Some fixed point theorems on ordered cone metric spaces

In the present work, two fixed point theorems for self maps on ordered cone metric spaces are proved motivated by [7, L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007) 1468–1476] and [15, A. C. M. Ran and M. C. B. Reuring, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Soc., 132, (2004), 1435–1443]      

二重福里哀级数系数的性质

<正> §1.引言.设 f(x)为以2π为周期的周期函数,其福里哀展开式为下列各事是大家熟知的:设 f(x)在一个基本区间(0,2π)不有界变差的函数,则     

Cyclic compositions and trisotopies

Cyclic compositions in the sense of Springer [T.A. Springer, Oktaven, Jordan-Algebren und Ausnahmegruppen, Universität Göttingen, 1963] and Knus, Merkurjev, Rost and Tignol [M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, Amer. Math. Soc. Colloq. Publ., vol. 44, Amer. Math. Soc., Providence, RI, 1998. With a preface in French by J. Tits] are investigated by means of cyclic trisotopies, a concept originally due to Albert [A.A. Albert, A construction of exceptional Jordan division algebras, Ann. of Math. (2) 67 (1958) 1–28]. Using the quadrupling of composition algebras, we enumerate cyclic trisotopies and compositions in a rational manner, i.e., without extending the base field. We relate cyclic trisotopies explicitly to simple associative algebras of degree 3 with involution and to the Tits process of cubic Jordan algebras.     

On the colored Tutte polynomial of a graph of bounded treewidth

We observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601-622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39-54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276-290], using a different algorithm based on logical techniques.     

Properties of (θ,s)-continuous functions

Joseph and Kwack [Proc. Amer. Math. Soc. 80 (1980) 341–348] introduced the notion of (θ,s)-continuous functions in order to investigate S-closed spaces due to Thompson [Proc. Amer. Math. Soc. 60 (1976) 335–338]. In this paper, further properties of (θ,s)-continuous functions are obtained and relationships between (θ,s)-continuity, contra-continuity and regular set-connectedness defined by Dontchev et al. [Internat. J. Math. Math. Sci. 19 (1996) 303–310 and elsewhere] are investigated.     

Fixed point theorems for generalized contractions in ordered metric spaces

The purpose of this paper is to present some fixed point results for self-generalized contractions in ordered metric spaces. Our results generalize and extend some recent results of A.C.M. Ran, M.C. Reurings [A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443], J.J. Nieto, R. Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239; J.J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205-2212], J.J. Nieto, R.L. Pouso, R. Rodríguez-López [J.J. Nieto, R.L. Pouso, R. Rodríguez-López, Fixed point theorem theorems in ordered abstract sets, Proc. Amer. Math. Soc. 135 (2007) 2505-2517], A. Petru?el, I.A. Rus [A. Petru?el, I.A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006) 411-418] and R.P. Agarwal, M.A. El-Gebeily, D. O'Regan [R.P. Agarwal, M.A. El-Gebeily, D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., in press]. As applications, existence and uniqueness results for Fredholm and Volterra type integral equations are given.     

An alternative definition of coarse structures

Roe [J. Roe, Lectures on Coarse Geometry, University Lecture Series, vol. 31, Amer. Math. Soc., Providence, RI, 2003] introduced coarse structures for arbitrary sets X by considering subsets of X×X. In this paper we introduce large scale structures on X via the notion of uniformly bounded families and we show their equivalence to coarse structures on X. That way all basic concepts of large scale geometry (asymptotic dimension, slowly oscillating functions, Higson compactification) have natural definitions and basic results from metric geometry carry over to coarse geometry.     

Double σ-multiplicative matrices

The class of σ-regular matrices was defined and characterized by Schaefer [P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972) 104-110] and further studied by Mursaleen [Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford 34 (1983) 77-86], Ahmad and Mursaleen [Z.U. Ahmad, Mursaleen, An application of Banach limits, Proc. Amer. Math. Soc. 103 (1988) 244-246]. In this paper we characterize four-dimensional σ-multiplicative matrices, and establish a core theorem.     

