Compact and precompact sets in asymmetric locally convex spaces

The aim of the present paper is to study precompactness and compactness within the framework of asymmetric locally convex spaces, defined and studied by the author in [S. Cobza?, Asymmetric locally convex spaces, Int. J. Math. Math. Sci. 2005 (16) (2005) 2585-2608]. The obtained results extend some results on compactness in asymmetric normed spaces proved by [L.M. García-Raffi, Compactness and finite dimension in asymmetric normed linear spaces, Topology Appl. 153 (2005) 844-853], and [C. Alegre, I. Ferrando, L.M. García-Raffi, E.A. Sánchez-Pérez, Compactness in asymmetric normed spaces, Topology Appl. 155 (6) (2008) 527-539].     

Hardy spaces for the conjugated Beltrami equation in a doubly connected domain

We consider Hardy spaces associated to the conjugated Beltrami equation on doubly connected planar domains. There are two main differences with previous studies (Baratchart et al., 2010 [2]). First, while the simple connectivity plays an important role in Baratchart et al. (2010) [2], the multiple connectivity of the domain leads to unexpected difficulties. In particular, we make strong use of a suitable parametrization of an analytic function in a ring by its real part on one part of the boundary and by its imaginary part on the other. Then, we allow the coefficient in the conjugated Beltrami equation to belong to W1,q for some q∈(2,+∞], while it was supposed to be Lipschitz in Baratchart et al. (2010) [2]. We define Hardy spaces associated with the conjugated Beltrami equation and solve the corresponding Dirichlet problem. The same problems for generalized analytic function are also solved.     

Modified block iterative algorithm for Quasi-?-asymptotically nonexpansive mappings and equilibrium problem in banach spaces

The purpose of this paper is to propose a modified block iterative algorithm for find a common element of the set of common fixed points of an infinite family of quasi-?-asymptotically nonexpansive mappings and the set of an equilibrium problem. Under suitable conditions, some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. As an application, at the end of the paper a numerical example is given. The results presented in the paper improve and extend the corresponding results in Qin et al. [Convergence theorems of common elements for equilibrium problems and fixed point problem in Banach spaces, J. Comput. Appl. Math., 225, 2009, 20-30], Zhou et al. [Convergence theorems of a modified hybrid algorithm for a family of quasi-?-asymptotically nonexpansive mappings, J. Appl. Math. Compt., 17 March, 2009, doi:10.1007/s12190-009-0263-4], Takahashi and Zembayshi [Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70, 2009, 45-57], Wattanawitoon and Kumam [Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal. Hybrid Syst., 3, 2009, 11-20] and Matsushita and Takahashi [A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory, 134, 2005, 257-266] and others.     

Approximation theorems for total quasi-?-asymptotically nonexpansive mappings with applications

The purpose of this article is to prove some approximation theorems of common fixed points for countable families of total quasi-?-asymptotically nonexpansive mappings which contain several kinds of mappings as its special cases in Banach spaces. In order to get the approximation theorems, the hybrid algorithms are presented and are used to approximate the common fixed points. Using this result, we also discuss the problem of strong convergence concerning the maximal monotone operators in a Banach space. The results of this article extend and improve the results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 134 (2005) 257-266], Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 149 (2007) 103-115], Li, Su [H. Y. Li, Y. F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72(2) (2010) 847-855], Su, Xu and Zhang [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3960], Wang et al. [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364-2371], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. 73 (2010) 2260-2270], Chang et al. [S.S. Chang, C.K. Chan, H.W. Joseph Lee, Modified block iterative algorithm for quasi-?-asymptotically nonexpansive mappings and equilibrium problem in Banach spaces, Appl. Math. Comput. 217 (2011) 7520-7530], Ofoedu and Malonza [E.U. Ofoedu, D.M. Malonza, Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type, Appl. Math. Comput. 217 (2011) 6019-6030] and Yao et al. [Y.H. Yao, Y.C. Liou, S.M. Kang, Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings, Appl. Math. Comput. 208 (2009) 211-218].     

Boundary conditions for linear manifolds. III. Infinite dimensional adjoints via solution spaces

A formula is given for the orthogonal complement of any vector subspace of l₂. Countably infinite adjoint subspaces in a Banach space are characterized via solution spaces. In particular, infinite dimensional self-adjoint subspaces in a reflexive Banach space are characterized via solution spaces, generalizing a result in Dunford and Schwartz [“Linear Operators, II,” Interscience, New York, 1963]. Applications are made to closed linear manifolds in l₂ ⊕ l₂ as well as infinite dimensional, generalized ordinary differential subspaces in a Hilbert space with the boundary conditions imposed on real sequences. The results are also expressed via solution spaces.     

