×

The freezing method for abstract nonlinear difference equations. (English) Zbl 1115.39010

The paper is concerned with the nonlinear difference equation \[ x_{k+1}=A_kx_k+F_k(x_k),\quad k=0,1,\dots,\text{ in }H,\tag{1} \] where \(H\) is a separable complex Hilbert space, \(\{A_k\}_{k\geq 0}\) is a family of bounded linear operators in \(H\) with \(D(A_k)\equiv D\subseteq H\), which verify the freezing condition \(\| A_k-A_j\| _{L(H)}\leq q| k-j| \), for all \(k,j=0,1,\dots,q>0\), and the functions \(F_k:\Omega_r\to H\) are continuous for each \(k\geq 0\), (\(\Omega_r=\{h\in H\); \(\| h\| _H\leq r\}\), \(0<r\leq +\infty\)) and satisfy the condition \(\| F_k(h)\| _H\leq \mu \| h\| _H+\xi\), for all \(h\in \Omega_r\), \(k=0,1,\dots\) (\(\mu\geq 0\), \(\xi\geq 0\)). Under some additional assumptions on the operators \(A_k\), \(k=0,1,\ldots\), the boundedness of the solution \(\{x_k\}_{k\geq 0}\) of the problem (1) with the initial condition \(x_0\in H\), is established. For this aim, the authors use some norm estimates for operator-valued functions and the multiplicative representation of solutions. In the case \(H=l^2(C)\), they obtain an explicit bound for the norm of solution to the problem (1).
For the linear problems \(y_{k+1}=A_ky_k\), \(k=0,1,\dots\), and \(y_{k+1}=A_ky_k+f_k\), \(k=0,1,\dots\), in \(H\), where \(A_k\) are Hilbert-Schmidt operators which satisfy the freezing condition and \(\{f_k\}_{k\geq 0}\) is a bounded sequence in \(H\), there are presented two sharp results. The stability of the linear delay difference equation \(x_{k+1}=Ax_k+Bx_{k-\sigma}\), \(k=0,1,\dots\), where \(\sigma>0\), \(A\) and \(B\) are bounded linear operators in a complex Banach space \(X\), is finally investigated.

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
47J05 Equations involving nonlinear operators (general)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agarwal, R.P.; Thompson, H.B.; Tisdell, C.C., Difference equations in Banach spaces, Advances in difference equations, IV, Comput. math. appl., 45, 1437-1444, (2003) · Zbl 1057.39007
[2] Bay, N.S.; Phat, V.N., Stability analysis of nonlinear retarded difference equations in Banach spaces, Comput. math. appl., 45, 951-960, (2003) · Zbl 1053.39004
[3] Bylov, B.F.; Grobman, B.M.; Nemickii, V.V.; Vinograd, R.E., The theory of Lyapunov exponents, (1966), Nauka Moscow, (in Russian) · Zbl 0144.10702
[4] Daleckii, Yu.L.; Krein, M.G., Stability of solutions of differential equations in Banach spaces, (1971), Amer. Math. Soc. Providence, RI
[5] Gel’fand, I.M.; Shilov, G.E., Some questions of differential equations, (1958), Nauka Moscow · Zbl 0091.11104
[6] Gil’, M.I., Operator functions and localization of spectra, Lecture notes in math., vol. 1830, (2003), Springer-Verlag Berlin · Zbl 1032.47001
[7] Gil’, M.I., Norm estimations for operator-valued functions and applications, (1995), Marcel Dekker New York · Zbl 0840.47006
[8] Gil’, M.I., Invertibility and spectrum of hille – tamarkin matrices, Math. nachr., 244, 78-88, (2002) · Zbl 1033.47004
[9] Gil’, M.I.; Medina, R., The freezing method for linear difference equations, J. difference equ. appl., 8, 5, 485-494, (2001) · Zbl 1005.39005
[10] Gonzalez, C.; Jimenez-Melado, A., An application of Krasnoselskii fixed point theorem to the asymptotic behavior of solutions of difference equations in Banach spaces, J. math. anal. appl., 247, 290-299, (2000) · Zbl 0962.39007
[11] Izobov, N.A., Linear systems of ordinary differential equations, (), 71-146, (in Russian) · Zbl 0342.34035
[12] Kato, T., Perturbation theory for linear operators, (1966), Springer-Verlag New York · Zbl 0148.12601
[13] Lakshmikantham, V.; Leela, S.; Matynyuk, A.A., Stability analysis of nonlinear systems, (1989), Marcel Dekker New York · Zbl 0676.34003
[14] Medina, R., Estimates for the norms of solutions of delay difference systems, Int. J. math. math. sci., 30, 11, 697-703, (2002) · Zbl 1002.39027
[15] Rodman, L., An introduction to operator polynomials, (1989), Birkhäuser Berlin · Zbl 0685.47011
[16] Vinograd, R., An improved estimate for the method of freezing, Proc. amer. math. soc., 89, 1, 125-129, (1983) · Zbl 0525.34040
[17] Kuruklis, S.A., The asymptotic stability of \(x_{n + 1} - a x_n + b x_{n - k} = 0\), J. math. anal. appl., 188, 719-731, (1994) · Zbl 0842.39004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.