Bounding Sets in Function Spaces with Applications to Nonlinear Operator Equations

Convergence of the two-step combined method and uniqueness of the solution of nonlinear operator equations

We consider the problem of approximate determination of isolated bounded-norm solutions of nonlinear operator equations in a Hilbert space. Closed balls are constructed, such that the existence and uniqueness conditions are satisfied in each ball.

Recurrence relations for a Newton-like method in Banach spaces

The Numerical Solution of Integral Equations on the Half-Line

Fixed points and approximate solutions for nonlinear operator equations

The purpose of Chap. 7 is to give a natural definition of generalized solution of non-solvable non-linear equation. In Sect. 7.1 we give the definition of generalized solution of equation with operator which acts in metric spaces. In Sect. 7.2 we give the definition of new concept of near-solution of non-linear operator equation. In Sect. 7.3 and 7.4 the existence, uniqueness, and correctness of generalized solution are proved. In Sect. 7.5 and 7.6 we investigate issues of embedding of metric spaces and extension of operators which are important for the theory of generalized solutions. Sections 7.7 and 7.8 contain examples of operators and describe numerical aspects of computation of generalized solutions. Section 7.9 is devoted to the development of the theory of generalized solutions of equations in uniform spaces and proximity spaces.KeywordsGeneralize SolutionOperator EquationBasic SpaceContinuous OperatorCauchy SequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Two theorems on an iteration method of order three -- usually referred to as Chebyshev's method -- for the approximate solution of nonlinear operator equations in Banach spaces are given. Very simple and weak assumptions are obtained as well as error estimates which are much better compared with known results. Some worked examples of operator equations of different types are given.

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.

In the present paper some Newton-like iteration methods are developed to enclose solutions of nonlinear operator equations of the kindF(x)=0. HereF maps a certain subset of a partially ordered vector space into another partially ordered vector space. The obtained results are proved without any special properties of the orderings by taking use of a new kind of a generalized divided difference operator, so that they even hold for nonconvex operators. Furthermore a method for constructing including starting points is presented and two examples are given.

Some theorems about the existence of solutions of nonlinear operator equation and the global homeomorphism in abstract space by the global existence theorem of an initial value problem are given.They are generalizations of some well-known results in these areas.

We consider free interpolation problems, the study of which was initiated by A.F. Leontiev. We obtain new criterions for the interpolation property of sets in the space of analytic in the upper half - plane functions of finite order. We provide examples of interpolation sets in the space of analytic in the upper half - plane functions of finite order. These examples are similar to interpolation sets in the space of analytic and bounded in the upper half - plane functions. In particular, we provide examples of sets satisfying the Newman condition and uniform Frostman condition.

Given a sequence $${\mathcal{U} =\{U_n: n \in \omega\}}$$ of non-empty open subsets of a space X, a set $${\{x_n : n \in \omega\}}$$ is a selection of $${\mathcal{U}}$$ if $${x_n \in U_n}$$ for every $${n \in \omega}$$ . We show that a space X is uncountable if and only if every sequence of non-empty open subsets of C p (X) has a closed discrete selection. The same statement is not true for $${C_p(X,[0,1])}$$ so we study when the above selection property (which we call discrete selectivity) holds in $${C_p(X,[0,1])}$$ . We prove, among other things, that $${C_p(X, [0,1])}$$ is discretely selective if X is an uncountable Lindelof $${\Sigma}$$ -space. We also give a characterization, in terms of the topology of X, of discrete selectivity of $${C_p(X,[0,1])}$$ if X is an $${\omega}$$ -monolithic space of countable tightness.

Previous papers have established sample-path versions of relations between marginal time-stationary and event-stationary (Palm) state probabilities for a process with an imbedded point process. This paper extends the use of sample-path analysis to provide relations between frequencies for arbitrary (measurable) sets in function space, rather than just marginal (one-dimensional) frequencies. We define sample-path analogues of the time-stationary and event-stationary (Palm) probability measures for a process with an imbedded point process, and then derive sample-path versions of the Palm transformation and inversion formulas.

Dense Sets in Spaces of Integrable Functions

The main objective of this paper is to compare – analytically as well as numerically – different approaches for obtaining optimal input currents in impedance tomography. Following the approaches described in, e.g. [Cheney and Isaacson, IEEE Trans. Biomed. Eng. 39: 852–860, 1992, Isaacson, IEEE Trans. Med. Imag. 5: 91–95, 1986, Ito and Kunisch, SIAM J. Contr. Optim. 33: 643–666., 1995, Knowles, An optimal current functional for electrical impedance tomography, 2004], we aim at constructing input currents j, which contain the most information about the difference between the unknown physical conductivity σ* and a given approximation σ 0. The differences can be measured by different discrepancy functionals and the optimal input currents which maximize these functionals depend on the function spaces chosen for defining j and on the norm for measuring the discrepancy. Moreover, the definition of the appropriately weighted Sobolev spaces depends on σ and this subsequently influences the iteration for maximizing the functionals. Numerical experiments illustrate features of the optimal input currents obtained for several combinations of function spaces. The reconstructions with these optimal currents are compared with those with standard input currents (sinusoid and dipole). The differences between the optimal currents obtained by different function space settings are significant. Two newly developed optimal currents can yield qualitatively better reconstructions.