A new contribution to a dynamic competitive equilibrium problem

The aim of this paper is to study the Walrasian equilibrium problem when the data are time dependent. For this model an existence result is provided using the variational inequality theory in infinite dimensional spaces. Our results are the generalization of some of the results obtained by several authors in the static case (see e.g. Donato et al. (2008) [5], Donato et al. (2008) [4] and Mordukhovich (2006) [11], Nagurney (1993) [2] and the references therein).     

Global attractors for gradient flows in metric spaces

We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope of Ambrosio et al. (2005) [5], we introduce the latter to include limits of time-incremental approximations constructed via the Minimizing Movements approach (De Giorgi, 1993; Ambrosio, 1995 [3], [15]).For both notions of solutions we prove the existence of the global attractor. Since the evolutionary problems we consider may lack uniqueness, we rely on the theory of generalized semiflows introduced in Ball (1997) [7].The notions of generalized and energy solutions are quite flexible, and can be used to address gradient flows in a variety of contexts, ranging from Banach spaces, to Wasserstein spaces of probability measures. We present applications of our abstract results, by proving the existence of the global attractor for the energy solutions, both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.     

Property C and closed maps

A standard theorem from dimension theory states that a closed (m+1) to 1 map defined on a finite dimensional space can raise dimension by at most m. Dimension raising maps on countable dimensional spaces and on weakly infinite dimensional spaces have been investigated by A.V. Arhangelskii, A.I. Vainstein and E.G. Sklyarenko. A typical theorem is that a closed map on such spaces raises dimension only if some point has an uncountable number of preimages. A class of infinite dimensional spaces closely related to the two types mentioned above is the class of C spaces. R. Pol's example in 1980 and work of F.D. Ancel have generated renewed interest in C spaces. We prove results about dimension raising closed maps defined on C spaces that are analogous to the results mentioned above.     

Strong convergence of iterative methods for k-strictly pseudo-contractive mappings in Hilbert spaces

In this paper, we introduce composite iterative schemes for finding fixed points of k-strictly pseudo-contractive mappings for some 0?k<1 in Hilbert spaces. Then, under certain different control conditions, we establish strong convergence theorems on the composite iterative schemes. The main theorems improve and generalize the recent corresponding results of Cho et al. [5] and Marino and Xu [9] as well as Halpern [6], Wittmann [12], Moudafi [10] and Xu [14].     

Basis problems for matrix valued functions

Basis problems for self-adjoint matrix valued functions are studied. We suggest a new and nonstandard method to solve basis problems both in finite and infinite dimensional spaces. Although many results in this paper are given for operator functions in infinite dimensional Hilbert spaces, but to demonstrate practicability of this method and to present a full solution of basis problems, in this paper we often restrict ourselves to matrix valued functions which generate Rayleigh systems on the n-dimensional complex space Cn. The suggested method is an improvement of an approach given recently in our paper [M. Hasanov, A class of nonlinear equations in Hilbert space and its applications to completeness problems, J. Math. Anal. Appl. 328 (2007) 1487-1494], which is based on the extension of the resolvent of a self-adjoint operator function to isolated eigenvalues and the properties of quadratic forms of the extended resolvent. This approach is especially useful for nonanalytic and nonsmooth operator functions when a suitable factorization formula fails to exist.     

Lineability criteria,with applications

Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer structures, then the more stringent notions of dense-lineability, maximal dense-lineability and spaceability arise naturally. In this paper, several lineability criteria are provided and applied to specific topological vector spaces, mainly function spaces. Sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature. Families of strict-order integrable functions, hypercyclic vectors, non-extendable holomorphic mappings, Riemann non-Lebesgue integrable functions, sequences not satisfying the Lebesgue dominated convergence theorem, nowhere analytic functions, bounded variation functions, entire functions with fast growth and Peano curves, among others, are analyzed from the point of view of lineability.     

Stochastic equations for linear random functionals and statistical problems

Linear Random Functionals have been introduced by the author [2] to develop the theory of Kalman filtering for infinite dimensional linear systems. It is reminiscent of the concept of stochastic integral, which it partly generalizes. We compare it to that of cylindrical Wiener processes, introduced by G. Da Prato- J. Zabczyk [4]. Like distributions, linearity limits the power of the tool. We can consider however some non-linear problems. We show that it is a powerful tool to deal with statistical problems in infinite dimensional spaces. For additional relevant references see [1], [6], [7], [3].     

Algebraic duality theorems for infinite LP problems

In this paper, we consider a primal-dual infinite linear programming problem-pair, i.e. LPs on infinite dimensional spaces with infinitely many constraints. We present two duality theorems for the problem-pair: a weak and a strong duality theorem. We do not assume any topology on the vector spaces, therefore our results are algebraic duality theorems. As an application, we consider transferable utility cooperative games with arbitrarily many players.     

