## 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)
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### References:

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