Absolutely continuous variational measures of Mawhin's type

Abstract In this paper we study absolutely continuous and σ-finite variational measures corresponding to Mawhin, F- and BV -integrals. We obtain characterization of these σ-finite variational measures similar to those obtained in the case of standard variational measures. We also give a new proof of the Radon-Nikodým theorem for these measures.

The Lr-Variational Integral

AbstractWe define the $$L^r$$ L r -variational integral and we prove that it is equivalent to the $$HK_r$$ H K r -integral defined in 2004 by P. Musial and Y. Sagher in the Studia Mathematica paper The$$L^{r}$$ L r -Henstock–Kurzweil integral. We prove also the continuity of $$L^r$$ L r -variation function.

Henstock type integral in compact zero-dimensional metric space and quasi-measures representations

Properties of a Henstock type integral defined on a compact zero-dimensional metric space are studied. Theorems on integral representation of so-called quasi-measures, i.e., linear functionals on the space of “polynomials” defined on the space of the above mentioned type, are obtained.

P-adic Henstock integral in the problem of representation of functions by multiplicative transforms

We introduce a path integral of Henstock type and use it to obtain inversion formulas for multiplicative integral transformations. The problem considered is a generalization of the problem of reconstruction of coefficients of a convergent orthogonal series from its sum.

Multidimensional P-adic Integrals in some Problems of Harmonic Analysis

The paper is a survey of results related to the problem of recovering the coefficients of some classical orthogonal series from their sums by generalized Fourier formulas. The method is based on reducing the coefficient problem to the one of recovering a function from its derivative with respect to an appropriate derivation basis. In the case of the multiple Vilenkin system the problem is solved by using a multidimensional P-adic integral.

Comparison of the P-integral with Burkill's integrals and some applications to trigonometric series

It is proved that the $P_r$-integral [9] which recovers a function from its derivative defined in the space $L^r$, 1 ≤r<∞, is properly included in Burkill’s trigonometric CP-and SCP-integrals. As an application to harmonic analysis, a de La Vallée-Poussin-type theorem for the $P_r$-integral is obtained: convergence nearly everywhere of a trigonometric series to a $P_r$-integrable function f implies that this series is the Pr-Fourier series of f.

MR2849946 Subramanian, N.; Krishnamoorthy, S.; Balasubramanian, S. A new double $\chi$ sequence space defined by a modulus function. Selçuk J. Appl. Math. 12 (2011), no. 1, 109--121.

P-adic Henstock integral in the problem of representation of functions by multiplicative transform

Henstock type integral in harmonic analysis on zero-dimensional groups

AbstractA Henstock type integral is defined on compact subsets of a locally compact zero-dimensional abelian group. This integral is applied to obtain an inversion formula for the multiplicative integral transform.

MR3085505 Reviewed Boonpogkrong, Varayu Stokes' theorem on manifolds: a Kurzweil-Henstock approach. Taiwanese J. Math. 17 (2013), no. 4, 1183–1196

On the Coefficients of Multiple Series with Respect to Vilenkin System

Abstract We give a sufficient condition for coefficients of double series Σ Σ n,m an,m χ n,m with respect to Vilenkin system to be convergent to zero when n + m → ∞. This result can be applied to the problem of recovering coefficients of a Vilenkin series from its sum.

Generality of Henstock-Kurzweil type integral on a compact zero-dimensional metric space

ABSTRACT A Henstock-Kurzweil type integral on a compact zero-dimensional metric space is investigated. It is compared with two Perron type integrals. It is also proved that it covers the Lebesgue integral.

Henstock-Kurzweil type integrals on zero-dimensional groups and its application in harmonic analysis

On the Almost Everywhere Convergence of Multiple Fourier-Haar Series

The paper deals with the question of convergence of multiple Fourier-Haar series with partial sums taken over homothetic copies of a given convex bounded set $$W\subset\mathbb{R}_+^n$$ containing the intersection of some neighborhood of the origin with $$\mathbb{R}_+^n$$ . It is proved that for this type sets W with symmetric structure it is guaranteed almost everywhere convergence of Fourier-Haar series of any function from the class L(ln+L)n−1.

