iterative approximation to a coincidence point of two mappings in this article two methods for approximating the coincidence point of two mappings are studied and moreover rates of convergence for both methods are given these results are illustrated by several examples in particular we apply such results to study the convergence and their rate of convergence of these methods to the solution of a nonlinear integral equation and of a nonlinear differential equation

quasi valuations extending a valuation abstract suppose f is a field with valuation v and valuation ring o v e is a finite field extension and w is a quasi valuation on e extending v we study quasi valuations on e that extend v in particular their corresponding rings and their prime spectra we prove that these ring extensions satisfy inc incomparability lo lying over and gd going down over o v in particular they have the same krull dimension we also prove that every such quasi valuation is dominated by some valuation extending v under the assumption that the value monoid of the quasi valuation is a group we prove that these ring extensions satisfy gu going up over o v and a bound on the size of the prime spectrum is given in addition a one to one correspondence is obtained between exponential quasi valuations and integrally closed quasi valuation rings given r an algebra over o v we construct a quasi valuation on r we also construct a quasi valuation on r o v f which helps us prove our main theorem the main theorem states that if r e satisfies r f o v and e is the field of fractions of r then r and v induce a quasi valuation w on e such that r o w and w extends v thus r satisfies the properties of a quasi valuation ring

symmetric spaces approach to some fixed point results abstract in this paper we present a simple and unified approach to the fixed point results on cone symmetric spaces and metric type spaces based on symmetric spaces fixed point theory we also give a new characterization of semi metric spaces with open balls

valuations on algebras with involution let a be a central simple algebra with involution of the first or second kind let v be a valuation on the fixed part f of z a a special v gauge g on a is a kind of value function on a extending v on f such that g x x 2g x for all x in a it is shown under certain restrictions if the residue characteristic is 2 that if v is henselian then there is a special v gauge g if and only if is anisotropic and g is unique if v is not henselian it is shown that there is a special v gauge g if and only if remains anisotropic after scalar extension from f to the henselization of f with respect to v when this occurs g is the unique invariant v gauge on a

common fixed point theorems for some new generalized contractive type mappings in this paper some new generalized contractive type conditions for a pair of mappings in metric space are defined some common fixed point results for these mappings are presented

a stabilization algorithm for a class of uncertain linear systems abstract this paper presents an algorithm for the stabilization of a class of uncertain linear systems the uncertain systems under consideration are described by state equations which depend on time varying unknown but bounded uncertain parameters the construction of the stabilizing controller involves solving a certain algebraic riccati equation furthermore the solution to this riccati equation defines a quadratic lyapunov function which is used to establish the stability of the closed loop system this leads to a notion of quadratic stabilizability it is shown that the stabilization procedure will succeed if and only if the given uncertain linear system is quadratically stabilizable the paper also deals with a notion of overbounding for uncertain linear systems this procedure enables the stabilization algorithm to be applied to a larger class of uncertain linear systems also included in the paper are results which indicate the degree of conservativeness introduced by this overbounding process

the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

an algebraic riccati equation approach to h optimization abstract this paper considers the general so called four block h optimal control problem with the assumption that system states are available for feedback it is shown that infimization of the h norm of the closed loop transfer function over all linear constant i e nondynamic stabilizing state feedback laws can be completely characterized via an algebraic riccati equation it is further shown that the optimal norm is not improved by allowing feedback to be dynamic thus the general state feedback h optimal control problem can be solved by iteratively solving one are and the controller can be chosen to be static gain

the defektsatz for central simple algebras let q be a central simple algebra finite dimensional over its center f and let v be a valuation ring of f the v has en extension to q i e there exists a dubrovin valuation ring b of q satisfying v f b generally the number of extensions of v to q is not finite and therefore the so called intersection property of dubrovin valuation rings b 1 b n is introduced this property is defined in terms of the prime ideals and the valuation overrings of the intersection b b n it is shown that there exists a uniquely determined natural number n depending only on v and having the following property if b 1 b k are extensions of v having the intersection property then k n and k n holds if and only if b 1 b k is integral over v

on the dec group of finite abelian galois extensions over global fields abstract if k f is a finite abelian galois extension of global fields whose galois group has exponent t we prove that there exists a short exact sequence 0 dec k f br t k f q p r q z z 0 where r q q and p is a finite set of primes of f that is empty if t is square free in particular we obtain that if t is square free then dec k f br t k f which we use to show that prime exponent division algebras over henselian valued fields with global residue fields are isomorphic to a tensor product of cyclic algebras finally we construct a counterexample to the result for higher exponent division algebras

robust h sub spl infin control of uncertain nonlinear system via state feedback this technical note presents a solution of robust h sub spl infin control problem for an affine nonlinear system with gain bounded uncertainty it is shown that the l sub 2 induced norm of a closed loop system with the uncertainty is less than one if an extended nonlinear system with no uncertainty is dissipative with respect to a supply rate in consequence of this result a state feedback law such that the closed loop system has h sub spl infin robust performance is derived based on the hamilton jacobi inequality it is also shown that the existing results for linear systems are special cases of the presented results

