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Approximating the fractional order differentiation and integration operators is a common approach in implementation of fractional order dynamics. This paper aims to investigate how the procedure of approximating the fractional order operators influences on the realizability of a fractional order impedance function by passive networks. To this aim, conditions for the possibility of passive realization of the approximations of the fractional order impedance functions by using RLC components are obtained. More precisely, considering two general forms for the filters approximating the fractional order operators, the open mapping theorem in complex analysis is applied to obtain the realizability conditions on the polar plots of the approximating filters. It is found that the approximated impedance function may be realizable by a passive RLC network, whereas the original fractional order impedance function cannot be realized by passive networks composed of resistors and fractional inductors and capacitors. Furthermore, for a class of impedance functions, the realizability condition is simplified as a condition on the phase of the filter approximating the fractional order differentiation operator. Some examples are presented to verify the usefulness of the obtained conditions.

A version of the classical Klee-And\^o Theorem states the following: For every Banach space $X$, ordered by a closed generating cone $C\subseteq X$, there exists some $\alpha>0$ so that, for every $x\in X$, there exist $x^{\pm}\in C$ so that $x=x^{+}-x^{-}$ and $\|x^{+}\|+\|x^{-}\|\leq\alpha\|x\|$. The conclusion of the Klee-And\^o Theorem is what is known as a conormality property. We prove stronger and somewhat more general versions of the Klee-And\^o Theorem for both conormality and coadditivity (a property that is intimately related to conormality). A corollary to our result shows that the functions $x\mapsto x^{\pm}$, as above, may be chosen to be bounded, continuous, and positively homogeneous, with a similar conclusion yielded for coadditivity. Furthermore, we show that the Klee-And\^o Theorem generalizes beyond ordered Banach spaces to Banach spaces endowed with arbitrary collections of cones. Proofs of our Klee-And\^o Theorems are achieved through an Open Mapping Theorem for cone-valued multi-functions/correspondences. We very briefly discuss a potential further strengthening of The Klee-And\^o Theorem beyond what is proven in this paper, and motivate a conjecture that there exists a Banach space $X$, ordered by a closed generating cone $C\subseteq X$, for which there exist no Lipschitz functions $(\cdot)^{\pm}:X\to C$ satisfying $x=x^{+}-x^{-}$ for all $x\in X$., Comment: Major rewrite. Large parts were removed which a referee pointed out can be proven through much easier methods

We investigate the validity of the Mean Ergodic Theorem in symmetric Banach function spaces E . The assertion of that theorem always holds when E is separable, whereas the situation is more delicate when E is non-separable. To describe positive results in the latter setting, we use the connections with the theory of singular traces.

We prove that if K and S are locally compact Hausdorff spaces and there exists a bijective coarse ( M , L ) -quasi-isometry T between the Banach spaces of real continuous functions C 0 ( K ) and C 0 ( S ) with M 2 , then K and S are homeomorphic. This nonlinear extension of Banach–Stone theorem (1933/1937) is in some sense optimal and improves some results of Amir (1965), Cambern (1967), Jarosz (1989), Dutrieux and Kalton (2005) and Gorak (2011). In the Lipschitz case, that is when L = 0 , we also improve the estimations of the distance of the map T from the isometries between the spaces C 0 ( K ) and C 0 ( S ) obtained by Gorak when K and S are compact spaces or not. As a consequence, we get a linear sharp refinement of the Amir–Cambern theorem.

We introduce the notions of extended topological vector spaces and extended seminormed spaces, following the main ideas of extended normed spaces, which were introduced by G. Beer and J. Vanderwerff. We provide a topological study of such structures, giving a unifying theory with main applications in the study of spaces of continuous functions. We also generalize classical results of functional analysis, as open mapping theorem and closed graph theorem.

