The authors discuss the perturbed projection solutions for linear compact operator equations by compactness principles in operator approximation theory introduced by P. M. Anselone and R. Ansorge [see Numer. Funct. Anal. Optimization 1, 589-618 (1979; Zbl 0466.65036)]. They mainly prove the following result:
Let \(K\in L(X)\) be sequentially compact, \((\lambda -K)^{-1}\) be bounded, \(S_ n\to^{ac}0\) and \(\| g_ n\| \to 0\) as \(n\to \infty\). Suppose projection operators \(P_ n:X\to X_ n\) satisfy \(P_ n\to^{p}I\) furthermore. Then the operator equation \(\lambda x=Kx+f\) and its perturbed projection equations \(\lambda x_ n=K_ nx_ n+P_ nf\) have unique solutions \(x_ 0\) and \(x_ n\), respectively, if for n large enough, also \(x_ n\to x_ 0\) as \(n\to \infty\), and the following estimations hold for large enough \(n:\) \[ \| (\lambda -K_ n)^{- 1}\| \leq (1+\| (\lambda -K)^{-1}\| \| K_ n\|)/(| \lambda | -\Delta_ n), \]
\[ \| x-x_ 0\| \leq \| (\lambda -K)^{-1}\| (\| (K-K_ n)K_ n\| \| x_ 0\| +\| (K- K_ n)P_ nf\| +| \lambda | \| P_ nf-f\|)/(| \lambda | -\Delta_ n), \]
\[ \| x_ n-x_ 0\| \leq \| (\lambda -K_ n)^{-1}\| ([\| P_ nK\| +\| \lambda -K_ n\|]\| P_ nx_ 0-x_ 0\| +\| S_ n\| \| P_ nx_ 0\| +\| g_ n\|), \]
\[ \| x_ n-x_ 0\| \leq \| (\lambda -K_ n)^{-1}\| \| (K-K_ n)x_ 0+f-P_ nf\|. \] Where \(\Delta_ n=\| (\lambda -K)^{-1}(K_ n-K)K_ n\|.\)
Here, X is a Banach space, \(X_ n\) are subspaces of X, L(X) is the usual space consisting of all bounded linear operators from X into X, \(x,f\in X,x_ n,g_ n\in X_ n\), \(K_ nx_ n=P_ nKx_ n+S_ nx_ n+g_ n\), \(T_ n\to^{ac}T\) denotes \(x_ n\to x\Rightarrow T_ nx_ n\to Tx\) and \(S\subset X\) bounded \(\Rightarrow\) every sequence \(\{x_ n:x_ n\in T_ n(S)\), \(n\in N'\subset N\) (the positive integer set)\(\}\) has convergent subsequences, \(T_ n\to^{p}T\) denotes \(T_ nx\to Tx\) for every \(x\in X\), I denotes the identical operator on X.