h sub infinity control and quadratic stabilization of systems with parameter uncertainty via output feedback the article concerns linear systems which are subject to both time varying norm bounded parameter uncertainty and exogenous disturbance it addresses the robust h sub infinity control problem of designing a linear dynamic output feedback controller such that the closed loop system is quadratically stable and achieves a prescribed level of disturbance attenuation for all admissible parameter uncertainties it is shown that such a problem is equivalent to a scaled h sub infinity control problem

extensions and applications of acf mappings abstract using a definition of asf sequences derived from the definition of asymptotic contractions of the final type of acf we give some new fixed point theorem for cyclic mappings and an alternating mapping which extend results from suzuki 2007 8 theorem 2 and zhang 2007 10 theorem 1

existence and uniqueness of common fixed points for two multivalued operators in ordered metric spaces in this paper we discuss some new fixed point theorems for a pair of multivalued operators which satisfy weakly generalized contractive conditions our results are the extension and improvement of corresponding results of j harjani and k sadarangani fixed point theorems for weakly contractive mappings in partially ordered sets nonlinear analysis 71 2009 3403 3410 and x zhang common fixed point theorems for some new generalized contractive type mappings j math anal appl 333 2007 780 786 finally some examples are given to illustrate the usability of our results

totally ramified valuations on finite dimensional division algebras division algebras d with valuation v are studied where d is lsnite dimensional and totally ramified over its center f i e the ramification index of v over wlf equals d f such division algebras have appeared in some important constructions but the structure of these algebras has not been systematically analyzed before when vlf is henselian a full classification of the f subalgebras of d is given when f has a henselian valuation v with separably closed residue field and a is any tame central simple f algebra an algorithm is given for computing the underlying division algebra of a from a suitable subgroup of a f some examples are constructed using this valuation theory including the first example of finitbdimensional f central division algebras d1 and d2 with d1 xf d2 not a division ring but d1 and d2 having no common subfield k d f valuation theory long a basic tool in commutative algebra has been relatively neglected in the study of division algebras until quite recently nontheless valuations are naturally present in a number of division algebras that have been constructed to exhibit special properties particularly algebras over iterated laurent power series fields for example such division algebras have been key ingredients in amitsur s noncrossed product construction am and in platonov s construction p of division algebras d with sk1 d 7 1 valuations are not so prevalent on division algebras as on fields but if a division algebra d does have a valuation this structure contains a substantial amount of information about d which would scarcely be accessible otherwise we consider here valued divisioll algebras d for which d is totqlly ramified and tame over its center f i e for which the ramification index ird rfi equals the dimension d f of d over f and the characteristic char d does not divide d f here rd is the value group of the valuation on d and d is the residue division algebra we assume throughout that d f oo valued division algebras of this type appear e g in amitsur s noncrossed product paper am 2 in saltman s work on indecomposable division algebras sa in certain of the malcevneumann division algebras considered by the first author and amitsur ta2 4 etc however the intrinsic structure of totally ramified tame division algebras has apparently not been examined closely before this may be because most past work on valued division algebras has concentrated on discrete valuations when rd z for such a valuation d is never totally ramified over its center f unless d f cf 3 2 below received by the editors june 26 1986 1980 mathematics subject clmsification 1985 reon primary 16a39 2supported in part by f n r s 2supported in part by the national science foundation 1987 american mathematlcal society 0002 9947 87 00 25 per page

hybrid fixed points of multivalued operators in metric spaces with applications abstract this paper deals with the existence of hybrid fixed points involving two multivalued operators in a complete metric space our claim is also illustrated with applications to a class of integral inclusions for proving the existence results

fixed point results for set valued contractions by altering distances in complete metric spaces nadler s contraction principle has led to fixed point theory of set valued contraction in non linear analysis inspired by the results of nadler the fixed point theory of set valued contraction has been further developed in different directions by many authors in particular by reich mizoguchi takahashi feng liu and many others in the present paper the concept of generalized contractions for set valued maps in metric spaces is introduced and the existence of fixed point for such a contraction are guaranteed by certain conditions our first result extends and generalizes the nadler feng liu and klim wardowski theorems and the second result is different from the reich and mizoguchi takahashi results as a consequence we derive some results related to fixed point of set valued maps satisfying certain conditions of integral type