This work presents an Ovsyannikov type theorem for an autonomous abstract Cauchy problem in a scale of decreasing Banach spaces, which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution. Then, using this theorem it studies the Cauchy problem for Camassa–Holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line, which is the main goal of this paper. Finally, it studies the continuity of the data-to-solution map in spaces of analytic functions.

We study the completeness properties of several different group topologies for the additive group of real numbers, and we also compute the corresponding dual groups. We first present two metrizable connected group topologies on R with topologically isomorphic dual groups, one of which is noncomplete and arcwise connected and the other one is compact (therefore complete), but not arcwise connected. Using a theorem about T -sequences and adapting a result about weakened analytic groups, we then describe a method for obtaining Hausdorff group topologies R that are strictly weaker than the usual topology and are complete. They are not Baire, and consequently not metrizable.

Classical Marchaud's theorem (1927) asserts that if f is a bounded function on [ a , b ] , k ∈ N , and the ( k + 1 ) th modulus of smoothness ω k + 1 ( f ; t ) is so small that η ( t ) = ∫ 0 t ω k + 1 ( f ; s ) s k + 1 d s + ∞ for t > 0 , then f ∈ C k ( ( a , b ) ) and f ( k ) is uniformly continuous with modulus cη for some c > 0 (i.e. in our terminology f is C k , c η -smooth). Using a known version of the converse of Taylor theorem we easily deduce Marchaud's theorem for functions on certain open connected subsets of Banach spaces from the classical one-dimensional version. In the case of a bounded subset of R n our result is more general than that of H. Johnen and K. Scherer (1973), which was proved by quite a different method. We also prove that if a locally bounded mapping between Banach spaces is C k , ω -smooth on every line, then it is C k , c ω -smooth for some c > 0 .

Let E be a Banach lattice and F a Banach space. A bounded linear operator T : E → F is an isomorphism on the positive cone of E if and only if T ∗ is almost surjective. A dual version of this theorem holds also. A bounded linear operator T : F → E is almost surjective if and only if T ∗ is an isomorphism on the positive cone of F ∗ .

Huang (1987) [8] proved the Erdős–Ko–Rado theorem for attenuated spaces. In this paper, we generalize Huangʼs result for singular linear spaces.

To extend the analysis of continuous-time general-equilibrium macro models we study 2 parameter variants Lp,q of the Lebesgue spaces, thus gaining separate control on the asymptotic behaviour (p) and the local behaviour (q): they behave w.r.t. p like the spaces lp and w.r.t. q like the spaces Lq on a probability space. Such spaces might naturally contain equilibrium variables (paths) as well as time-dependent policies of a macro model. Convolution behaves very well on those spaces, which can be used as a basis for the classical “comparative statics” (see e.g. Mertens and Rubinchik (2011)). Finally, we generalise the classical implicit function theorem (ift) for a family of Banach spaces, with the resulting implicit function having derivatives that are locally Lipschitz to very strong operator norms.

We generalize from the real line to normed groups (i.e., the invariantly metrizable, right topological groups) a result on infinite combinatorics, valid for 'large' subsets in both the category and measure senses, which implies both Steinhaus's Theorem and many of its descendants. We deduce the inherent measure-category duality in this setting directly from properties of analytic sets. We apply the result to extend the Pettis Theorem and the Continuous Homomorphism Theorem to normed groups, i.e., beyond metrizable topological groups, and in a companion paper deduce automatic joint-continuity results for group operations.

Fortʼs theorem states that if F : X → 2 Y is an upper (lower) semicontinuous set-valued mapping from a Baire space ( X , τ ) into the nonempty compact subsets of a metric space ( Y , d ) then F is both upper and lower semicontinuous at the points of a dense G δ subset of X. In this paper we show that a variant of Fortʼs theorem holds, without the assumption of the compactness of the images, provided we restrict the domain space of the mapping to a large class of “nice” Baire spaces.

