Extended iterative methods for the solution of operator equations

Summary Given an iterative methodM ₀, characterized byx ^(k+1=G ₀(x( ^k )) (k0) (x(⁰) prescribed) for the solution of the operator equationF(x)=0, whereF:XX is a given operator andX is a Banach space, it is shown how to obtain a family of methodsM _p characterized byx ^(k+1=G _p (x( ^k )) (k0) (x(⁰) prescribed) with order of convergence higher than that ofM _o. The infinite dimensional multipoint methods of Bosarge and Falb [2] are a special case, in whichM ₀ is Newton's method.Analogues of Theorems 2.3 and 2.36 of [2] are proved for the methodsM _p, which are referred to as extensions ofM ₀. A number of methods with order of convergence greater than two are discussed and existence-convergence theorems for some of them are proved.Finally some computational results are presented which illustrate the behaviour of the methods and their extensions when used to solve systems of nonlinear algebraic equations, and some applications currently being investigated are mentioned.     

The existence of a nontrivial solution to p‐Laplacian equations in with supercritical growth

In this paper, we consider the p‐Laplacian equations in with supercritical growth where △ _pu = div( | ? u | ^{p ? 2} ? u),1 < p < N is the p‐Laplacian operator. Under certain assumptions on V (x) and f(u) that will be given in Section 1, we prove that the problem has at least a nontrivial solution by using variational methods combined with perturbation arguments. The solutions to subcritical p‐Laplacian equations are estimated applying the L ^∞ norm. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal Control of Nonlinear Fredholm Integral Equations

Optimal control problems with nonlinear equations usually do not possess optimal solutions, so that their natural (i.e., continuous) extension (relaxation) must be done. The relaxed problem may also serve to derive first-order necessary optimality condition in the form of the Pontryagin maximum principle. This is done here for nonlinear Fredholm integral equations and problems coercive in an L ^p-space of controls with p<+. Results about a continuous extension of the Uryson operator play a key role.     

On a Liouville phenomenon for entire weak supersolutions of elliptic partial differential equations

We study a new Liouville-type phenomenon for entire weak supersolutions of elliptic partial differential equations of the form A(u)=0 on , n2. Typical examples of the operator A(u) are the p-Laplacian for p>1, the mean curvature operator, and their well-known modifications.     

The indentation of two rigid stamps into an elastic sphere with a spheroidal inclusion

On the basis of the expansion formulas of the vector solutions of the Lamé equations in spherical coordinates with respect to the solutions of the Lamé equations in oblate spheroidal coordinates and on the basis of their inverse formulas, one solves the problem of the compression of an elastic ball with an absolutely rigid inclusion in the form of an oblate spheroid. The problem is reduced to an infinite system of linear algebraic equations of the second kind with a completely continuous operator in ₂. Results of the numerical solution of the infinite system are given and the obtained results are analyzed.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 9–13, 1989.     

Second-Order Elliptic Equation of Divergence Form Having a Compactly Supported Solution

An equation of the form -div (gu) = u has a solution u of class C ₀ , where g is a real positive definite matrix-valued function belonging to the Holder classes with exponent less than 1. From the spectral point of view, this means that there exists a Schrodinger operator with periodic metric and the spectrum of this operator contains an eigenvalue of infinite multiplicity. Bibliography: 9 titles.     

On the Fredholm theory of Wiener-Hopf equations and the coupling method

The Fredholm properties of the Wiener-Hopf operator onL _p(⁺,^m) are investigated using the coupling method for solving operator equations. The theory applies to equations whose kernel is an element ofL ₁(,^mxm). As usual the determinant of the symbol is assumed to have no zeros on the real line. The method of analysis is independent of the realization theory for symbols that are analytic in a strip containing the real axis although in some sense closely related to it. The connection between the two methods is briefly analysed in the paper.     

Existence of bounded solutions for nonlinear elliptic equations in unbounded domains

In this paper we study the existence of bounded weak solutions for some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the p-laplacian operator, and the lower order terms, which depend on the solution u and its gradient u, have a power growth of order p–1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity.     

Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with infinite delay

In this article, we investigate the existence and asymptotic stability in p-th moment of a mild solution to a class of neutral stochastic integro-differential equation of fractional order involving non-instantaneous impulses with infinite delay in a Hilbert space. A new set of sufficient conditions proving existence and asymptotic stability of mild solution is derived by utilizing solution operator, functional analysis, stochastic analysis and fixed point technique. Finally, an example is provided to illustrate the obtained abstract result.     

