A discrete “three-particle” Schrödinger operator in the Hubbard model

In the space L ₂(T ^ν ×T ^ν ), where T ^ν is a ν-dimensional torus, we study the spectral properties of the “three-particle” discrete Schrödinger operator ? = H₀ + H₁ + H₂, where H₀ is the operator of multiplication by a function and H₁ and H₂ are partial integral operators. We prove several theorems concerning the essential spectrum of ?. We study the discrete and essential spectra of the Hamiltonians H^t and h arising in the Hubbard model on the three-dimensional lattice.     

The Fine Spectra of the Difference Operator △ Over the Sequence Space bvp, （1 ≤ p 〈 ∞）^＊

The sequence space bvp consisting of all sequences （xk） such that （xk -xk-1） belongs to the space gp has recently been introduced by Basar and Altay [Ukrainian Math. J., 55（1）, 136-147（2003）]; where 1 ≤ p ≤ ∞. In the present paper, some results concerning with the continuous dual and f-dual, and the AD-property of the sequence space bvp have been given and the norm of the difference operator A acting on the sequence space bvp has been found. The fine spectrum with respect to the Goldberg＇s classification of the difference operator △ over the sequence space bvp has been determined, where 1≤p〈∞.     

On the functions of one selfadjoint integral operator

We construct a selfadjoint Akhiezer integral operator S on L ₂(?1, 1) with the spectrum {?1; 0; 1} and the property that every function φ(S) that is not a multiple of S fails to be an Akhiezer integral operator.     

On the Integral Operators of the Third Kind

We establish some necessary and sufficient conditions for an operator in the space of square summable functions to be representable as a sum of multiplication by a bounded function and an integral operator.     

Existence of Solutions for a Class of Nonvariational Quasilinear Periodic Problems

We consider a nonlinear nonvariational periodic problem with a nonsmooth potential. Using the spectrum of the asymptotic (as |x| →?∞) differential operator and degree theoretic methods based on the degree map for multivalued perturbations of (S)_?+? operators, we establish the existence of a nontrivial smooth solution.     

A Trotter-Kato type result for a second order difference inclusion in a Hilbert space

A Trotter-Kato type result is proved for a class of second order difference inclusions in a real Hilbert space. The equation contains a nonhomogeneous term f and is governed by a nonlinear operator A, which is supposed to be maximal monotone and strongly monotone. The associated boundary conditions are also of monotone type. One shows that, if An is a sequence of operators which converges to A in the sense of resolvent and fn converges to f in a weighted l²-space, then under additional hypotheses, the sequence of the solutions of the difference inclusion associated to An and fn is uniformly convergent to the solution of the original problem.     

On the Action of a Linear Operator over Sequences in a Banach Space

In this paper we study the action of a bounded linear operator over different kinds of sequences of a Banach space. Our work is mainly devoted to minimal and M - basic sequences. Plans and García Castellón have characterized the boundedness of a linear operator T by requiring the minimality of any sequence whose image is a minimal sequence (e.g. [P, 1969], [GC, 1990]). We extend these results to other types of sequences like M-basic, basic, strong M-basic, etc., We are also interested on conditions that ensure the minimality of the image of a given minimal sequence. Thus in Corollary 3.7 we characterize semi - Fredholm operators as those which transform every p-minimal sequence into q-minimal. In the last section we deal with M - basis whose image is M - basis or norming M - basis or basis or in general the “best” possible sequence.     

Time periodic solutions of compressible Navier-Stokes equations

In this paper, we study the Navier-Stokes equations with a time periodic external force in Rn. We show that a time periodic solution exists when the space dimension n?5 under some smallness assumption. The main idea is to combine the energy method and the spectral analysis for the optimal decay estimates on the linearized solution operator. With the optimal decay estimates, we prove the existence and uniqueness of time periodic solution in some suitable function space by the contraction mapping theorem. In addition, we also study the time asymptotic stability of the time periodic solution.     

Nonlinear homogeneous perturbation of the discrete spectrum of a self-adjoint operator and a new Constrained Saddle Point Theorem

We prove a Critical Point Theorem for C¹ functionals on the unit sphere of a separable Hilbert space H which improves a previous result of ours. This is applied in nonlinear eigenvalue theory to study the effect of suitably restricted homogeneous perturbations upon the discrete spectrum of a bounded self-adjoint operator in H.     

Dynamics of a class of ODEs more general than almost periodic

A temporally global solution, if it exists, of a nonautonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. Assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts, the existence of a special bounded entire solution, as well the existence of stable and unstable manifolds of this solution are established. Moreover, it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation. In the stable case it is shown that this special solution is the pullback attractor of the system. A class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a C₀-semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations.     