In this paper, we extend the concepts of linearizing cone, regularity assumption and Lagrange multiplier rule due to Maurer and Zowe [H. Maurer, J. Zowe, First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Math. Program. 16 (1979) 98-110] to an optimization problem under inclusion constraints. By virtue of the Robinson-Ursescu open mapping theorem, we obtain a Kuhn-Tucker necessary optimality condition. Moreover, we propose a Lagrangian by using the support function for set-valued maps, and establish some second-order sufficient and necessary optimality conditions for a strict local minimizer of order 2 based on the second-order derivative of the Lagrangian. As applications, we also investigate some second-order optimality conditions for the strict local minimizer of order 2 of a smooth scalar optimization problem with equality and inequality constraints.

Let X and Y be Banach spaces and let α be a tensor norm. The principal result is the following theorem. If either X ⁎ ⁎ ⁎ or Y has the approximation property, then each α-nuclear operator T : X ⁎ → Y such that T ⁎ ( Y ⁎ ) ⊂ X can be approximated in the α-nuclear norm by finite-rank operators of type X ⊗ Y . In the special case of (Grothendieck) nuclear operators, the theorem provides a strengthening for the classical theorem on the nuclearity of operators with a nuclear adjoint. The hypotheses about the approximation property are essential. The main application yields an affirmative answer to [C. Pineiro, J.M. Delgado, p-Convergent sequences and Banach spaces in which p-compact sets are q-compact, Proc. Amer. Math. Soc. 139 (2011) 957–967]: for p ⩾ 1 , a sequence ( x n ) ⊂ X is p-null if and only if lim x n = 0 and ( x n ) is relatively p-compact in X.

A theorem proved by Fort in 1951 states that an upper semi-continuous mapping from a Baire space X into non-empty compact subsets of a metric space is both lower and upper semi-continuous at the points of a dense residual set of X . In this paper, we present several results concerning the generic continuity of set-valued mappings. First we demonstrate that generic continuity, in the sense of Baire category, cannot be extended to almost everywhere continuity (with respect to some measure). Then we show that Baireness of the domain space is a necessary condition for Fort’s theorem to hold. Finally, we show that Fort’s theorem holds without any compactness assumption on the images of the set-valued mapping.

We study isometries between normed spaces over a non-Archimedean valued field K . We show the failure of a Mazur–Ulam theorem in the framework of non-Archimedean spaces. Considering Aleksandrov problem, we prove that a surjective Lipschitz map E → E with the strong distance one preserving property, where E is a finite-dimensional normed space, is an isometry if and only if K is locally compact. We prove also that every isometry E → E for finite-dimensional E is surjective if and only if K is spherically complete and card ( k ) is finite.

We introduce the class of α -nonexpansive mappings in Banach spaces. This class contains the class of nonexpansive mappings and is related to the class of firmly nonexpansive mappings in Banach spaces. In addition, we obtain a fixed point theorem for α -nonexpansive mappings in uniformly convex Banach spaces.

We prove a generalization of Gravesʼ Open Mapping Theorem for a class of mappings which can be approximated at a reference point by a set-valued one having particular properties. The nonlinear mapping is restricted to a closed convex subset of a Banach space. We apply the results to derive necessary and sufficient conditions ensuring the existence of a differentiable selection for the inverse mapping. A slight generalization of a sufficient condition by J. Klamka on so-called constrained exact local controllability of nonlinear and semi-linear dynamic systems is also proved.

In this article, we give the sufficient and necessary conditions for which several important mapping spaces are weak* dense in the corresponding completely bound mapping spaces.

