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.


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


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