### A convergence result for discrete steepest descent in weighted Sobolev spaces.

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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 $z{\u2099}^{\delta}$ 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...

In this paper we obtain a general fixed point theorem for an affine mapping in Banach space. As an application of this theorem we study existence of periodic solutions to the equations of the linear elasticity theory.

The paper generalizes the instruction, suggested by B. Sz.-Nagy and C. Foias, for operatorfunction induced by the Cauchy problem $${T}_{t}:\left\{\begin{array}{c}t{h}^{\text{'}\text{'}}\left(t\right)+(1-t){h}^{\text{'}}\left(t\right)+Ah\left(t\right)=0\hfill \\ h\left(0\right)={h}_{0}\left(t{h}^{\text{'}}\right)\left(0\right)={h}_{1}\hfill \end{array}\right.$$ A unitary dilatation for ${T}_{t}$ is constructed in the present paper. then a translational model for the family ${T}_{t}$ is presented using a model construction scheme, suggested by Zolotarev, V., [3]. Finally, we derive a discrete functional model of family ${T}_{t}$ and operator $A$ applying the Laguerre transform $$f\left(x\right)\to {\int}_{0}^{\infty}f\left(x\right)\phantom{\rule{0.166667em}{0ex}}{P}_{n}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{e}^{-x}dx$$ where ${P}_{n}\left(x\right)$ are Laguerre polynomials [6, 7]. We show that the Laguerre transform...

Si considera l’equazione astratta $B{A}_{1}u+{A}_{0}u=h$, dove ${A}_{i}$