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**Stability in linear neutral difference equations with variable delays.**
*(English)*
Zbl 1289.39028

A scalar linear neutral difference equation of the form
\[
\Delta x(n)=\sum _{j=1}^N a_j(n)x(n-\tau _j(n))+\sum _{j=1}^N c_j(n)\Delta x(n-\tau _j(n)) \tag{1}
\]
is considered, where \(\Delta x(k):= x(k+1)-x(k)\) and the delays \(\tau _j(n)\), \(j=1,\dots, N\), are all positive sequences.

The authors are motivated by the results of three earlier papers by Y. N. Raffoul [J. Math. Anal. Appl. 324, No. 2, 1356–1362 (2006; Zbl 1115.39015)], E. Yankson [Electron. J. Qual. Theory Differ. Equ. 2009, Paper No. 8, 7 p. (2009; Zbl 1181.39017)] and M. Islam and E. Yankson [Electron. J. Qual. Theory Differ. Equ. 2005, Paper No. 26, 18 p. (2005; Zbl 1103.39009)], where the stability of the zero solution was established for several partial cases of equation (1). They use the same method of reduction of the stability problem of the zero solution to a fixed point problem for an appropriately constructed contraction operator on the set of all forward solutions. The idea of the method and its applications to various classes of equations can be found in the monograph by T. A. Burton [Stability by fixed point theory for functional differential equations. Mineola, NY: Dover Publications (2006; Zbl 1160.34001)].

Sufficient conditions for the stability of the zero solution of equation (1) are given in terms of the coefficients \(a_j(n)\), \(c_j(n)\), the delays \(\tau _j(n)\), and the sequence \(H(n)=1-\sum_{j=1}^N h_j(n)\), where \(h_j(n)\) are generally arbitrary but appropriately chosen sequences (\(j=1,\dots, N\)). Those conditions generalize the stability results established in the three previously published papers. The conditions are quite convoluted and their applicability to particular cases is not straightforward (especially in view of the arbitrary nature of the sequences \(h_j(n)\) involved). The paper does not contain any example that would demonstrate the applicability of the derived results.

The authors are motivated by the results of three earlier papers by Y. N. Raffoul [J. Math. Anal. Appl. 324, No. 2, 1356–1362 (2006; Zbl 1115.39015)], E. Yankson [Electron. J. Qual. Theory Differ. Equ. 2009, Paper No. 8, 7 p. (2009; Zbl 1181.39017)] and M. Islam and E. Yankson [Electron. J. Qual. Theory Differ. Equ. 2005, Paper No. 26, 18 p. (2005; Zbl 1103.39009)], where the stability of the zero solution was established for several partial cases of equation (1). They use the same method of reduction of the stability problem of the zero solution to a fixed point problem for an appropriately constructed contraction operator on the set of all forward solutions. The idea of the method and its applications to various classes of equations can be found in the monograph by T. A. Burton [Stability by fixed point theory for functional differential equations. Mineola, NY: Dover Publications (2006; Zbl 1160.34001)].

Sufficient conditions for the stability of the zero solution of equation (1) are given in terms of the coefficients \(a_j(n)\), \(c_j(n)\), the delays \(\tau _j(n)\), and the sequence \(H(n)=1-\sum_{j=1}^N h_j(n)\), where \(h_j(n)\) are generally arbitrary but appropriately chosen sequences (\(j=1,\dots, N\)). Those conditions generalize the stability results established in the three previously published papers. The conditions are quite convoluted and their applicability to particular cases is not straightforward (especially in view of the arbitrary nature of the sequences \(h_j(n)\) involved). The paper does not contain any example that would demonstrate the applicability of the derived results.

Reviewer: Anatoli F. Ivanov (Lehman)