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UDC 515.12 We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones. Specially, we prove some relations between an open linear operator and its adjoint in uc-cones (locally convex cones which their convex quasi-uniform structures are generated by one element).

The theory of X-normed spaces over non-Archimedean valued fields with valuations of higher rank was introduced by H. Ochsenius and W. H. Schikhof in [9] and further developed in [10–12, 16, 17] and [13]. In order to obtain results like the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem, H. Ochsenius and W. H. Schikhof used 1st countability conditions in the value group of the based field. In this article the author develops a new tool to work with transfinite induction simplifying the techniques employed in X-normed spaces, thus accomplishing a Generalized Baire Category Theorem that allows the proof of an Open Mapping Theorem for X-normed spaces without restrictions on the value group of the based field. Additionally, some contributions to the theory of X-normed spaces are presented regarding quotient spaces.

In this manuscript, first we study the Hyers–Ulam stability of linear impulsive Volterra integral equations with the help of open mapping theorem approach. Then we give an existence and uniqueness theorem for the solutions of a class of nonlinear impulsive integral equations with a bounded variable delay. Moreover, with the help of integral inequality of Gronwall type, we investigate the Hyers–Ulam stability and Hyers–Ulam–Rassias stability of the same class of integral equations. We also present examples to support our main results.

In this paper, we address a fundamental question about the nature of bilinear operators first posed by Walter Rudin. While it has been known that there is not an open mapping theorem for bilinear operators in general, we show that bilinear operators enjoy an open mapping theorem, when the range has dimension three or less, and we address the simple, but crucial property that differentiates bilinear operators from their linear counterparts. We also give an example which shows that polynomials, in general do not enjoy an open mapping theorem, even when the range has dimension three.

In this paper we develop the theory of anti-compact sets we introduced earlier. We describe the class of Frechet spaces where anti-compact sets exist. They are exactly the spaces that have a countable set of continuous linear functionals. In such spaces we prove an analogue of the Hahn–Banach theorem on extension of a continuous linear functional from the original space to a space generated by some anti-compact set. We obtain an analogue of the Lyapunov theorem on convexity and compactness of the range of vector measures, which establishes convexity and a special kind of relative weak compactness of the range of an atomless vector measure with values in a Frechet space possessing an anti-compact set. Using this analogue of the Lyapunov theorem, we prove the solvability of an infinite-dimensional analogue of the problem of fair division of resources. We also obtain an analogue of the Lyapunov theorem for nonadditive analogues of measures that are vector quasi-measures valued in an infinite-dimensional Frechet space possessing an anti-compact set. In the class of Frechet spaces possessing an anti-compact set, we obtain analogues of the Krein–Milman theorem on extreme points for convex bounded sets that are not necessarily compact. A special place is occupied by analogues of the Krein–Milman theorem in terms of extreme sequences introduced in the paper (the so-called sequential analogues of the Krein–Milman theorem).

In 1955, M. A. Krasnosel’skii proved a fixed-point theorem for a single-valued map which is a completely continuous contraction (a hybrid theorem). Subsequently, his work was continued in various directions. In particular, it has stimulated the development of the theory of condensing maps (both single-valued and set-valued); the images of such maps are always compact. Various versions of hybrid theorems for set-valued maps with noncompact images have also been proved. The set-valued contraction in these versions was assumed to have closed images and the completely continuous perturbation, to be lower semicontinuous (in a certain sense). In this paper, a new hybrid fixed-point theorem is proved for any set-valued map which is the sum of a set-valued contraction and a compact set-valued map in the case where the compact set-valued perturbation is upper semicontinuous and pseudoacyclic. In conclusion, this hybrid theorem is used to study the solvability of operator inclusions for a new class of operators containing all surjective operators. The obtained result is applied to solve the solvability problem for a certain class of control systems determined by a singular differential equation with feedback.

We study the notion of a non-Archimedean fuzzy anti-2-normed space over a non-Archimedean field and prove that Mazur-Ulam theorem holds under some conditions in non-Archimedean fuzzy anti-2-normed spaces.

We provide a new and significantly shorter optimality proof of recent quantified Tauberian theorems, both in the setting of vector-valued functions and of $C_0$-semigroups, and in fact our results are also more general than those currently available in the literature. Our approach relies on a novel application of the open mapping theorem., Comment: To appear in the Israel Journal of Mathematics

This paper summarizes the research work on functional space theory, including the infinite matrix method, characterization of $({op})$ type space,invariance, nonlinear open mapping theorem, nonlinear closed graph theorem, demi-linear equicontinuity theorem, demi-linear duality theory, anddemi-linear generalized function theory.

We prove a Banach–Stone type theorem for surjective linear isometries of quaternion-valued continuous function spaces.

We generalize a recent global implicit function theorem from [8] to the case of a mapping acting between Banach spaces. Considerations related to duality mapping and to certain auxiliary functional are used in the proof together with the local implicit function theorem and mountain pass geometry. An application to integro-differential systems is given.

The abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

In this paper, we study Baire property of a family of spaces which contains properly the family of all topological spaces and generalize the existing results. Also, we study the images and inverse images of such spaces.

We prove a closed graph theorem for Baire locally convex spaces (for Baire linear topological spaces) in the domain and weakly C-Suslin locally convex spaces (respectively, for C-Suslin linear topological spaces) in the range which improves some classic closed graph theorems and other, more recent, related results.

We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.

Tchakaloff’s theorem gives a quadrature formula for polynomials of a given degree with respect to a compactly supported positive measure which is absolutely continuous with respect to Lebesgue measure. We study the validity of two possible analogues of Tchakaloff’s theorem in an infinite-dimensional Banach space E E : a weak form valid when E E has a Schauder basis, and a stronger form requiring conditions on the support of the measure as well as on the space E E .

