Math

The purpose of this paper is to study the convergence of the Ishikawa and Mann iterative sequences with mixed errors to approximate the solutions of nonlinear operator equations with perturbed maccretive mappings in arbitrary Banach spaces. The results presented in this paper extend and improve some recent results in Jung and Morales [Nonlinear Anal. TMA 46 (2001) 231-243], Jung et al.

the present paper, we study the quasiequilibrium problem and generalized qussiequilibrium problem of generalized quasi-variational inequality in H-spaces by a new method. Some new equilibrium existence theorems are given. Our results are different from corresponding given results or contain some recent results as their special cases.

The purpose of this paper is to prove new existence theorems of equilibria in generalized SSC without assuming the interior condition of the state correspondence nor the strong continuity assumption on agent's preference correspondence by using the Fan-Glicksberg fixed-point theorem, and also give some corollaries including the Nash equilibrium existence for an abstract economy. Our results generalize the corresponding results due to Keiding, Vind and Kim and Yuan in several ways.

In this survey, a new minimax inequality and one equivalent geometric form are proved. Next, a theorem concerning the existence of maximal elements for an L C-majorized correspondence is obtained. By the maximal element theorem, existence theorems of equilibrium point for a noncompact one-person game and for a noncompact qualitative game with L C-majorized correspondences are given. Using the last result and employing 'approximation approach', we prove the existence of equilibria for abstract economies in which the constraint correspondence is lower (upper) semicontinuous instead of having lower (upper) open sections or open graphs in infinitedimensional topological spaces. Then, as the applications, the existence theorems of solutions for the quasi-variational inequalities and generalized quasi-variational inequalities for noncompact cases are also proven. Finally, with the applications of quasi-variational inequalities, the existence theorems of Nash equilibrium of constrained games with noncompact are given. Our results include many results in the literature as special cases.

In this note, the characterization for a set-valued mapping with finitely metrically open values being a generalized metric KKM mapping in hyperconvex metric spaces is established. This result could be regarded as a dual form of corresponding results for the Fan᎐KKM principle in hyperconvex metric spaces obtained recently w Ž .

The aim of this paper is to establish general existence results of maximal elements for L L-majorized mappings, which are, in turn, used to establish the Ž general existence theorems of equilibria for generalized games resp., abstract. economics without lower semicontinuity for both constraint and preference map-Ž. Ž. pings correspondences and in which strategic resp., commodity spaces may not Ž. be compact, the set of players resp., agents may not be countable, and underlying spaces are either finite or infinite dimensional locally topological vector spaces. Our results unify or improve corresponding results in the literature.

Some sufficient and necessary conditions for the iterative sequence converging to a common fixed points for a family of nonexpansive mappings in Banach spaces are obtained. The results presented in this paper not only give an affirmative answer to Halpern's open question and a partial answer to the Reich's open question but also extend and improve some recent results of Bauschke [

In this paper, Ky Fan's KKM mapping principle is used to establish the existence of solutions for simultaneous variational inequalities. By applying our earlier results together with Fan-Glicksberg fixed point theorem, we prove some existence results for implicit variational inequalities and implicit quasi-variational inequalities for set-valued mappings which are either monotone or upper semi-continuous.

The aim of this paper is to develop the general generic stability theory for nonlinear complementarity problems in the setting of infinite dimensional Banach spaces. We first show that each nonlinear complementarity problem can be approximated arbitrarily by a nonlinear complementarity problem which is stable in the sense that the small change of the objective function results in the small change of its solution set; and thus we say that almost all complementarity problems are stable from viewpoint of Baire category. Secondly, we show that each nonlinear complementarity problem has, at least, one connected component of its solutions which is stable, though in general its solution set may not have good behaviour (i.e., not stable). Our results show that if a complementarity problem has only one connected solution set, it is then always stable without the assumption that the functions are either Lipschitz or differentiable.

The purpose of this article is to give a characterization of an upper hemicontinuous mapping with non-empty convex values being upper demicon- tinuous, i.e., we show that an upper hemicontinuous set-valued mapping with non-empty convex values (not necessarily compact-valued) is upper demicon- tinuous if and only if the set-valued mapping has no interior asymptotic plane.

This paper presents adaptive wavelet collocation methods for the numerical solutions to partial integro-differential equations (PIDEs) arising from option pricing in a market driven by jump-diffusion process. The first contribution of this paper lies in the formulation of the wavelet collocation schemes: the integral and differential operators are formulated in the collocation setting exactly and efficiently in both adaptive and non-adaptive wavelet settings. The wavelet compression technique is employed to replace the full matrix corresponding to the nonlocal integral term by a sparse matrix. An adaptive algorithm is developed, which automatically obtains the solution on a near-optimal grid. The second contribution of this paper is the theoretical analysis of the wavelet collocation schemes: due to the possible degeneracy of the parabolic operators, classical solutions of the jump-diffusion models may not exist. In this paper we first prove the convergence and stability of the proposed numerical schemes under the framework of viscosity solution theory, and then the numerical experiments demonstrate the accuracy and computational efficiency of the methods we developed.

In this paper we introduce the (I, J) similar method for incompressible two-dimensional Euler equations, and obtain a series of explicit (I, J) similar solutions to the incompressible two-dimensional Euler equations. These solutions include all of the twin wave solutions, some new singularity solutions, and some global smooth solutions with a finite energy. We also reveal that the twin wave solution and an affine solution to the two-dimensional incompressible Euler equations are, respectively, a plane wave and constant vector. We prove that the initial boundary value problem of the incompressible two-dimensional Euler equations admits a unique solution and discuss the stability of the solution. Finally, we supply some explicit piecewise smooth solutions to the incompressible three-dimensional Euler case and an example of the incompressible three-dimensional Navier-Stokes equations which indicates that the viscosity limit of a solution to the Navier-Stokes equations does not need to be a solution to the Euler equations.