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

value functions on central simple algebras in this paper we study noncommutative valuation rings as defined by dubrovin while there is in general no valuation associated to a dubrovin valuation ring we show that there is a value function associated to any dubrovin valuation ring integral over its center by using value functions we obtain information on dubrovin valuation rings in a tensor product both generalizing and giving a much simpler proof of a result about valued division algebras by being able to work directly with central simple algebras we gain new information about division algebras over henselian fields

fixed points of multivalued operators in ordered metric spaces with applications abstract the purpose of this paper is to present some new hybrid fixed point theorems involving multivalued operators which satisfy weakly generalized contractive conditions in an ordered complete metric space an example of the existence of solutions for a perturbed impulsive hyperbolic differential inclusion with variable times is given to illustrate the usability of our results

valuation rings in finite dimensional division algebras a subring b of a division algebra d is called a valuation ring of d if x or x 1 is contained in b for every x in d x 0 such a ring is called an extension of the valuation ring b of k the centre of d if b k b let d be a division algebra finite dimensional over its centre k d k n2 b a valuation ring of k and b bi iei the set of all extension of b in d theorem 1 b possesses at most n extensions in d i e b n theorem 2 any two extensions of b in d are conjugate in d theorems 3 and 4 the set t of elements in d which are integral over b is a subring of d if and only if b 1 in this case t bi bi in b

correspondences between valued division algebras and graded division algebras abstract if d is a tame central division algebra over a henselian valued field f then the valuation on d yields an associated graded ring gd which is a graded division ring and is also central and graded simple over gf after proving some properties of graded central simple algebras over a graded field including a cohomological characterization of its graded brauer group it is proved that the map d gd g yields an index preserving isomorphism from the tame part of the brauer group of f to the graded brauer group of gf this isomorphism is shown to be functorial with respect to field extensions and corestrictions and using this it is shown that there is a correspondence between f subalgebras of d with center tame over f and graded gf subalgebras of gd

equivalent conditions for generalized contractions on ordered metric spaces abstract we establish a geometric lemma giving a list of equivalent conditions for some subsets of the plane as its application we get that various contractive conditions using the so called altering distance functions coincide with classical ones we consider several classes of mappings both on metric spaces and ordered metric spaces in particular we show that unexpectedly some very recent fixed point theorems for generalized contractions on ordered metric spaces obtained by harjani and sadarangani j harjani k sadarangani generalized contractions in partially ordered metric spaces and applications to ordinary differential equations nonlinear anal 72 2010 1188 1197 and amini harandi and emami a amini harandi h emami a fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations nonlinear anal 72 2010 2238 2242 do follow from an earlier result of o regan and petrusel d o regan and a petrusel fixed point theorems for generalized contractions in ordered metric spaces j math anal appl 341 2008 1241 1252

value functions and associated graded rings for semisimple algebras we introduce a type of value function y called a gauge on a finite dimensional semisimple algebra a over a field f with valuation v the filtration on a induced by y yields an associated graded ring gr y a which is a graded algebra over the graded field gr v f key requirements for y to be a gauge are that gr y a be graded semisimple and that dim gr v f gr y a dim f a it is shown that gauges behave well with respect to scalar extensions and tensor products when v is henselian and a is central simple over f it is shown that gr y a is simple and graded brauer equivalent to gr w d where d is the division algebra brauer equivalent to a and w is the valuation on d extending v on f the utility of having a good notion of value function for central simple algebras not just division algebras and with good functorial properties is demonstrated by giving new and greatly simplified proofs of some difficult earlier results on valued division algebras

discriminants of symplectic graded involutions on graded simple algebras abstract in this article we define and study the discriminant of symplectic graded involutions on non inertially split graded simple algebras with simple 0 component in particular we show that if f is a graded field of characteristic different from 2 d is a graded central division algebra over f with exp d 2 and ker d 4 see the preliminaries below a m n d and is a graded involution of symplectic type on a then there is only a finite number of values for the discriminants where describes all graded involutions of symplectic type on a see proposition 2 11 consequently for any graded central simple algebra c over f with c 0 simple non split exp c 2 ker c 4 and deg c ind c even we have 0 for any graded involutions of symplectic type and on c see corollary 2 12 we prove also that if e is a henselian valued field with residue characteristic different from 2 d is a central division algebra of exponent 2 over e with ker d 4 and b m n d with n even then for any symplectic involutions on b preserving a tame gauge defined on b we have 0 see corollary 3 5

extensions of valuation rings in central simple algebras certain subrings r of simple algebras q finite dimensional over their center k are studied these rings are called q valuation rings since they share many properties with commutative valuation rings let v be a valuation ring of k the center of q and let m be the set of q valuation rings r in q with r n k v then 1s 1 this extension theorem which does not hold if one considers only total valuation rings was proved by n i dubrovin here first a somewhat different proof of this result is given and then information about the set m is obtained theorem the elements in j are conjugate if v has finite rank theorem the elements in m are total valuation rings if m contains one total valuation ring in this case q is a division ring theorem i i 1 if 5 contains an invariant total valuation ring

