Operator Equations Of Type Research Articles

Krasnoselskii-type results for equiexpansive and equicontractive operators with application in radiative transfer equations

<abstract><p>In this paper, we study the results of fixed points for the operator equations of type $ x = H(\digamma x, x) $ using the idea of measure of noncompactness and assuming that the operator $ \digamma $ is $ k $ -set contractive (strictly $ k $ -set contractive, or a continuous) and the family $ \{H(u, .):u\} $ is equiexpansive or equicontractive. The obtained results are generalization of Krasnoselskii type fixed point results. Some examples are given to elaborate new concepts. We use the main result to find the existence of solutions for the stationary radiative transfer equation in a channel. We demonstrate our theory with an example by comparison of an approximate solution with the exact solution.</p></abstract>

THE PROBLEM OF RESTORING A FUNCTION BY FAMILIES OF SPHERES IN SPACE

In this chapter, we study the question of reducing the problem of integral geometry for a special class of surfaces to the Cauchy problem for some equation of evolutionary type. Fourier transform methods are also used. Methods are obtained that allow reducing the problem of integral geometry for special families of curves and surfaces to the Cauchy problem for equations of evolutionary type, and classes of such problems are distinguished. Uniqueness theorems are proved for some new classes of operator equations of Voltaire type in three-dimensional space.

A note on Furuta type operator equation

A note on Furuta type operator equation

On Rubel type operator equations on the space of analytic functions

The purpose of this work is to describe all pairs of linear operators that act in spaces of analytic functions in domains and satisfy the Rubel’s type operator equations. We describe all generalized derivation pairs of linear operators on the space of functions analytic in domains. Using this result we describe all pairs of linear operators that act in the spaces of functions analytic in domains and satisfying the operator analog of Kannappan–Nandakumar’s equation.

Cubic convergence order yielding iterative regularization methods for ill-posed Hammerstein type operator equations

For the solution of nonlinear ill-posed problems, a Two Step Newton-Tikhonov methodology is proposed. Two implementations are discussed and applied to nonlinear ill-posed Hammerstein type operator equations $$KF(x)=y$$ , where K defines the integral operator and F the function of the solution x on which K operates. In the first case, the Fr $$\acute{e}$$ chet derivative of F is invertible in a neighbourhood which includes the initial guess $$x_0$$ and the solution $${\hat{x}}$$ . In the second case, F is monotone. For both cases, local cubic convergence is established and order optimal error bounds are obtained by choosing the regularization parameter according to the the balancing principle of Pereverzev and Schock (2005).We also present the results of computational experiments giving the evidence of the reliability of our approach.

Different Levels of Perturbations of the Operators of Hammerstien’s Type Operator Equations

We have studied perturbations of Hammerstein’s Type Operator Equations in general Banach Spaces. In this paper, two different levels of perturbations have been studied in Hilbert spaces. We prove that these levels satisfy the regularization conditions for Hammerstein type operator equations.

Integrable hydrodynamic equations for initial chiral currents and infinite hydrodynamic chains from WZNW model and string model of WZNW type with SU(2), SO(3), SP(2), SU(∞), SO(∞), SP(∞) constant torsions

The WZNW and string models are considered in terms of the initial and invariant chiral currents assuming that the internal and external torsions coincide (anticoincide) and they are the structure constants of the SU(n), SO(n), SP(n) Lie algebras. These models are the auxiliary problems in order to construct integrable equations of hydrodynamic type. It was shown that the WZNW and string models in terms of invariant chiral currents are integrable for the constant torsion associated with the structure constants of the SU(2), SO(3), SP(2) and SU(3) algebras only. The equation of motion for the density of the first Casimir operator was obtained in the form of the inviscid Burgers equation. The solution of this equation is presented through the Lambert function. Also, a new equation of motion for the initial chiral current was found. The integrable infinite hydrodynamic chains obtained from the WZNW and string models are given in terms of invariant chiral currents with the SU(2), SO(3), SP(2) and with SU(∞), SO(∞), SP(∞) constant torsions. Also, the equations of motion for the density of any Casimir operator and new infinite-dimensional equations of hydrodynamic type for the initial chiral currents through the symmetric structure constant of SU(∞), SO(∞), SP(∞) algebras are obtained.

A two step newton type iteration for ill-posed Hammerstein type operator equations in Hilbert scales

In this paper regularized solutions of ill-posed Hammerstein type operator equation $KF(x)=y$, where $K:X\rightarrow Y$ is a bounded linear operator with non-closed range and $F:X\rightarrow X$ is non-linear, are obtained by a two step Newton type iterative method in Hilbert scales, where the available data is $y^{\delta}$ in place of actual data $y$ with $\|y-y^\delta\|\leq\delta$. We require only a weaker assumption $\|F^{\p}(x_0)x\|\sim\|x\|_{-b}$ compared to the usual assumption $\|F^{\p}(\widehat{x})x\|\sim\|x\|_{-b}$, where $\widehat{x}$ is the actual solution of the problem, which is assumed to exist, and $x_0$ is the initial approximation. Two cases, viz-a-viz, (i)~when $F^{\p}(x_0)$ is boundedly invertible and (ii)~$F^{\p}(x_0)$ is non-invertible but $F$ is monotone operator, are considered. We derive error bounds under certain general source conditions by choosing the regularization parameter by an a~priori manner as well as by using a modified form of the adaptive scheme proposed by Perverzev and Schock [{\bf14}].

