On nonlinear ill-posed accretive equations
Let \(X\) be a real reflexive Banach space with strictly convex dual \(X^*\). Let \(J: X \to X^*\) and \(Q: X^* \to X\) be dual mappings and \(A: X \to X\) a smooth \(m\)-accretive operator, i.e., \(\langle A(x+h)-A(x), J(h) \rangle \geq 0\) for all \(x, h \in X\) and \(R(A+\lambda I)=X\) for each \(\lambda>0\). Suppose that \(A\) is weakly continuous and satisfies \(\| A(y)-A(x_0)-Q A^{\prime}(x_0)^* J(y-x_0) \| \leq \tau \| A(y)-A(x_0)\|\) for all \(y \in X\) for some solution \(x_0\) of the equation \(A(x)=f\) \((f \in X)\). The author studies Lavrentiev's regularization method \(A(x)+\alpha (x-x_*)=f_{\delta}\), \(x=x_{\alpha}^{\delta}\), where \(f_{\delta}\) is an approximation to \(f\) such that \(\| f_{\delta} -f \| \leq \delta\). Let \(\alpha=\alpha(\delta)\) be determined by the discrepancy principle \(\| A(x_{\alpha(\delta)}^{\delta})-f_{\delta} \|=K \delta^p\) with \(K>1\) and \(p \in (0,1)\). Suppose that the sourcewise representation \(x_0-x_*=A^{\prime}(x_0) z\) \((z \in X)\) holds. Under these assumptions, \(\| x_{\alpha(\delta)}^{\delta}-x_0 \|=O(\delta^{\mu})\), \(\mu=\min \{ 1-p, p/2 \}\).
scientific article; zbMATH DE number 2145550