state feedback performance recovery via an observer in h robust control abstract the paper is concerned with h robust suboptimal control problem for linear time invariant systems with structural uncertainties the main result shows that if there exists a state feedback gain matrix which ensures h robust suboptimally of the closed loop system then the robust performance achieved by the state feedback can be asymptotically recovered by a controller with high gain observer

robust h sub infinity control for linear systems with norm bounded time varying uncertainty robust h sub infinity control design for linear systems with uncertainty in both the state and input matrices is treated a state feedback control design which stabilizes the plant and guarantees an h sub infinity norm bound constraint on disturbance attenuation for all admissible uncertainties is presented the robust h sub infinity control problem is solved via the notion of quadratic stabilization with an h sub infinity norm bound necessary and sufficient conditions for quadratic stabilization with an h sub infinity norm bound are derived the results can be regarded as extensions of existing results on h sub infinity control and robust stabilization of uncertain linear systems

common fixed points for jℋ operators and occasionally weakly biased pairs under relaxed conditions abstract some common fixed point theorems due to 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 jungck and rhoades g jungck and b e rhoades fixed point theorems for occasionally weakly compatible mappings fixed point theory 7 2 2006 287 296 fixed point theory 9 2008 383 384 erratum and imdad and soliman m imdad a h soliman some common fixed point theorems for a pair of tangential mappings in symmetric spaces appl math lett 23 2010 351 355 are extended to two new classes of non commuting selfmaps which contain the occasionally weakly compatible and weakly biased selfmaps as proper subclasses some illustrative examples are also provided to highlight the realized improvements

springer s theorem for tame quadratic forms over henselian fields a quadratic form over a henselian valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tame extension the witt group of tame quadratic forms is shown to be canonically isomorphic to the witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation hence also isomorphic to a direct sum of copies of the witt group of the residue field indexed by the value group modulo 2

robust h control for linear time invariant systems with norm bounded uncertainty in the input matrix abstract this paper focuses on the problem of robust h control design for a class of linear time invariant systems with uncertainty in the state space model we consider uncertain systems with norm bounded parameter uncertainty in the input matrix the paper presents a state feedback control design which stabilizes the plant for all admissible uncertainties and also guarantees an h norm bound constraint on disturbance attenuation paralleling to the theory of robust control the robust h control problem is solved via the notion of quadratic stabilization with an h norm bound constraint necessary and sufficient conditions for quadratic stabilization with an h norm bound are derived it is shown that the solution to this problem involves solving a parameter dependent algebraic riccati equation the results can be regarded as extensions of existing results on robust stabilization of linear uncertain systems and h optimal control

asymptotic h sup infinity disturbance attenuation based on perfect observation a design method of controllers which ensure internal stability and attain asymptotically h sup infinity disturbance attenuation is presented the design procedure consists of two steps 1 to design an h sup infinity state feedback control via an algebraic riccati equation approach and 2 under a certain minimum phase condition to recover the achievable performance asymptotically by applying high gain observers it makes use of the perfect observation for the design of high gain observers it is shown that the asymptotic recovery can be attained by using reduced order observers provided there is no direct feedthrough of controls and disturbances in observations

remarks on contractive conditions of integral type abstract we show that most contractive conditions of integral type given recently by many authors coincide with classical ones this is done with the help of a geometric lemma on subsets of the quadrant 0 2 in particular we extend a recent result of suzuki who observed that an integral version of the banach contraction principle given by branciari could be derived from a classical fixed point theorem of meir and keeler nevertheless we also give a new contractive condition of integral type which is independent of classical ones since it does not force the nonexpansiveness of a mapping

common fixed point theorems of gregus type for weakly compatible mappings satisfying generalized contractive conditions abstract we prove a common fixed point theorem of gregus type for four mappings satisfying a generalized contractive condition in metric spaces using the concept of weak compatibility which generalizes theorems of i altun d turkoglu b e rhoades fixed points of weakly compatible mappings satisfying a general contractive condition of integral type fixed point theory appl 2007 2007 article id 17301 a djoudi l nisse gregus type fixed points for weakly compatible mappings bull belg math soc 10 2003 369 378 a djoudi a aliouche common fixed point theorems of gregus type for weakly compatible mappings satisfying contractive conditions of integral type j math anal appl 329 1 2007 31 45 p vijayaraju b e rhoades r mohanraj a fixed point theorem for a pair of maps satisfying a general contractive condition of integral type int j math math sci 15 2005 2359 2364 x zhang common fixed point theorems for some new generalized contractive type mappings j math anal appl 333 2 2007 780 786 we prove also a common fixed point theorem which generalizes theorem 3 5 of h k pathak m s khan t rakesh a common fixed point theorem and its application to nonlinear integral equations comput math appl 53 2007 961 971 and common fixed point theorems of gregus type using a strict generalized contractive condition a property e a and a common property e a