Ill-posed problems, \(C_{0}\)-semigroups and the Showalter regularization (Q1766727)

The authors study linear ill-posed equations \(Ax=y\) in Banach space \(X\), where \(-A\) generates a bounded \(C_0\)-semigroup. The Showalter method known for bounded operators \(A\) is here generalized to unbounded operators. Consider the abstract Cauchy problem \(u^{\prime}(t)=-Au(t)+y\), \(t \geq 0\), \(u(0)=0\), \(u \in C^1([0,\infty),X)\). Let \(T: [0,\infty) \to {\mathcal L}(X)\) be a semigroup generated by \(-A\). The unknown solution \(x\) is approximated by the elements \(\int_0^t T(s)y ds\) and \(t^{-1} \int_0^t u(s)ds\) \((t \to \infty)\). Rate of convergence estimates have the form \(O(t^{-\mu})\), provided that \(x \in \text{ Im}A^{\mu}\), \(\mu \in (0,1)\).