Examples of Saturated Convergence Rates for Tikhonov Regularization

Tikhonov regularization is one of the most popular methods for solving linear operator equations of the first kind Au = f with bounded operator, which are ill-posed in general (Fredholm's integral equation of the first kind is a typical example). For problems with inexact data (both the operator and the right-hand side) the rate of convergence of regularized solutions to the generalised solution u ₊ (i.e.the minimal-norm least-squares solution) can be estimated under the condition that this solution has the source form: u ₊ im(A*A). It is well known that for Tikhonov regularization the highest-possible worst-case convergence rates increase with only for some values of , in general not greater than one. This phenomenon is called the saturation of convergence rate. In this article the analysis of this property of the method with a criterion of a priori regularization parameter choice is presented and illustrated by examples constructed for equations with compact operators.This revised version was published online in October 2005 with corrections to the Cover Date.     

Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay

In this paper, we obtain some results on the existence and uniqueness of solutions to stochastic functional differential equations with infinite delay at phase space BC((-∞,0];R^d) which denotes the family of bounded continuous R^d-value functions defined on (-∞,0] with norm under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition. The solution is constructed by the successive approximation.     

On nonnegative solvability of linear operator equations

LetE be a Banach lattice having order continuous norm. Suppose, moreover,T is a nonnegative reducible operator having a compact iterate and which mapsE into itself. The purpose of this work is to extend the previous results of the authors, concerning nonnegative solvability of (kernel) operator equations on generalL ^p-spaces. In particular, we provide necessary and sufficient conditions for the operator equation x=T x+y to possess a nonnegative solutionxE wherey is a given nonnegative and nontrivial element ofE and is any given positive parameter.     

Regularity of a Boundary Point for Degenerate Parabolic Equations with Measurable Coefficients

We investigate the continuity of solutions of quasilinear parabolic equations in the neighborhood of the nonsmooth boundary of a cylindrical domain. As a special case, one can consider the equation with the p-Laplace operator p. We prove a sufficient condition for the regularity of a boundary point in terms of C _p-capacity.     

Error Bounds for the Truncation of Infinite Linear Differential Systems

An infinite system of linear differential equations of the form = Ax + f, x(0) = y is considered, wherex, y and f are infinite column vectors in E, and A is a constantinfinite matrix defining a bounded operator on E, where E isl₁, (c₀), or l. Explicit error bounds are obtained for the approximationof the solution of the infinite system by the solutions of thefinite truncated systems.     

On coercive solvability of parabolic equations

In this paper we study the coercive solvability of abstract differential equations of parabolic type in the spaces of L. N. Slobodetskii W _p . It is established that the solution of equations with a constant operator A which generates an analytic semigroup belongs to the trace space E(, p, A). The results obtained are applied to the study of equations with a variable operator.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 409–419, April, 1972.     

An infinite family of p-adic non-Haar wavelet bases and pseudo-differential operators

In this paper an infinite family of new compactly supported non-Haar p-adic wavelet bases in is constructed. We also study the connections between wavelet analysis and spectral analysis of p-adic pseudo-differential operators. A criterion for a multidimensional p-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We prove that these wavelets are eigenfunctions of the fractional operator. Since many p-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in these models. The text was submitted by the authors in English.     

A population dynamics model with nonautonomous past

Abstract

In this paper we study a two-phase population model, which distinguishes the population by two different stages

By the standard technique of characteristics, this population equation is transformed as the ordinary differential equation with nonautonomous past

where 1 ≤ p < ∞ and I = [?r, 0] (finite delay) or I = (?∞, 0] (infinite delay), E a Banach space, Φ : W^1,p(I, E) → E a linear delay operator and B a nonlinear operator on E. The main result of this paper is the well-posedness of this delay equation by using the (right) multiplicative perturbation result of Desch and Schappacher in [8].     

Algorithms for inversion of diagonal plus semiseparable operator matrices

LetH be a Hilbert space andRHH be a bounded linear operator represented by an operator matrix which is a sum of a diagonal and of a semiseparable type of order one operator matrices. We consider three methods for solution of the operator equationRx=y. The obtained results yields new algorithms for solution of integral equations and for inversion of matrices.     

Explicit solution of the operator equation

In this paper we find the explicit solution of the equation

A^*X+X^*A=B