Asymptotic error estimates of a linearized projection-difference method for a differential equation with a monotone operator

We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A(t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A(t). The method is applied to a model parabolic problem.     

Composition operators on a Banach space of analytic functions

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL _α ¹ (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range. Supported by NNSFC No.19671036     

On a Class of Analytic Operator Functions and Their Linearizations

We consider an operator function T in a Krein space which can formally be written as (0.1)but the last term on the right of (0.1) is replaced by a relatively form‐compact perturbation of a similar form. We study relations between the operator function T, a selfadjoint operator M in some Krein space, associated with T, and an operator which can be constructed with the help of the operator function –T^–1. The results are applied to a Sturm‐Liouville problem with a coefficient depending rationally on the eigenvalue parameter.     

Nonlinear Boundary Value Problems for Second Order Differential Inclusions

In this paper we study two boundary value problems for second order strongly nonlinear differential inclusions involving a maximal monotone term. The first is a vector problem with Dirichlet boundary conditions and a nonlinear differential operator of the form x ↦ a(x, x′)′. In this problem the maximal monotone term is required to be defined everywhere in the state space ℝ^N. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form x ↦ (a(x)x′)′. In this case the maximal monotone term need not be defined everywhere, incorporating into our framework differential variational inequalities. Using techniques from multivalued analysis and from nonlinear analysis, we prove the existence of solutions for both problems under convexity and nonconvexity conditions on the multivalued right-hand side.     

在再生核空间中方程AuBu+Cu＝f的求解方法

$ 1 引言 本文研究下面一类非线性算子方程求解问题 AμBμ Cμ=f, (1.1)其中f,μ∈W(Ω),μ(O)=1,||f ||=1,A,B,C∈(W(Ω)→W(Ω)),(W(Ω)→W(Ω))是W(Ω)到W(Ω)的连续线性算子空间,W(Ω)是定义在Ω域上的(Ω是实数域R的有界域)再生核空间。 本文是在再生核空间上,通过将一维非线性算子方程(1.1)转化为二维线性算子方     

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.     

Ill-posedness and local ill-posedness concepts in hilbert spaces *

In this paper, we study ill-posedness concepts of nonlinear and linear operator equations in a Hilbert space setting. Such ill-posedness information may help to select appropriate optimization approaches for the stable approximate solution of inverse problems, which are formulated by the operator equations. We define local ill-posedness of a nonlinear operator equation F(x) = y ₀ in a solution point x ₀:and consider the interplay between the nonlinear problem and its linearization using the Fréchet derivative F′(x ₀). To find a corresponding ill-posedness concept for the linearized equation we define intrinsic ill-posedness for linear operator equations A x = y and compare this approach with the ill-posedness definitions due to Hadamard and Nashed     

Continuous dependence on data for bilocal difference equations

The continuous dependence on data is studied for a class of second order difference equations governed by a maximal monotone operator A in a Hilbert space. A nonhomogeneous term f appears in the equation and some bilocal boundary conditions a, b are added. One shows that the function which associates to {a, b, A, f} the solution of this boundary value problem is continuous in a specific sense. One uses the convergence of a sequence of operators in the sense of the resolvent. The problem studied here is the discrete variant of a problem from the continuous case.     

Eigenvalue problems and fixed point theorems for a class of positive nonlinear operators

We study the eigenvalue problems for a class of positive nonlinear operators defined on a cone in a Banach space. Using projective metric techniques and Schauder’s fixed-point theorem, we establish existence, uniqueness, monotonicity and continuity results for the eigensolutions. Moreover, the method leads to a result on the existence of a unique fixed point of the operator. Applications to nonlinear boundary-value problems, to differential delay equations and to matrix equations are considered.      

Exact and analytical solutions for a nonlinear sigma model

We consider wave solutions to nonlinear sigma models in n dimensions. First, we reduce the system of governing PDEs into a system of ODEs through a traveling wave assumption. Under a new transform, we then reduce this system into a single nonlinear ODE. Making use of the method of homotopy analysis, we are able to construct approximate analytical solutions to this nonlinear ODE. We apply two distinct auxiliary linear operators and show that one of these permits solutions with lower residual error than the other. This demonstrates the effectiveness of properly selecting the auxiliary linear operator when performing homotopy analysis of a nonlinear problem. From here, we then obtain residual error‐minimizing values of the convergence control parameter. We find that properly selecting the convergence control parameter makes a drastic difference in the magnitude of the residual error. Together, appropriate selection of the auxiliary linear operator and of the convergence control parameter is shown to allow approximate solutions that quickly converge to the true solution, which means that few terms are needed in the construction of such solution. This, in turn, greatly improves computational efficiency. Copyright © 2013 John Wiley & Sons, Ltd.