Analogue of Dini-Riemann theorem for non-absolutely convergent integrals

An analogue of classical Dini-Riemann theorem related to non-absolutely convergent series of real number is proved for the Lebesgue improper integral.

Regularity of some method of summation for double sequences

Some generalization of Toeplitz method of summation is introduced for double sequences and condition of regularity of it is obtained.

MR2896126 Selmanogullari, T.; Savas, E.; Rhoades, B. E. On $q$-Hausdorff matrices. Taiwanese J. Math. 15 (2011), no. 6, 2429--2437

Least energy solutions to the Dirichlet problem for the equation −D(u) = f (x, u)

Let be a bounded smooth domain in RN. We prove a general existence result of least energy solutions and least energy nodal ones for the problem −u = f(x, u) in u = 0 on ∂ (P) where f is a Carathéodory function. Our result includes some previous results related to special cases of f . Finally, we propose some open questions concerning the global minima of the restriction on the Nehari manifold of the energy functional associated with (P) when the nonlinearity is of the type f(x, u) = λ|u| s−2u − μ|u| r−2u, with s, r ∈ (1, 2) and λ,μ > 0.

Generalized Hake property for integrals of Henstock type

An integral of Henstock-Kurzweil type is considered relative to an abstract differential basis in a topological space. It is shown that under certain conditions posed onto the basis this integral satisfies the generalized Hake property.

Multidimensional dyadic Kurzweil–Henstock- and Perron-type integrals in the theory of Haar and Walsh series

Abstract The problem of recovering the coefficients of rectangular convergent multiple Haar and Walsh series from their sums, by generalized Fourier formulas, is reduced to the one of recovering a function (the primitive) from its derivative with respect to the appropriate derivation basis. Multidimensional dyadic Kurzweil–Henstock- and Perron-type integrals are compared and it is shown that a Perron-type integral, defined by major and minor functions having a special continuity property, solves the coefficients problem for series which are convergent everywhere outside some uniqueness sets.

Kurzweil-Henstock type integral on zero-dimensional group and some of its application

A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case.

Representation of quasi-measure by a Henstock-Kurzweil type integral on a compact zero-dimensional metric space

A derivation basis is introduced in a compact zero-dimensional metric space X. A Henstock-Kurzweil type integral with respect to this basis is defined and used to represent the so-called quasi-measure on X.

Existence of viscosity solutions to two-phase problems for fully nonlinear equations with distributed sources

In this paper we construct a viscosity solution of a two-phase free boundary problem for a class of fully nonlinear equation with distributed sources, via an adaptation of the Perron method. Our results extend those in [Caffarelli, 1988], [Wang, 2003] for the homogeneous case, and of [De Silva, Ferrari, Salsa, 2015] for divergence form operators with right hand side.

MR3106093 Reviewed Łochowski, Rafał M. On a generalisation of the Hahn-Jordan decomposition for real càdlàg functions. Colloq. Math. 132 (2013), no. 1, 121–138

HENSTOCK INTEGRAL AND DINI-RIEMANN THEOREM

In [5] an analogue of the classical Dini-Riemann theorem related to non-absolutely convergent series of real number is obtained for the Lebesgue improper integral. Here we are extending it to the case of the Henstock integral.

MR2805472 Tsagareishvili, V. Fourier-Haar coefficients of continuous functions. 132 (2011), no. 1-2, 1--14.

MR2824899 Kayaduman, Kuddusi; Çakan, Celal The Cesáro core of double sequences. Abstr. Appl. Anal. 2011, Art. ID 950364, 9 pp

A version of Hake’s theorem for Kurzweil–Henstock integral in terms of variational measure

Abstract We introduce the notion of variational measure with respect to a derivation basis in a topological measure space and consider a Kurzweil–Henstock-type integral related to this basis. We prove a version of Hake’s theorem in terms of a variational measure.

A Hake-Type Theorem for Integrals with Respect to Abstract Derivation Bases in the Riesz Space Setting

Abstract A Kurzweil-Henstock type integral with respect to an abstract derivation basis in a topological measure space, for Riesz space-valued functions, is studied. A Hake-type theorem is proved for this integral, by using technical properties of Riesz spaces.