Selections and countably-approachable points

The present paper improves a result of Gutev [V. Gutev, Approaching points by continuous selections, J. Math. Soc. Japan (4) (2006) 1203–1210] by characterizing the countably-approachable points in sense of [V. Gutev, Approaching points by continuous selections, J. Math. Soc. Japan (4) (2006) 1203–1210] by a natural extreme-like condition in the spirit of [V. Gutev, T. Nogura, Vietoris continuous selections and disconnectedness-like properties, Proc. Amer. Math. Soc. 129 (2001) 2809–2815; V. Gutev, T. Nogura, Selection pointwise-maximal spaces, Topology Appl. 146–147 (2005) 397–408]. This demonstrates the natural relationship between different extreme-like points with respect to continuous selections for the Vietoris hyperspace of nonempty closed subsets.     

Hilbert C*-Modules over Monotone Complete C*-Algebras

The aim of the present paper is to describe self-duality and C*-reflexivity of Hilbert A-modules ?? over monotone complete C*-algebras A by the completeness of the unit ball of ?? with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results ofH. Widom [Duke Math. J. 23, 309-324, MR 17 # 1228] and W. L. Paschke [Trans. Amer. Mat. Soc. 182 , 443-468, MR 50 # 8087, Canadian J. Math. 26, 1272-1280, MR 57 # 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. M. Ozawa [J. Math. Soc. Japan 36, 589-609, MR 85 # 46068]). Especially, one derives that for a C*-algebra A the A-valued inner product of every Hilbert A-module ?? can be continued to an A-valued inner product on it's A-dual Banach A-module ??' turning ??' to a self-dual Hilbert A-module if and only if A is monotone complete (or, equivalently, additively complete) generalizing a result of M. Hamana [Internat. J. Math. 3 (1992), 185 - 204]. A classification of countably generated self-dual Hilbert A-modules over monotone complete C*-algebras A is established. The set of all bounded module operators End ′(??) on self-dual Hilbert A-modules ?? over monotone complete C*-algebras A is proved again to be a monotone complete C*-algebra. Applying these results a Weyl-Berg type theorem is proved.     

Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces

The iteration scheme for families of nonexpansive mappings, essentially due to Halpern [Bull. Amer. Math. Soc. 73 (1967) 957-961], is established in a Banach space. The main theorem extends a recent result of O'Hara et al. [Nonlinear Anal. 54 (2003) 1417-1426] to a Banach space setting. For the same iteration scheme, with finitely many mappings, a complementary result to a result of Jung and Kim [Bull. Korean Math. Soc. 34 (1997) 93-102] (also Bauschke [J. Math. Anal. Appl. 202 (1996) 150-159]) is obtained by imposing other condition on the sequence of parameters. Our results also improve results in [C. R. Acad. Sci. Sér A-B Paris 284 (1977) 1357-1359; J. Math. Anal. Appl. 211 (1997) 71-83; Arch. Math. 59 (1992) 486-491] in framework of a Hilbert space.     

Hille and Nehari type criteria for third-order dynamic equations

In this paper, we extend the oscillation criteria that have been established by Hille [E. Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948) 234-252] and Nehari [Z. Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc. 85 (1957) 428-445] for second-order differential equations to third-order dynamic equations on an arbitrary time scale T, which is unbounded above. Our results are essentially new even for third-order differential and difference equations, i.e., when T=R and T=N. We consider several examples to illustrate our results.     

Bézout rings with almost stable range 1

Elementary divisor domains were defined by Kaplansky [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949) 464-491] and generalized to rings with zero-divisors by Gillman and Henriksen [L. Gillman, M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956) 362-365]. In [M.D. Larsen, W.J. Lewis, T.S. Shores, Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc. 187 (1) (1974) 231-248], it was also proved that if a Hermite ring satisfies (N), then it is an elementary divisor ring. The aim of this article is to generalize this result (as well as others) to a much wider class of rings. Our main result is that Bézout rings whose proper homomorphic images all have stable range 1 (in particular, neat rings) are elementary divisor rings.     