Lie symmetries for the orbital linearization of smooth planar vector fields around singular points

This work is a generalization of the method proposed in [I.A. García, S. Maza, Linearization of analytic isochronous centers from a given commutator, J. Math. Anal. Appl. 339 (1) (2008) 740-745] of linearization of analytic isochronous centers from a given commutator. In this paper we propose a constructive procedure to get the change of variables that orbitally linearizes a smooth planar vector field on C² around an elementary singular point (i.e., a singular point with associated eigenvalues λ,μ∈C satisfying λ≠0) or a nilpotent singular point from a given infinitesimal generator of a Lie symmetry.     

Eine kartesisch abgeschlossene Kategorie glatter Abbildungen zwischen beliebigen lokalkonvexen Vektoräumen

This paper is a continuation of [6], in which I identified thec ^∞-complete bornological locally convex spaces (in short: 1cs) as the right ones for infinite dimensional analysis. Here I discuss smooth mappings between arbitrary 1cs, where a mapping is called smooth iff its compositions with smooth curves are smooth. The 1^st part is mainly devoted to prove the cartesian closedness of the category of (bornological,c ^∞-complete) 1cs together with the smooth mappings between them. In the 2^nd part I discuss the bornology of function spaces and furthermore demonstrate the smoothness of the differentiation process. Finally, in the 3^rd part, I compare this concept of smoothness with several others, discussed byKeller in [5], and show it to be the weakest that fulfills the chainrule.     

Improvement in estimating the population mean in simple random sampling using information on auxiliary attribute

This paper proposes some estimators for the population mean by adapting the estimator in Singh et al. (2008) [5] to the ratio estimators presented in Kadilar and Cingi 2006 [2]. We obtain mean square error (MSE) equation for all proposed estimators, and show that all proposed estimators are always more efficient than ratio estimator in Naik and Gupta (1996) [3], and Singh et al. (2008) [5]. The results have been illustrated numerically by taking some empirical population considered in the literature.     

On some nonlinear evolution equations with strong dissipation,III

In this paper we continue the existence theories of classical solutions of nonlinear evolution equations with strong dissipation studied in previous papers [5, 6], where we proved the existence of global classical solutions with small data applying small energy techniques. This time, we prove the existence of a set of initial values which guarantees the solution to be global. We know the set is not bounded in the escalated energy spaces (Sobolev spaces). For the purpose, we establish approximate equations with another dissipative term which give a devised penalty to the solutions and lead the solutions to be bounded for all t > 0. Therefore we give an improvement to existence theories of equations describing a local statement of balance of momentum for materials for which the stress is related to strain and strain rate. These have been studied by many authors (cf. Greenberg et al. 19], Greenberg [10], Davis [3], Clements [2], Andrews [1], Yamada [12], Webb [13], etc.).     

On factorization of Hilbert-Schmidt mappings

We present some results on factorization of Hilbert-Schmidt multilinear mappings and polynomials through infinite dimensional Banach spaces, L₁ and L_∞ spaces. We conclude this work with a result on factorization of holomorphic mappings of Hilbert-Schmidt type.     

Common fixed point results for two new classes of hybrid pairs in symmetric spaces

Some common fixed point theorems due to Abbas and Khan [M. Abbas, A.R. Khan, Common fixed points of generalized contractive hybrid pairs in symmetric spaces, Fixed Point Theor. Appl. 2009 (2009) 11, Article ID 869407, doi:10.1155/2009/869407], and Abbas and Rhoades [M. Abbas, B.E. Rhoades, Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings defined on symmetric spaces, Pan. Amer. Math. J. 18 (1) (2008) 55-62] are proved for two new classes of hybrid pair of mappings which contain occasionally weakly compatible hybrid pairs as a proper subclass. Consequently, some results proved by Hussain et al. [N. Hussain, M.A. Khamsi, A. Latif, Common fixed points for JH-operators and occasionally weakly biased pairs under relaxed conditions, Nonlinear Anal. 74 (2011) 2133-2140], Bhatt et al. [A. Bhatt, et al., Common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions, Nonlinear Anal. 73 (2010) 176-182] and many others are extended to hybrid pair of mappings. Examples are also presented to support the concepts defined in the paper.     

On some nonlinear evolution equations with the strong dissipation,II

In this paper we continue the existence theories of classical solutions of nonlinear evolution equations with the strong dissipation studied in a previous paper [5]. In particular, we give sufficient conditions under which some of the equations have global solutions and at the same time we find steady state solutions of these equations which are exponentially stable as t → ∞. In the application, we improve the existence results to the equations which describe a local statement of balance of momentum for materials for which the stress is related to strain and strain rate through some constitutive equation (cf. Greenberg et al. [6], Greenberg [7], Davis [2], Clements [1], etc.).