Nonlinear differential problems with first- or second-order differences in Hilbert spaces

In a real Hilbert space, we investigate the existence, uniqueness and asymptotic properties of the generalized solutions of some nonlinear differential systems with first- or second-order differences, subject to extreme conditions and initial data. Some generalizations of Minty’s theorem for the characterization of maximal monotone operators are also presented.     

Commutators and accretive operators

In this paper, singular values of commutators of Hilbert space operators are estimated. To this aim the accretivity of a transform of the operators is applied. Some recent results of Kittaneh [F. Kittaneh, Singular value inequalities for commutators of Hilbert space operators, Linear Algebra Appl. 430 (2009) 2362-2367] are extended.     

A quasi-Newton method for solving fixed point problems in Hilbert spaces

Summary We study the convergence properties of a projective variant of Newton's method for the approximation of fixed points of completely continuous operators in Hilbert spaces. We then discuss applications to nonlinear integral equations and we produce some numerical examples.     

A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for α$\alpha$-inverse-strongly monotone mappings in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings. The results presented in the paper extend and improve some recent results of Yao and Yao [Y.Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2) (2007) 1551–1558], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings, Appl. Math. Comput. (2007) doi:10.1016/j.amc.2007.07.075], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for Equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006) 506–515], Su, Shang and Qin [Y.F. Su, M.J. Shang, X.L. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. (2007) doi:10.1016/j.na.2007.08.045] and Chang, Cho and Kim [S.S. Chang, Y.J. Cho, J.K. Kim, Approximation methods of solutions for equilibrium problem in Hilbert spaces, Dynam. Systems Appl. (in print)].     

Linear dependence of operators characterized by trace functionals

linear dependence property of two Hilbert space operators is expressed in terms of trace functionals.     

Existence theorems for monotone operators in reflexive Banach spaces

In this paper, we obtain an existence theorem for single-valued monotone operators in a reflexive Banach space. Using this result, we prove a fixed point theorem for nonexpansive mappings in a Hilbert space and an existence theorem for maximal monotone operators in a Banach space. Received: 3 July 2006 Revised: 15 January 2007     

Strong convergence of the CQ method for fixed point iteration processes

Strong convergence theorems are obtained for the CQ method for an Ishikawa iteration process, a contractive-type iteration process for nonexpansive mappings, and the proximal point algorithm for maximal monotone operators in Hilbert spaces.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.     

Strong convergence theorems for 2-generalized hybrid mappings in Hilbert spaces

In this paper, we prove a strong convergence theorem of Halpern’s type for 2-generalized hybrid mappings in a Hilbert space. We also deal with strong convergence theorems by hybrid methods for these nonlinear mappings in a Hilbert space.     

A class of concave operators with applications

In this paper we provide some properties of a class of concave operators and apply these results in discussing three-point boundary value problems for differential equations and nonlinear Volterra integral equations.     

Einige Existenz- und Einschließungssätze für Fixpunkte expandierender Integraloperatoren

Some existence theorems for fixed points of nonlinear integral operators expanding a cone in a Banach space in the sense of Krasnoselskii [5] are given. Using special cones we find pointwise inclusions for the fixed points. The question of finding the best bounds for the fixed points leads to a nonlinear optimization problem with infinitely many restrictions.

     

Solution of finite systems of equations by interval iteration

In actual practice, iteration methods applied to the solution of finite systems of equations yield inconclusive results as to the existence or nonexistence of solutions and the accuracy of any approximate solutions obtained. On the other hand, construction of interval extensions of ordinary iteration operators permits one to carry out interval iteration computationally, with results which can give rigorous guarantees of existence or nonexistence of solutions, and error bounds for approximate solutions. Examples are given of the solution of a nonlinear system of equations and the calculation of eigenvalues and eigenvectors of a matrix by interval iteration. Several ways to obtain lower and upper bounds for eigenvalues are given.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.     

