Sense Of Jordan Research Articles

Some remarks on Vainikko integral operators in BV type spaces

In this paper we study Vainikko integral operators which are similar to so-called cordial integral operators and contain the classical Hardy operator, the Schur operator, and the Hilbert transform as special cases. For such operators we obtain norm estimates and equalities, mainly in BV type spaces in the sense of Jordan, Wiener, Riesz, and Waterman. Several examples are also discussed.

Types of Convergence Which Preserve Continuity

In the first part of this paper we investigate four types of convergence of sequences of functions in metric spaces which preserve continuity. Besides a consideration of locally, quasi and continuously uniform convergence, we introduce the notion of semi uniform convergence. We discuss how all types of convergence are related both to each other and to pointwise convergence, and illustrate their behavior by examples. Moreover, we show how some of the types of convergence can be used to characterize compactness of the domains the functions under consideration live in. In the second part we investigate sequences of composition operators in the space \(BV\) of functions of bounded variation in the sense of Jordan. We give criteria under which such sequences converge locally uniformly and semi uniformly and present a new and short proof for the fact that composition operators which map the space \(BV\) into itself are automatically continuous.

On the characterization of compactness in the space of functions of bounded variation in the sense of Jordan

The main goal of this paper is to prove a compactness criterion for subsets of the Banach space of functions of bounded variation in the sense of Jordan. In comparison to the compactness criterion contained in Dunford–Schwartz's monograph, the conditions which appear in our criterion seems to be easier to verify. Moreover, using this criterion we define a quasimeasure of noncompactness in that space. Some applications of quasimeasures of noncompactness to fixed point theorems are also presented.

Some Remarks on Multiplier Spaces I: Classical Spaces

The aim of this note is to characterize the multiplier class X/Y of functions g such that fg belongs to X whenever f belongs to Y for certain given classes X and Y of real valued functions on [0, 1]. This paper is the first of two connected parts and deals with classical spaces X and Y of continuous, bounded and Darboux functions, as well as functions of bounded variation in the sense of Jordan and functions which have a primitive. Moreover, we give a new and elementary proof for the fact that D/D contains only constant functions.

Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations

AbstractIn this paper we are going to apply the Henstock-Kurzweil integrals defined on an unbounded intervals to differential and integral equations defined on such intervals. To deal with linear differential equations we examine convolution involving functions integrable in Henstock-Kurzweil sense. In the case of nonlinear Hammerstein integral equation as well as Volterra integral equation we look for solutions in the space of functions of bounded variation in the sense of Jordan.

On the continuity of superposition operators in the space of functions of bounded variation

In the paper we present results on the continuity of nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan. It is shown that the continuity of an autonomous superposition operator is automatically guaranteed if the acting condition is met. We also give a simple proof of the fact that a nonautonomous superposition operator generated by a continuously differentiable function is uniformly continuous on bounded sets. Moreover, we present necessary and sufficient conditions for the continuity of a superposition operator (autonomous or nonautonomous) in a general setting. Thus, we give the answers to two basic open problems mentioned in the monograph (Appell et al. in Bounded variation and around, series in nonlinear analysis and application, De Gruyter, Berlin, 2014).

Solvability of Hammerstein Integral Equations with Applications to Boundary Value Problems

In this paper we present some new results regarding the solvability of nonlinear Hammerstein integral equations in a special cone of continuous functions. The proofs are based on a certain fixed point theorem of Leggett and Williams type. We give an application of the abstract result to prove the existence of nontrivial solutions of a periodic boundary value problem. We also investigate, via a version of Krasnosel'skiĭ's theorem for the sum of two operators, the solvability of perturbed Hammerstein integral equations in the space of continuous functions of bounded variation in the sense of Jordan. As an application of these results, we study the solvability of a boundary value problem subject to integral boundary conditions of Riemann–Stieltjes type. Some examples are presented in order to illustrate the obtained results.

Nonautonomous superposition operators in the spaces of functions of bounded variation

The main goal of this paper is to give an answer to the main problem of the theory of nonautonomous superposition operators acting in the space of functions of bounded variation in the sense of Jordan. Namely, we give necessary and sufficient conditions which guarantee that nonautonomous superposition operators map that space into itself and are locally bounded. Moreover, special attention is drawn to nonautonomous superposition operators generated by locally bounded mappings as well as to superposition operators generated by functions with separable variables.