This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem., Comment: 24 pages, published online in Proc. Roy. Soc. Edinburgh Sect. A

This paper generalizes the Aleksandrov problem, the Mazur-Ulam theorem and Benz theorem on n-normed spaces. It proves that a one-distance preserving mapping is an n- isometry if and only if it has the zero-distance preserving property, and two kinds of n-isometries on n-normed spaces are equivalent.

Under some additional assumptions on an unbounded sequence of operators and the geometry of the spaces, it is shown that, in the classical Banach-Steinhaus resonance theorem, the set of divergence contains an infinite-dimensional space, excluding zero.

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 ∗ .

This study introduces the Baire One-Stieltjes integral. The integral is shown to possess the desired elementary integral properties, including Cauchy Criterion and Henstock Lemma. It is known that every Riemann-Stieltjes integrable function is Baire One-Stieltjes integrable, while every Baire One-Stieltjes integrable function is Henstock-Stieltjes integrable. Between the Riemann-Stieltjes and the Baire One-Stieltjes, strict inclusion is exhibited in the space of regulated functions. We also formulate convergence theorems for this Stieltjes integral such as Uniform Convergence Theorem, Equi-integrability Convergence Theorem and Monotone Convergence Theorem. Also, some results leading to Riesz Representation Theorem are presented for this Stiletjes integral.

In this paper a very easy proof of the fundamental theorem of finite linear spaces is given.

The aim of this note is twofold. First, we prove an analogue of the well-known Robinson–Ursescu Theorem on the relative openness with a linear rate (restrictive metric regularity) of a multivalued mapping. Second, we prove a generalization of Graves Open Mapping Theorem for a class of mappings which can be approximated at a reference point by a bunch of linear mappings. The approximated non-linear mapping is restricted to a closed convex subset of a Banach space.

Nadler’s Fixed Point Theorem for Multi-valued functions (see [6]) was generalized by H. Aydi et al in their recent paper [1]. We use the approach of Gordji et. al. [3] to further generalize this result for a larger class of multifunctions (with a more general contraction condition) on complete partial metric spaces. The result yields the main theorem in [1] as a special case. Mathematics Subject Classification: 54H25

In this paper, the concept of I-Baire spaces is introduced, and characterizations and properties of these spaces are given. It is shown that (X; ) is Baire if and only if (X;; I) is I-Baire for any ideal I on X.

We consider the possibility of generalizing the averaging theorem from the case of sets from n-dimensional Euclidean space to the case of sets from Banach spaces. The result is a cornerstone for constructing the theory of the Riemann integral for non-convex-valued multivalued mappings and for proving the convexity of this multivalued integral. We obtain a generalization of the averaging theorem to the case of sets from uniformly smooth Banach spaces as well as some corollaries.

For X a compact Hausdorff topological space and F a strictly convex and reflexive Banach space we characterize the surjective isometries of certain subspaces of C(X, F). It follows from this characterization that surjective isometries on spaces of vector valued Lipschitz functions equipped with a p-norm are also weighted composition operators.

We study some classes of summing operators between spaces of integrable functions with respect to a vector measure in order to prove a factorization theorem for 1-summing operators between Banach spaces.

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.

Some sufficient conditions are provided that guarantee that the difference of a compact mapping and a proper mapping defined between any two Banach spaces over \( \mathbb{K} \) has at least one zero. When conditions are strengthened, this difference has at most a finite number of zeros throughout the entire space. The proof of the result is constructive and is based upon a continuation method.

The notion of quasicontinuity was perhaps the first time used by R. Baire in [2]. Let X, Y be topological spaces and Q(X,Y) be the space of quasicontinuous mappings from X to Y. If X is a Baire space and Y is metrizable, in Q(X,Y) we can approach each (x, y) in the graph Grf of f along some trajectory of the form {(xk, fnk (xk )): k??} if and only if we can approach most points along a vertical trajectory. This result generalizes Theorem 5 from [3]. Moreover in the class of topological spaces with the property QP we give a characterization of Baire spaces by the above mentioned fact. We also study topological spaces with the property QP.

In this paper we obtain a general fixed point theorem for an affine mapping in Banach space. As an application of this theorem we study existence of periodic solutions to the equations of the linear elasticity theory.

We establish that if X and Y are metric compacta and f: X → Y is a continuous surjective mapping, then the openness of the mapping OH(f): OH(X) → OH(Y) of spaces of weakly additive homogeneous functionals is equivalent to the openness of f.

A two-function minimax theorem is proved. In this result, the concavity-convexity conditions of both functions involve monotone transforms and mixing of functional values, and the “wupwardness/w-downwardness” conditions; both spaces are required to be compact topological spaces but without linear structure. By this result, an open question proposed by Forgo and Joo in 1998 is answered.

The conformal invariance of Brownian motion is used to give a short proof of the Open Mapping theorem for analytic functions.

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

The main results of the paper include (a) a theorem containing estimates for the surjection modulus of a “partial composition” of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutin’s theorem about additive perturbation of a mapping into a Banach space by a Lipschitz mapping; (b) a “double fixed point” theorem for a couple of mappings, one from X into Y and another from Y to X which implies a fairly general version of the set-valued contraction mapping principle and also a certain (different) version of the first theorem.

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.

We prove a complex analog of Kurzweil’s theorem on the approximation of continuous functions on separable Banach spaces admitting a separating polynomial and obtain a complex analog of new results due to Boiso and Hajek.