ITERATIVE SOLUTIONS FOR SYSTEMS OF NONLINEAR OPERATOR EQUATIONS IN BANACH SPACE

By using partial order method, the existence, uniqueness and iterative approximation of solutions for a class of systems of nonlinear operator equations in Banach space are discussed. The results obtained in this paper extend and improve recent results.

Multilevel Iteration Methods for Solving Linear Operator Equations of the First Kind

In this paper we develop two multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for linear ill-posed problems. The two algorithms and their convergence analyses are presented in an abstract framework.

SOLUTIONS TO THE SYSTEM OF OPERATOR EQUATIONS AXB = C = BXA

In this paper, we present some necessary and sufficient conditions for the ex- istence of solutions, hermitian solutions and positive solutions to the system of operator equations AXB = C = BXA in the setting of bounded linear operators on a Hilbert space. Moreover, we obtain the general forms of solutions, hermitian solutions and positive solutions to the system above.

A Principal Theorem of Normal Discretization Schemes for Operator Equations of the First Kind

The authors announce a newly-proved theorem of theirs. This theorem is of principal significance to numerical computation of operator equations of the first kind.

Splitting of Operators and Operator Equations

In this paper,we consider the splitting operators into the product of two positive operators.For given operators A and T∈B(H)with A≥0 and R(A)=R(T),if T can be splitted as T=AX for some positive operator X,then the existence of X and the properties of T are studied.The related research are the solutions of operator equations T=XAX and TX=XAX,which have some particular properties and broad applications in many fields.The conditions for the existence of solutions or idempotent solutions of these kinds of equations are studied and new representations of the general solutions are given.

Lipschitz Operators and the Solvability of Non-linear Operator Equations

Let U and V be Banach spaces, L and N be non-linear operators from U into V. L is said to be Lipschitz if L 1(L) := sup{∥Lx - Ly∥ · ∥x - y∥-1 : x ≠ y} is finite. In this paper, we give some basic properties of Lipschitz operators and then discuss the unique solvability, exact solvability, approximate solvability of the operator equations Lx = y and Lx + Nx = y. Under some conditions we prove the equivalence of these solvabilities. We also give an estimation for the relative-errors of the solutions of these two systems and an application of our method to a non-linear control system.

The Reflexive Selfadjoint Solutions to Some Operator Equations

In this paper,we study the existence of the reflexive,reflexive selfadjoint and reflexive positive solutions to some operator equations with respect to the generalized reflection operator dual(P,Q).We derive necessary and sufficient conditions for the solvability of these equations and provide a detailed description of the solutions in the solvable case by using the Moore-Penrose inverses.

（Φ，△）－TYPE PROBABILISTIC CONTRACTOR AND SOLUTIONS FOR A NONLINEAR OPERATOR EQUATIONS IN MENGER PN－SPACES

The purpose of this paper is to introduce the coneept of (Φ,△)-type probabilistic contractor in Menger PN-spaces and to study the existence and uniqueness of solutions for the nonlinear operator equations with such probabilistic contractor in Menger PN-spaces.The results presented in this paper improve and extend the corresponding results in [1] and [4-8].

A Posteriori Parameter Choice Strategy for Nonlinear Monotone Operator Equations

In this paper we present an a posteriori parameter choice strategy for nonlinear ill-posed operator equations involving monotone operators. Under certain conditions, this a posteriori parameter choice strategy guarantees the optimal convergence rate O (δ1/2) for Tikhonov-Browder regularization, where δ denotes the noise level of the data perturbation.

EXISTENCE AND NONEXISTENCE FOR THE INITIAL BOUNDARY VALUE PROBLEM OF ONE CLASS OF SYSTEM OF MULTIDIMENSIONAL NONLINEAR SCHRDINGER EQUATIONS WITH OPERATOR AND THEIR SOLITON SOLUTIONS

The nonlinear interactions between the monochromatic wave have been considered by K. Matsunchi, who also proposed one class of the nonlinear Schrdinger equation system with wave operator. We also obtain the same type of equations, which are satisfied by transverse velocity of higher frequency electron, as we study soliton in plasma physics. In this paper, under some condition we study the existence and nonexistence for this equations in the cases possessing different signs in nonlinear term.

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR A NONHOMOGENEOUS BIHARMONIC EQUATION WITH A BOUNDARY OPERATOR OF A FRACTIONAL ORDER

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.

MIKUSINSKI＇S OPERATORS SOLUTION OF THREE－ORDER LINEAR DIFFERENCE EQUATION WITH VARIABLE COEFFICIENTS

This paper proceeds the papers [3] [4], we make use of the idea of the variable ,number operators and some concepts and conclusions of the shifting operators serieswith variable coefficients in the operational field of Mikusinski, it is devoted to thesolution of the general three-order linear difference equation with variable,coefficients,and it is also devoted to the better solution .formula for the some special three-orderlinear difference equations with variable coefficients, in addition, we try to provide theidea and method for realizing solution of the more than three-order linear differenceequation with variable coefficients.