A survey on curvatures of Hessian manifolds

This paper deals with the relationship of curvatures between Hessian and Riemannian manifolds. By making use of the methods of Shima [Shima H. Hessian manifolds of constant Hessian sectional curvature. J Math Soc Jpn 1995;47(4):735–53] we investigate a new approach to the curvatures of a manifold. Finally, some results are mentioned in the concluding remarks.     

Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities

We study the existence, uniqueness and asymptotic behavior, as well as the stability of a special kind of traveling wave solutions for competitive PDE systems involving intrinsic growth, competition, crowding effects and diffusion. The traveling waves are exclusive in the sense that as the variable goes to positive or negative infinity, different species are close to extinction or carrying capacity. We perform an appropriate affine transformation of the traveling wave equations into monotone form and construct appropriate upper and lower solutions. By this means, we reduce the existence proof to application of well-known theory about monotone traveling wave systems (cf. [A. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering, MIA, Kluwer, Boston, 1989; J. Wu, X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations 13 (2001) 651-687] and [I. Volpert, V. Volpert, V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994]). Then, by using spectral analysis of the linearization over the profile, we prove the orbital stability of the traveling wave in some Banach spaces with exponentially weighted norm. Furthermore, we show that the introduction of some weight is necessary in the sense that, in general, traveling wave solutions with initial perturbations in the (unweighted) space C₀ are unstable (cf. [I. Volpert, V. Volpert, V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994] and [D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981]).     

Pseudocompact groups: progress and problems

Several months ago the speaker and Jan van Mill gave a proof of this result [W.W. Comfort, J. van Mill, Extremal pseudocompact abelian groups are compact metrizable, Abstracts Amer. Math. Soc. 27 (2006) 78 (Abstract #1014-22-958); W.W. Comfort, J. van Mill, Extremal pseudocompact abelian groups are compact metrizable, Proc. Amer. Math. Soc. 135 (2007) 4039-4044]: A pseudocompact abelian group of uncountable weight admits both a proper dense pseudocompact subgroup and a strictly larger pseudocompact group topology.This presentation will describe both the necessary new details of the argument and the historical development (useful tools, special cases). Among those who contributed essentially are: K.A. Ross (1964, 1966); T. Soundararajan (1982); L.C. Robertson (1982, 1988); J. van Mill (1989); J. van Mill and H. Gladdines (1994); J. Galindo (2002).Several related unsolved problems will be cited.     

Existence of multidimensional shock waves and phase boundaries

In this paper, a kind of Riemann problem for the Euler equations in a van der Waals fluid is considered. We constructed the weak solution in multidimensional space which contains one shock front and one subsonic phase boundary. We mainly follow the arguments of Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95; A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1-93] and Métivier's [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479] work. The linear stability results are based on Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95] work for the single shock front and Wang and Xin's [Y.-G. Wang, Z. Xin, Stability and existence of multidimensional subsonic phase transitions, Acta Math. Appl. Sin. 19 (2003) 529-558] work for the single phase boundary. The initial boundary value problem concerned in this paper is different from the boundary value problem for double shock fronts concerned in [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479], we slightly modified Métivier's frame work to establish the existence for the solution to the nonlinear problem.     

极值Beltrami系数的Hamilton序列

考虑了Strebel点与Hamilton序列之间的关系.这个问题是Gardiner F.P.最早研究的(见[Approximation of infinite-dimensional Tcichmiiller spaces,Trans.Amer.Math.Soc.,1984,282(1):367-383]).在无限小Teichmiiller空间中,证明了范金华在[On infinitesimal Teichmüller space,Bull.Austral Math.Soc.,2008,78:293-300]中得到的使{φ_n}成为Hamilton序列的充分条件不是必要的.     

Super-efficiency of vector optimization in Banach spaces

Using variational analysis, in terms of the Clarke normal cone, we consider super-efficiency of vector optimization in Banach spaces. We establish some characterizations for super-efficiency. In particular, dropping the assumption that the ordering cone has a bounded base, we extend a result in Borwein and Zhuang [J.M. Borwein, D. Zhuang, Super-efficiency in vector optimization, Trans. Amer. Math. Soc. 338 (1993) 105-122] to the nonconvex setting.