Motivated by recent work, we establish the Baire Theorem in the broad context afforded by weak forms of completeness implied by analyticity and,kappa-analyticity, thereby adding to the 'Baire space recognition literature' (cf. Aarts and Lutzer (1974) [1], Haworth and McCoy (1977) [43]). We extend a metric result of van Mill, obtaining a generalization of Oxtoby's weak alpha-favourability conditions (and therefrom variants of the Baire Theorem), in a form in which the principal role is played by kappa-analytic (in particular analytic) sets that are 'heavy' (everywhere large in the sense of some sigma-ideal). From this perspective fine-topology versions are derived, allowing a unified view of the Baire Theorem which embraces classical as well as generalized Gandy-Harrington topologies (including the Ellentuck topology), and also various separation theorems. A multiple-target form of the Choquet Banach-Mazur game is a primary tool, the key to which is a restatement of the Cantor Theorem, again in kappa-analytic form.

Let X and Y be Banach spaces and T : Y → X be a bounded operator. In this note, we show first some operator versions of the dual relation between q-convexity and p-smoothness of Banach spaces case. Making use of them, we prove then the main result of this note that the two notions of uniform q-convexity and uniform p-smoothness of an operator T introduced by J. Wenzel are actually equivalent to that the corresponding T-modulus δ T of convexity and the T-modulus ρ T of smoothness introduced by G. Pisier are of power type q and of power type p, respectively. This is also an operator version of a combination of a Hoffman's theorem and a Figiel–Pisier's theorem. As their application, we show finally that a recent theorem of J. Borwein, A.J. Guirao, P. Hajek and J. Vanderwerff about q-convexity of Banach spaces is again valid for q-convexity of operators.

Our object is to present an independent proof of the extension of V.A. Markov's theorem to Gâteaux derivatives of arbitrary order for continuous polynomials on any real normed linear space. The statement of this theorem differs little from the classical case for the real line except that absolute values are replaced by norms. Our proof depends only on elementary computations and explicit formulas and gives a new proof of the classical theorem as a special case. Our approach makes no use of the classical polynomial inequalities usually associated with Markov's theorem. Instead, the essential ingredients are a Lagrange interpolation formula for the Chebyshev nodes and a Christoffel–Darboux identity for the corresponding bivariate Lagrange polynomials. We use these tools to extend a single variable inequality of Rogosinski to the case of two real variables. The general Markov theorem is an easy consequence of this.

We prove a general theorem on transitive topological group actions that give rise to almost Cech-complete homogeneous spaces. This theorem implies the known open mapping theorems whose proof is a Baire category argument. As an application, we prove the uniform generalized Schonflies theorem generalizing Wright (1969) [19, Theorem 3] .

In the present work we prove a coincidence point theorem in Menger spaces with a t-norm T which satisfies the condition sup{T(t,t):t

Recently, Mursaleen, Lohani and Mohiuddine [Chaos Solitons & Fractals (2009), accepted] have introduced the notions of intuitionistic fuzzy 2-metric space. In this paper, we study various topological properties and prove Baire’s Theorem and Cantor’s Intersection Theorem in this new setup.

Article history: Received 24 July 2008 Available online 24 December 2008 Submitted by J.D.M. Wright A (not necessarily linear) mapping Φ from a Banach space X to a Banach space Y is said to be a 2-local isometry if for any pair x, y of elements of X ,t here is as urjective linear isometry T : X → Y such that Tx = Φx and Ty = Φ y. We show that under certain conditions on locally compact Hausdorff spaces Q , K and a Banach space E ,e very 2-local isometry on C0(Q , E) to C0(K , E) is linear and surjective. We also show that every 2-local isometry on � p is linear and surjective for 1 p < ∞, p �= 2, but this fails for the Hilbert space � 2 .

We consider a strong lattice property for a Banach function space B on a compact Hausdorff space, which gives a general Stone–Weierstrass theorem for B. We also study the relation of this theorem and its proof to a certain decomposition of an associated compactification, and to another lattice-like property.

We analyze a nonlinear equation in Banach spaces, with the nonlinearity composed of multiple terms of different degrees. We prove a theorem regarding the existence of solutions for such equations. Moreover, we show how this result may be applied to obtain the well-posedness of various parabolic initial value problems.

The aim of this article is to generalize the Mazur–Ulam theorem to the case of linear n -normed spaces.

