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Let X₁ and X₂ be complex Banach spaces, and let A₁ ∈ BL(X₁), A₂ ∈ BL(X₂), A₃ ∈ BL(X₁,X₂) and A₄ ∈ BL(X₂,X₁). We propose an iterative procedure which is a modified form of Newton's iterations for obtaining approximations for the solution R ∈ BL(X₁,X₂) of the Riccati equation A₂R - RA₁ = A₃ + RA₄R, and show that the convergence of the method is quadratic. The advantage of the present procedure is that the conditions imposed on the operators A₁, A₂, A₃, A₄ are weaker than the corresponding conditions...

Many discrepancy principles are known for choosing the parameter α in the regularized operator equation $(T*T+\alpha I)x_{\alpha }^{\delta }=T*y^{\delta }$, $|y-y^{\delta }|\le \delta$, in order to approximate the minimal norm least-squares solution of the operator equation Tx = y. We consider a class of discrepancy principles for choosing the regularization parameter when T*T and $T*y^{\delta }$ are approximated by Aₙ and $z{ₙ}^{\delta }$ respectively with Aₙ not necessarily self-adjoint. This procedure generalizes the work of Engl and Neubauer (1985), and particular cases of the results are applicable...

It is known that the nonlinear nonhomogeneous backward Cauchy problem $u_t\left(t\right)+Au\left(t\right)=f(t,u\left(t\right))$, $0\le t<\tau$ with $u\left(\tau \right)=\phi$, where $A$ is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on $\phi$ and $f$, that a solution of the above problem satisfies an integral equation involving the spectral representation of $A$, which is also ill-posed. Spectral truncation is used...