Dark Sharma-Tasso-Olver Equations and Their Recursion Operators

A complete scalar classification for dark Sharma-Tasso-Olver＇s（STO＇s） equations is derived by requiring the existence of higher order differential polynomial symmetries. There are some free parameters for every class of dark STO systems, thus some special equations including symmetry equation and dual symmetry equation are obtained by selecting a free parameter. Furthermore, the recursion operators of STO equation and dark STO systems are constructed by a direct assumption method.

INVARIANTS-PRESERVING DU FORT-FRANKEL SCHEMES AND THEIR ANALYSES FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH WAVE OPERATOR

Du Fort-Frankel finite difference method(FDM)was firstly proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953.It is an explicit and unconditionally von Neumann stable scheme.However,there has been no research work on numerical solutions of nonlinear Schrödinger equations with wave operator by using Du Fort-Frankel-type finite difference methods(FDMs).In this study,a class of invariants-preserving Du Fort-Frankel-type FDMs are firstly proposed for one-dimensional(1D)and two-dimensional(2D)nonlinear Schrödinger equations with wave operator.By using the discrete energy method,it is shown that their solutions possess the discrete energy and mass conservative laws,and conditionally converge to exact solutions with an order of for ofο(T^(2)+h_(x)^(2)+(T/h_(x))^(2))1D problem and an order ofο(T^(2)+h_(x)^(2)+h_(Y)^(2)+(T/h_(X))^(2)+(T/h_(y))^(2))for 2D problem in H1-norm.Here,τdenotes time-step size,while,hx and hy represent spatial meshsizes in x-and y-directions,respectively.Then,by introducing a stabilized term,a type of stabilized invariants-preserving Du Fort-Frankel-type FDMs are devised.They not only preserve the discrete energies and masses,but also own much better stability than original schemes.Finally,numerical results demonstrate the theoretical analyses.

A NOTE ON PROBABILISTIC CONTRACTOR COUPLE AND SOLUTIONS FOR A SYSTEM OF NONLINEAR EQUATIONS IN N A MENGER PN-SPACES

The concept of (Phi, Delta)-type probabilistic contractor couple was introduced which simplifies and weakens the definition of probabilistic contractor couple given by Zhang Shisheng. The existence and uniqueness of the solutions for a system of nonlinear operator equations with this kind of propabilistic contractor couple in N. A. Menger PN-spaces were studied. The works improve and extend the corresponding results by M. Altman, A. C. Lee, W. J. Padgett et al.

Fixed Point Theory for 1-Set Weakly Contractive Operators in Banach Spaces

In this work, using an analogue of Sadovskii＇s fixed point result and several important inequalities we investigate and give new existence theorems for the nonlinear operator equation F（x） =μx, （μ≥1） for some weakly sequentially continuous, weakly condensing and weakly 1-set weakly contractive operators with different boundary conditions. Correspondingly, we can obtain some applicable fixed point theorems of Leray-Schauder, Altman and Furi-Pera types in the weak topology setting which generalize and improve the corresponding results of [3,15,16].

SOLUTIONS OF THE GENERALn－TH ORDER VARIABLE COEFFICIENTS LINEAR DIFFERENCE EQUATION

In this paper.variable operator and its product with shifting operator are studied.The product of power series of shifting operator with variable coefficient is defined andits convergence is proved under Mikusinski’s sequence convergence.After turning ageneral variable coefficient linear difference equation of order n into a set of operatorequations.we can obtain the solutions of the general n-th order variable coefficientlinear difference equation.

ON THE ASYMPTOTIC ORDER OF TIKHONOV REGULARIZATION WITH OPERATORS ANDDATA ERROR

It is considered that the Tikhonov regularization with closed operators for solving linear operator equations of the first kind in the presence of perturbed operators. A class of the regularization parameter choice strategies that lead to optimal convergence rates are proposed.

AN APPLICATION OF HARDY-BOEDEWADT'S THEOREM TO ITERATED FUNCTIONAL EQUATIONS

In this paper an iterated functional equation of polynomial type which does not possess the firt order iterative term g(x) is to be discussed. The difficulties resulted from loss of the first order term are overcome by utilization of Hardy-Boedewadt's theorem.

Numerical Solution for Fractional Partial Differential Equation with Bernstein Polynomials

A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational matrix of Bernstein polynomials is derived. With the operational matrix, the equation is transformed into the products of several dependent matrixes which can also be regarded as the system of linear equations after dispersing the variable. By solving the linear equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.