Given a program of a linear bounded and bijective operator T , does there exist a program for the inverse operator T − 1 ? And if this is the case, does there exist a general algorithm to transfer a program of T into a program of T − 1 ? This is the inversion problem for computable linear operators on Banach spaces in its non-uniform and uniform formulation, respectively. We study this problem from the point of view of computable analysis which is the Turing machine based theory of computability on Euclidean space and other topological spaces. Using a computable version of Banach’s Inverse Mapping Theorem we can answer the first question positively. Hence, the non-uniform version of the inversion problem is solvable, while a topological argument shows that the uniform version is not. Thus, we are in the striking situation that any computable linear operator has a computable inverse while there exists no general algorithmic procedure to transfer a program of the operator into a program of its inverse. As a consequence, the computable version of Banach’s Inverse Mapping Theorem is a powerful tool which can be used to produce highly non-constructive existence proofs of algorithms. We apply this method to prove that a certain initial value problem admits a computable solution. As a preparation of Banach’s Inverse Mapping Theorem we also study the Open Mapping Theorem and we show that the uniform versions of both theorems are limit computable, which means that they are effectively Σ 2 0 -measurable with respect to the effective Borel hierarchy.

In the paper finite-dimensional dynamical control systems described by second order semilinear stationary ordinary differential state equations with delay in control are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local controllability in a given time interval are formulated and proved. These conditions require verification of constrained global controllability of the associated linear first-order dynamical control system. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero. Moreover, several remarks and comments on the existing results for controllability of semilinear dynamical control systems are also presented. Finally, simple numerical example which illustrates theoretical considerations is also given. It should be pointed out, that the results given in the paper extend for the case of semilinear second-order dynamical systems constrained controllability conditions, which were previously known only for linear second-order systems.

The Tietze-Urysohn Theorem states that every continuous real-valued function defined on a closed subspace of a normal space can be extended to a continuous function on the whole space. We prove an effective version of this theorem in the Type Two Model of Effectivity (TTE). Moreover, we introduce for qcb-spaces a slightly weaker notion of normality than the classical one and show that this property suffices to establish an Extension Theorem for continuous functions defined on functionally closed subspaces. Qcb-spaces are known to form an important subcategory of the category Top of topological spaces. QCB is cartesian closed in contrast to Top.

In this paper, a new KKM theorem is established in L-convex spaces. As applications, a Ky Fan matching theorem for compactly open covers, a Fan–Browder coincidence theorem, a Fan–Browder fixed point theorem and a maximal element theorem are obtained in L-convex spaces. These theorems unify, improve and generalize some recent results in the references therein. Finally, the equilibrium existence theorems for abstract economies and qualitative games in L-convex spaces are yielded.

In this paper, among other results we investigate open mapping, closed graph and uniform boundedness theorems in the IF-normed spaces.

In this paper, using a fixed point theorem which was recently obtained by the authors for firmly nonexpansive type mappings, we prove a strong convergence theorem of Browder’s type in Banach spaces. Using our result, we study the approximation of zero points of monotone operators in Banach spaces.

In this article, the generalized Roper—Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in Cn are extended to Hilbert spaces or Banach spaces.

We present a Stone duality for bitopological spaces in analogy to the duality between topological spaces and frames, and discuss the resulting notions of sobriety and spatiality. Under the additional assumption of regularity, we prove a characterisation theorem for subsets of a bisober space that are compact in one and closed in the other topology. This is in analogy to the celebrated Hofmann–Mislove theorem for sober spaces. We link the characterisation to Taylor’s and Escardo’s reading of the Hofmann–Mislove theorem as continuous quantification over a subspace.

This paper details an existence and uniqueness theorem for solving an operator equation of the form F ( x ) = 0 , where F is a Gateaux differentiable operator defined on an open convex subset of a Banach space proved. From the main theorem, an earlier theorem of Argyros follows as a consequence. Other corollaries constitute the semilocal versions of the theorems due to Ozban and Weerakoon and Fernando in a general Banach space. Our main theorem leads to the existence of solutions for a class of nonlinear Urysohn-type integral equations in the n-dimensional Euclidean space.

