(P_3(i,j)\neq -2\) and \( \triangle_1u(i,j)=u(i,j+1)-u(i,j)\), \(\triangle_2u(i,j)=u(i,j+1)-u(i,j) \). Several motivations are given for this equation. The main purpose of the paper is to find oscillatory solutions i.e. solutions that eventually will not keep constant sign. The approach is mainly based on some properties of the discrete sets; it is introduced the upper frequency measure on lattice planes, the lower one and the frequency measure which all rely on set’s cardinal. Further there are defined sequences with unsaturate positive parts using these measures. Conditions for the solutions of the equation above to be with unsaturated upper positive part are given and this property ensures the oscillatory character.

MSC:

39A11 Stability of difference equations (MSC2000)
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References:

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