On integral operators and nonlinear integral equations in the spaces of functions of bounded variation

In this paper we introduce new conditions on a kernel of a linear Fredholm integral operator which turn out to be sufficient and necessary for that operator to map the space of functions of bounded variation in the sense of Jordan into itself. Furthermore, we apply those conditions to deal with the problem of the existence of solutions to nonlinear Hammerstein equations in that space. To achieve our goals we will use a topological degree approach as well as a fixed point approach.

On continuity and compactness of some nonlinear operators in the spaces of functions of bounded variation

In this paper, we deal with one of the basic problems of the theory of autonomous superposition operators acting in the spaces of functions of bounded variation, namely the problem concerning their continuity. We basically consider autonomous superposition operators generated by analytic functions or functions of \(C^1\)-class. We also investigate the problem of compactness of some classical linear and nonlinear operators acting in the space of functions of bounded variation in the sense of Jordan. We apply our results to the examination of the existence and the topological properties of solutions to nonlinear equations in those spaces.

LOCAL BOUNDEDNESS OF NONAUTONOMOUS SUPERPOSITION OPERATORS IN

The main goal of this paper is to give the answer to one of the main problems of the theory of nonautonomous superposition operators acting in the space of functions of bounded variation in the sense of Jordan. Namely, we prove that if the superposition operator maps the space $BV[0,1]$ into itself, then it is automatically locally bounded, provided its generator is a locally bounded function.

Uniformly continuous composition operators in the space of bounded Φ-variation functions in the Schramm sense

We prove that any uniformly continuous Nemytskii composition operator in the space of functions of bounded generalized \(\Phi\)-variation in the Schramm sense is affine. A composition operator is locally defined. We show that every locally defined operator mapping the space of continuous functions of bounded (in the sense of Jordan) variation into the space of continous monotonic functions is constant.

On mappings of higher order and their applications to nonlinear equations

One of the main goals of this paper is to investigate mappings of higher order which possess ``good'' properties, in particular, when we treat them as perturbations of nonlinear differential as well as integral equations. We draw a particular attention to nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan or in the sense of Young. We provide sufficient conditions which guarantee that nonlinear Hammerstein operators are of higher order in such spaces. We also prove a few extensions of Lovelady's fixed point theorem in Archimedean as well as non-Archimedean setting. Finally, we apply our results to prove the existence and uniqueness results to some commonly known nonlinear equations with perturbations.

On the superposition operator in the space of functions of bounded variation, revisited

In this paper we provide sufficient conditions under which a nonautonomous superposition operator maps the space of functions of bounded variation in the sense of Jordan or Young into itself. We apply these results to examine the existence and uniqueness of solutions to Hammerstein and Volterra–Hammerstein integral equations in those spaces.

Modular metric spaces, II: Application to superposition operators

Applying the theory of modular metric spaces developed in the first part of this paper [V.V. Chistyakov (2009) [1]] we define a metric semigroup and an abstract convex cone of functions of finite generalized variation in the approach of Schramm [M. Schramm, Functions of Φ -bounded variation and Riemann–Stieltjes integration, Trans. Amer. Math. Soc. 287 (1) (1985) 49–63], which are significantly larger as compared to the spaces of bounded variation in the sense of Jordan, Wiener–Young and Waterman. We present a complete description of generators of Lipschitz continuous, bounded and some other classes of superposition Nemytskii operators mapping in these semigroups and cones, which extends recent results by Matkowski and Miś [J. Matkowski, J. Miś, On a characterization of Lipschitzian operators of substitution in the space BV 〈 a , b 〉 , Math. Nachr. 117 (1984) 155–159], Maligranda and Orlicz [L. Maligranda, W. Orlicz, On some properties of functions of generalized variation, Monatsh. Math 104 (1987) 53–65], Zawadzka [G. Zawadzka, On Lipschitzian operators of substitution in the space of set-valued functions of bounded variation, Rad. Mat. 6 (1990) 279–293] and Chistyakov [V.V. Chistyakov, Mappings of generalized variation and composition operators, J. Math. Sci. (New York) 110 (2) (2002) 2455–2466, V.V. Chistyakov, Lipschitzian Nemytskii operators in the cones of mappings of bounded Wiener φ -variation, Folia Math. 11 (1) (2004) 15–39].