The maximum number of edges spanned by a subset of given diameter in a Hamming space with alphabet size at least three is determined. The binary case was solved earlier by Ahlswede and Khachatrian [A diametric theorem for edges, J. Combin. Theory Ser. A 92(1) (2000) 1–16].

In this paper, a new maximal element theorem is proved in H -space and, as applications, the Browder–Hartman–Stampacchia variational inequalities and the complementarity problem are discussed in Banach spaces and locally convex space, respectively.

In this paper, by using a KKM type theorem in FC-space, a new Himmelberg type fixed point theorem for compact and closed set-valued mappings and an existence theorem for equilibrium points for a generalized game in locally FC-uniform spaces without any convexity structure are proved for the first time. By applying the equilibrium existence theorem, some new existence results for solutions for several classes of systems of generalized vector quasi-equilibrium problems are obtained in locally FC-uniform spaces. These results generalize some known results from the literature.

In the paper finite-dimensional dynamical control systems described by second order semilinear stationary ordinary differential state equations are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local controllability in a given time interval are formulated and proved. These conditions require verification of constrained global controllability of the associated linear first-order dynamical control system. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero. Moreover, several remarks and comments on the existing results for controllability of semilinear dynamical control systems are also presented. Finally, simple numerical example, which illustrates theoretical considerations is also given. It should be pointed out, that the results given in the paper extend for the case of semilinear second-order dynamical systems constrained controllability conditions, which were previously known only for linear second-order systems.

The paper establishes a sufficient condition for the controllability of the first-order impulsive functional differential systems with infinite delay in Banach spaces. We use Schauder’s fixed point theorem combined with a strongly continuous operator semigroup. An example is given to illustrate our results.

We prove a category version of the Poincare recurrence theorem for a non-invertible map of a Baire space which is continuous, nearly feebly open and has no nonempty open wandering set.

We present a Stone duality for bitopological spaces in analogy to the duality between topological spaces and frames, and discuss the resulting notions of sobriety and spatiality. Under the additional assumption of regularity, we prove a characterisation theorem for subsets of a bisober space that are compact in one and closed in the other topology. This is in analogy to the celebrated Hofmann-Mislove theorem for sober spaces. We link the characterisation to Taylor's and Escardó's reading of the Hofmann-Mislove theorem as continuous quantification over a subspace. As an application, we define locally compact d-frames and show that these are always spatial.

In the present paper finite-dimensional dynamical control systems described by semilinear ordinary differential state equations with multiple point delays in control are considered. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero. Using generalized open mapping theorem, sufficient conditions for constrained local relative controllability near the origin are formulated and proved. Roughly speaking, it will be proved that under suitable assumptions constrained global relative controllability of a linear associated approximated dynamical system implies constrained local relative controllability near the origin of the original semilinear dynamical system. This is a generalization to constrained controllability case some previous results concerning controllability of linear systems with multiple point delays in the control and with unconstrained controls. Moreover, necessary and sufficient conditions for constrained global relative controllability of an associated linear dynamical system with multiple point delays in control are discussed. Simple numerical example which illustrates theoretical considerations is also given. Finally, some remarks and comments on the existing results for controllability of nonlinear dynamical systems are also presented.

In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X=In+A∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0,∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A.

This paper is concerned with the controllability of the first-order impulsive functional differential systems in a Banach space. Sufficient conditions for controllability are obtained by using the Schaefer fixed-point theorem. An example is provided to illustrate the theory.

In this article the concept of local conjugation of a C1 mapping between two Banach manifolds is introduced. Then a rank theorem for nonlinear semi-Fredholm operators between two Banach manifolds and a finite rank theorem are established in global analysis.