About a deficit in low-order convergence rates on the example of autoconvolution

We revisit in -spaces the autoconvolution equation with solutions which are real-valued or complex-valued functions defined on a finite real interval, say . Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity, deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low-order convergence rates for regularized solutions of nonlinear ill-posed operator equations with solutions in a Hilbert space setting. So, for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex -functions with a fixed and uniformly bounded modulus function.

Riccati type operator equation and Furuta's question

As a continuation of the Riccati type operator equation K = THT, this article is to consider the Riccati type equation K p = H δ 2 T 1 2 (T 1 2 H δ+r T 1 2 ) 1 w −1 T 1 2 H δ 2

Some Properties of Furuta Type Inequalities and Applications

This work is to consider Furuta type inequalities and their applications. Firstly, some Furuta type inequalities underA≥B≥0are obtained via Loewner-Heinz inequality; as an application, a proof of Furuta inequality is given without using the invertibility of operators. Secondly, we show a unified satellite theorem of grand Furuta inequality which is an extension of the results by Fujii et al. At the end, a kind of Riccati type operator equation is discussed via Furuta type inequalities.

Newton type iteration for Tikhonov regularization of non-linear ill-posed Hammerstein type equations

An iterative method is investigated for a nonlinear ill-posed Hammerstein type operator equation KF(x)=f. We use a center-type Lipschitz condition in our convergence analysis instead of the usual Lipschitz condition. The adaptive method of Pereverzev and Schock (SIAM J. Numer. Anal. 43(5):2060–2076, 2005) is used for choosing the regularization parameter. The optimality of this method is proved under a general source condition involving the Frechet derivative of F at some initial guess x 0. A numerical example of nonlinear integral equation shows the efficiency of this procedure.

PROJECTION METHOD FOR NEWTON-TIKHONOV REGULARIZATION FOR NON-LINEAR ILL-POSED HAMMERSTEIN TYPE OPERATOR EQUATIONS

An iteratively regularized projection scheme for the ill-posed Hammerstein type operator equation KF(x) = f has been considered. Here F : D(F) ⊆ X → X is a non-linear operator, K : X → Y is a bounded linear operator, X and Y are Hilbert spaces. The method is a combination of dis- cretized Tikhonov regularization and modified Newton's method. It is assumed that the Frechet derivative of F at x0 is invertible i.e., the ill-posedness of the operator KF is due to the ill-posedness of the linear operator K. Here x0 is an initial approximation to the solution ˆ x of KF(x) = f. Adaptive choice of the parameter suggested by Perverzev and Schock(2005) is employed in select- ing the regularization parameter �. A numerical example is given to test the reliability of the method.

Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type

It is shown that any \(\mu \in \mathbb{C}\) is an infinite multiplicity eigenvalue of the Steklov smoothing operator \(S_h\) acting on the space \(L^1_{loc}(\mathbb{R})\). For \(\mu \neq 0\) the eigenvalue-eigenfunction problem leads to studying a differential-difference equation of mixed type. An existence and uniqueness theorem is proved for this equation. Further a transformation group is defined on a countably normed space of initial functions and the spectrum of the generator of this group is studied. Some possible generalizations are pointed out.

On Improving the Semi-Local Convergence of Newton-Type Projection Method for Ill-Posed Hammerstein Type Operator Equations

  Abstract—A Two Step Newton-Tikhonov Projection method is presented for obtaining a stable approximate solution of nonlinear ill-posed Hammerstein type operator equations KF(x) =f. The regularization parameter is chosen according to the adaptive parameter choice strategy suggested by Perverzev and Schock (2005). The error estimates obtained with respect to the general source conditions are of optimal order. We also give the numerical example which confirms the efficiency of the proposed method.

About convergence rates in regularization for ill-posed operator equations of hammerstein type.

About convergence rates in regularization for ill-posed operator equations of hammerstein type.

Continuous regularization method for illposed operator equations of Hammerstein type.

Continuous regularization method for illposed operator equations of Hammerstein type.

A Newton-like method for the numerical solution of nonlinear Fredholm-type operator equations

In a recent paper, Hernández and Salanova proposed an interesting Newton-like method for the solution of nonlinear Fredholm integral equations. The aim of the present paper is to carry over this method to nonlinear operator equations of similar type. By using a functional analytic approach, the presentation becomes much clearer and more general. Apart from this, some new results in the Banach space setting as well as in the applications are obtained.

Integrable solutions of a mixed type operator equation

In this paper we prove the existence of integrable solutions for a generalized mixed type operator equation, which contains many key integral and functional equations appearing frequently in Mathematical literature. Our main tool is a Krasnosel'skii type fixed point theorem recently proved by Latrach and Taoudi, the first author. An existence theory for a class of nonlinear transport equations is also developed.

An iterative regularization method for ill-posed Hammerstein type operator equation

An iterative regularization method for ill-posed Hammerstein type operator equation