Asymptotic behavior of resolvents of coaccretive operators in the Hilbert ball

We study the resolvents of coaccretive operators in the Hilbert ball, with special emphasis on the asymptotic behavior of their compositions and metric convex combinations. We consider the case where the given coaccretive operators share a common fixed point inside the ball, as well as the case where they share a common sink point on its boundary. We establish weak convergence in the former case and strong convergence in the latter. We also present two related convergence results for a continuous implicit scheme.     

Fluctuations of interacting Markov chain Monte Carlo methods

We present a multivariate central limit theorem for a general class of interacting Markov chain Monte Carlo algorithms used to solve nonlinear measure-valued equations. These algorithms generate stochastic processes which belong to the class of nonlinear Markov chains interacting with their empirical occupation measures. We develop an original theoretical analysis based on resolvent operators and semigroup techniques to analyze the fluctuations of their occupation measures around their limiting values.     

Samuel multiplicity and the structure of semi-Fredholm operators

Two numerical invariants refining the Fredholm index are introduced for any semi-Fredholm operator in such a way that their difference calculates the Fredholm index. These two invariants are inspired by Samuel multiplicity in commutative algebra, and can be regarded as the stabilized dimension of the kernel and cokernel. A geometric interpretation of these invariants leads naturally to a 4×4 uptriangular matrix model for any semi-Fredholm operator on a separable Hilbert space. This model can be regarded as a refined, local version of the Apostol's 3×3 triangular representation for arbitrary operators. Some classical results, such as Gohberg's punctured neighborhood theorem, can be read off directly from our matrix model. Banach space operators are also considered.     

Existence and uniqueness for a nonlinear discrete hyperbolic system

The existence, uniqueness and asymptotic behaviour of the solutions to a nonlinear discrete hyperbolic system subject to some extreme conditions and initial data are investigated in a real Hilbert space.     

Measures of weak noncompactness,nonlinear leray-schauder alternatives in banach algebras satisfying condition (p) and an application

In this paper, we establish some new nonlinear Leray-Schauder alternatives for the sum and the product of weakly sequentially continuous operators in Banach algebras satisfying certain sequential condition (P). The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness. As an application, our results are used to prove the existence of solutions for a nonlinear integral equation.     

On Landweber iteration for nonlinear ill-posed problems in Hilbert scales

Summary. In this paper we derive convergence rates results for Landweber iteration in Hilbert scales in terms of the iteration index for exact data and in terms of the noise level for perturbed data. These results improve the one obtained recently for Landweber iteration for nonlinear ill-posed problems in Hilbert spaces. For numerical computations we have to approximate the nonlinear operator and the infinite-dimensional spaces by finite-dimensional ones. We also give a convergence analysis for this finite-dimensional approximation. The conditions needed to obtain the rates are illustrated for a nonlinear Hammerstein integral equation. Numerical results are presented confirming the theoretical ones. Received May 15, 1998 / Revised version received January 29, 1999 / Published online December 6, 1999     

Two abstract approaches in vectorial fixed point theory

In this paper a fixed point theory is established for operators defined on Cartesian product spaces. Two abstract approaches are presented in terms of closure operators and of general functionals called measures of deviations from zero resembling the measures of noncompactness. In particular, we give vectorial versions to Mönch’s fixed point theorems. An application is included to illustrate the theory.     

Left-definite theory with applications to orthogonal polynomials

In the past several years, there has been considerable progress made on a general left-definite theory associated with a self-adjoint operator A that is bounded below in a Hilbert space H; the term ‘left-definite’ has its origins in differential equations but Littlejohn and Wellman [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280-339] generalized the main ideas to a general abstract setting. In particular, it is known that such an operator A generates a continuum {H_r}_r>0 of Hilbert spaces and a continuum of {A_r}_r>0 of self-adjoint operators. In this paper, we review the main theoretical results in [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280-339]; moreover, we apply these results to several specific examples, including the classical orthogonal polynomials of Laguerre, Hermite, and Jacobi.