On nonlinear integral equations and Λ-bounded variation

We discuss the existence or the existence and uniqueness of global and local Λ-bounded variation (ΛBV) solutions as well as continuous ΛBV-solutions of nonlinear Hammerstein and Volterra-Hammerstein integral equations formulated in terms of the Lebesgue integral. Since the space of functions of bounded variation in the sense of Jordan is a proper subspace of functions of Λ-bounded variation and for some class of functions φ, the space of functions of bounded φ-variation in the sense of Young is also a proper subspace of the space under consideration, our results extend known results in the literature.

Multi-Valued Mappings of Bounded Generalized Variation

We study the mappings taking real intervals into metric spaces and possessing a bounded generalized variation in the sense of Jordan--Riesz--Orlicz. We establish some embeddings of function spaces, the structure of the mappings, the jumps of the variation, and the Helly selection principle. We show that a compact-valued multi-valued mapping of bounded generalized variation with respect to the Hausdorff metric has a regular selection of bounded generalized variation. We prove the existence of selections preserving the properties of multi-valued mappings that are defined on the direct product of an interval and a topological space, have a bounded generalized variation in the first variable, and are upper semicontinuous in the second variable.

On the Hardy--Littlewood Theorem for Functions of Bounded Variation

We derive here an improved version of the well-known Hardy--Littlewood theorem for functions of bounded variation in the classical sense of Jordan. Results for the positive and negative variations are also discussed.

Neuropathy from hexane

Applying the theory of modular metric spaces developed in the first part of this paper [V.V. Chistyakov (2009) [1]] we define a metric semigroup and an abstract convex cone of functions of finite generalized variation in the approach of Schramm [M. Schramm, Functions of Φ-bounded variation and Riemann–Stieltjes integration, Trans. Amer. Math. Soc. 287 (1) (1985) 49–63], which are significantly larger as compared to the spaces of bounded variation in the sense of Jordan, Wiener–Young and Waterman. We present a complete description of generators of Lipschitz continuous, bounded and some other classes of superposition Nemytskii operators mapping in these semigroups and cones, which extends recent results by Matkowski and Miś [J. Matkowski, J. Miś, On a characterization of Lipschitzian operators of substitution in the space BV〈a,b〉, Math. Nachr. 117 (1984) 155–159], Maligranda and Orlicz [L. Maligranda, W. Orlicz, On some properties of functions of generalized variation, Monatsh. Math 104 (1987) 53–65], Zawadzka [G. Zawadzka, On Lipschitzian operators of substitution in the space of set-valued functions of bounded variation, Rad. Mat. 6 (1990) 279–293] and Chistyakov [V.V. Chistyakov, Mappings of generalized variation and composition operators, J. Math. Sci. (New York) 110 (2) (2002) 2455–2466, V.V. Chistyakov, Lipschitzian Nemytskii operators in the cones of mappings of bounded Wiener φ-variation, Folia Math. 11 (1) (2004) 15–39].

Concerning simple continuous curves

Various definitions of simple continuous arcs and closed curves have been given.t The definitions of arcs usually contain the requirement that the point-set in question should be bounded. In attempting to prove that every interval t of an open curve as defined in a recent papert is a simple continuous arc, while I found it easy to prove that t satisfies all the other requirements of Janiszewski's definition (modified as indicated below) it was only by a rather lengthy and complicated argument that I succeeded in proving that it satisfies the requirement of boundedness. In Lennes' definition the requirement of boundedness is superfluous.? However I found it difficult to prove that t satisfies a certain one of the other requirements of this definition, namely that the point-set in question should contain no proper connected subset that contains both A and B. In the present paper I will give a definition l of a simple continuous arc which stipulates neither that the set 111 should be bounded nor that it should contain no proper connected subset containing both A and B. I will show that, in a euclidean space of two dimensions, every point-set that satisfies this definition is an arc in the sense of Jordan. It is easy to prove? that every interval of an open curve satisfies this definition.