Existence and quasilinearization in Banach spaces

We establish some existence results for the nonlinear problem Au=f in a reflexive Banach space V, without and with upper and lower solutions. We then consider the application of the quasilinearization method to the above mentioned problem. Under fairly general assumptions on the nonlinear operator A and the Banach space V, we show that this problem has a solution that can be obtained as the strong limit of two quadratically convergent monotone sequences of solutions of certain related linear equations.     

Stochastic evolution equations in Hilbert spaces

In this paper we consider Ito's stochastic differential equation in Hilbert spaces. A strong solution is generated by the difference approximation. A regularity result is obtained for solutions to a class of parabolic stochastic partial differential equations. Hyperbolic stochastic evolution equations are also discussed.     

Convergence analysis of non-quadratic proximal methods for variational inequalities in Hilbert spaces

We consider a general approach for the convergence analysis of proximal-like methods for solving variational inequalities with maximal monotone operators in a Hilbert space. It proves to be that the conditions on the choice of a non-quadratic distance functional depend on the geometrical properties of the operator in the variational inequality, and –- in particular –- a standard assumption on the strict convexity of the kernel of the distance functional can be weakened if this operator possesses a certain `reserve of monotonicity'. A successive approximation of the `feasible set' is performed, and the arising auxiliary problems are solved approximately. Weak convergence of the proximal iterates to a solution of the original problem is proved.     

Method of quasilinearization and positivity of solutions in abstract cones

In this paper, the method of quasilinearization is extended to differential equations in Banach spaces using abstract cones. Frequently, it is useful to apply this technique to a comparison system. Several applications are given indicating the flexibility obtained by appropriate choice of the comparison system and cone.     

Existence, upper and lower solutions and quasilinearization for singular differential equations

In this paper, we discuss existence theorems in the presenceof upper and lower solutions as well as the method of quasilinearization(QSL) for general non-linear second-order singular ordinarydifferential equations. We show the existence of solutions underthe assumption of weak continuity of the non-linear part. Ifthe non-linear part is monotone decreasing, a solution may beobtained by the QSL method as the strong limit of a quadraticallyconvergent sequence of approximate solutions. Under strongerassumptions on the linear and the non-linear parts, a solutionis quadratically bracketed between two monotone sequences ofapproximate solutions of certain related linear equations.     

A quasilinearization method for elliptic problems with a nonlinear boundary condition

We study a nonlinear elliptic second order problem with a nonlinear boundary condition. Assuming the existence of an ordered couple of a supersolution and a subsolution, we develop a quasilinearization method in order to construct an iterative scheme that converges to a solution. Furthermore, under an extra assumption we prove that the convergence is quadratic.     

Strong convergence of projection-like methods in Hilbert spaces

Many mathematical and applied problems can be reduced to finding a common point of a system of convex sets. The aim of this paper is twofold: first, to present a unified framework for the study of all the projection-like methods, both parallel and serial (chaotic, mostremote set, cyclic order, barycentric, extrapolated, etc.); second, to establish strong convergence results for quite general sets of constraints (generalized Slater, generalized uniformly convex, made of affine varieties, complementary, etc.). This is done by introducing the concept of regular family. We proceed as follows: first, we present definitions, assumptions, theorems, and conclusions; thereafter, we prove them.     

Numerical solution of the Falkner-Skan equation based on quasilinearization

We present two iterative methods for solving the Falkner-Skan equation based on the quasilinearization method. We formulate the original problem as a new free boundary value problem. The truncated boundary depending on a small parameter is an unknown free boundary and has to be determined as part of solution. Using a change of variables, the free boundary value problem is transformed to a problem defined on [0, 1]. We apply the quasilinearization method to solve the resulting nonlinear problem. Then we propose two different iterative algorithms by means of a cubic spline solver. Numerical results for various instances are compared with those reported previously in the literature. The comparisons show the accuracy, robustness and efficiency of the presented methodology.     

Products of EP operators on Hilbert spaces

A Hilbert space operator is called the EP operator if the range of is equal to the range of its adjoint . In this article necessary and sufficient conditions are given for a product of two EP operators with closed ranges to be an EP operator with a closed range. Thus, a generalization of the well-known result of Hartwig and Katz (Linear Algebra Appl. 252 (1997), 339-345) is given.

     

A quasilinearization approach to heat transfer in MHD visco-elastic fluid flow

The paper deals with the study of the flow of an incompressible electrically conducting visco-elastic fluid referred to as Walter’s liquid B over a porous non-isothermal stretching sheet using quasilinearization technique adopting a numerical approach. The sheet is assumed to stretch uni-directionally and the fluid above the sheet is at rest if there were no stretching. The temperature profiles are obtained numerically and these are displayed through graphs for diverse values of Prandtl number, visco-elastic parameter, magnetic parameter, source/sink parameter, wall temperature parameter and wall heat flux parameter. The results are compared with those available in literature obtained through analytical procedures and are seen to be in good agreement.     

Existence and Uniqueness Results for Neutral SDEs in Hilbert Spaces

Abstract

In this paper using a Picard type approximation, we present the existence and uniqueness theorems for mild solutions of the semilinear neutral SDEs in Hilbert spaces whose coefficients satisfy non-Lipschitz condition.     

Hilbert spaces expanded with a unitary operator

We study Hilbert spaces expanded with a unitary operator with a countable spectrum. We show that the theory of such a structure is ω ‐stable and admits quantifier elimination. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dissipativity of Runge-Kutta methods in Hilbert spaces

This paper concerns the discretization by Runge-Kutta methods of the initial value problemu _t =f(u), under the dissipative structural condition that there exist α≥0, β>0, such thatf:W→H, ℜe, ∀w∈W, for complex Hilbert spacesW⊆H. It is shown that strong A-stability is necessary to ensure the dissipativity of the method, whilst algebraic stability plus |R(∞)|<1 is a sufficient condition in the case of DJ-irreducible methods.     

On a class of completely continuous operators in Hilbert spaces

We introduce a class G of completely continuous operators and prove theorems on the spectral structure of these operators. In particular, operators of this class are similar to model operators in de Branges spaces.     

Estimates for norms of resolvents of~operators on tensor products of Hilbert spaces

A class of linear operators on tensor products of Hilbert spaces is considered. That class contains integro-differential operators arising in various applications. Estimates for the norm of the resolvent of considered operators are derived. By virtue of the obtained estimates, the spectrum of perturbed operators is investigated. These results are new even in the finite-dimensional case. Applications to integro-differential operators are also discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

非线性奇异扩散方程解的存在性与唯一性

本文讨论奇异扩散方程Ｃａｕｃｈｙ问题解的存在性与唯一性,得到的主要结果是：存在唯一的正则解。     

Strong convergence theorems of hybrid methods for two asymptotically nonexpansive mappings in Hilbert spaces

In this paper, we introduce two modifications of the Ishikawa iteration, by using the hybrid methods, for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups in a Hilbert space. Then, we prove that such two sequences converge strongly to common fixed points of two symptotically nonexpansive mappings and asymptotically nonexpansive semigroups, respectively. Our main result is connected with the results of Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups, Nonlinear. Anal. 67(2007) 2306-2315], Martinez-Yanes and Xu [C. Martinez-Yanes, H.K. Xu, Strong convergence of CQ method for fixed point iteration processes, Nonlinear. Anal. 64 (2006) 2400-2411] and many others.