Generalized Henstock integrals in the theory of series in multiplicative systems

Properties of a Henstock type integral defined by means of a differential basis generated by P-adic paths ae studied. It is proved that this integral solves the problem of coefficients reconstruction by using generalized Fourier formulas for a series over multiplivative systems.

HENSTOCK- AND PERRON-TYPE INTEGRAL ON A COMPACT ZERO-DIMENSIONAL METRIC SPACE

P-adic Henstock integral in the theory of series over systems of characters of zero-dimensional groups.

We introduce a path integral of Henstock type and use it to obtain inversion formulas for multiplicative integral transformations. The problem considered is a generalization of the problem of reconstruction of coefficients of a convergent orthogonal series from its sum.

On the Ward theorem for P-adic-path bases associated with a bounded sequence

In this paper we prove that each differentiation basis associated with a $\mathcal P$-adic path system defined by a bounded sequence satisfies the Ward Theorem.

MR2865796 Riecan, Beloslav; Tkacik, Štefan A note on the Kluvánek integral. Tatra Mt. Math. Publ. 49 (2011), 59--65

The arithmetic decomposition of central Cantor sets

Abstract Every central Cantor set of positive Lebesgue measure is the arithmetic sum of two central Cantor sets of Lebesgue measure zero. Under some mild condition this result can be strengthened by stating that the summands can be chosen to be C s regular if the initial set is of this class.

An Inversion Formula for the Multiplicative Integral Transform

Integration of functions ranging in complex Riesz space and some applications in harmonic analysis

The theory of Henstock—Kurzweil integral is generalized to the case of functions ranging in complex Riesz space R and defined on any zero-dimensional compact Abelian group. The constructed integral is used to solve the problem of recovering the R-valued coefficients of series in systems of characters of these groups by using generalized Fourier formulas.

Existence and multiplicity of solutions for Dirichlet problems involving nonlinearities with arbitrary growth.

In this article we study the existence and multiplicity of solutions for the Dirichlet problem $$\displaylines{ -\Delta_p u=\lambda f(x,u)+ \mu g(x,u)\quad\hbox{in }\Omega,\cr u=0\quad\hbox{on } \partial \Omega }$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $f,g:\Omega \times \mathbb{R}\to \mathbb{R}$ are Caratheodory functions, and $\lambda,\mu$ are nonnegative parameters. We impose no growth condition at $\infty$ on the nonlinearities f,g. A corollary to our main result improves an existence result recently obtained by Bonanno via a critical point theorem for $C^1$ functionals which do not satisfy the usual sequential weak lower semicontinuity property.

Representation of Quasi-Measure by Henstock–Kurzweil Type Integral on a Compact-Zero Dimensional Metric Space

Abstract A derivation basis is introduced in a compact zero-dimensional metric space 𝑋. A Henstock–Kurzweil type integral with respect to this basis is defined and used to represent the so-called quasi-measure on 𝑋.

Kurzweil-Henstock type integral in fourier analysis on compact zero-dimensional group

Abstract A Kurzweil-Henstock type integral defined on a zero-dimensional compact abelian group is studied and used to obtain a generalization of some results related to the problem of recovering, by generalized Fourier formulae, the coefficients of convergent series with respect to the characters of such a group.

Integration of both the derivatives with respect to P-paths and approximative derivatives

In the present paper, in terms of generalized absolute continuity, we present a descriptive characteristic of the primitive with respect to a system of P-paths and study the relationship between the Denjoy-Khinchin integral and the Henstock H P-integral. © 2009 Pleiades Publishing, Ltd.

MR2876776 Orhan, C.; Tas, E.; Yurdakadim, T. The Buck-Pollard property for $p$-Cesàro matrices. Numer. Funct. Anal. Optim. 33 (2012), no. 2, 190--196.

Dual of the Class of HKr Integrable Functions

We define for 1 <= r < infinity a norm for the class of functions which are Henstock-Kurzweil integrable in the L-r sense. We then establish that the dual in this norm is isometrically isomorphic to L-r' and is therefore a Banach space, and in the case r = 2, a Hilbert space. Finally, we give results pertaining to convergence and weak convergence in this space.

MR2968982 Boonpogkrong, Varayu; Chew, Tuan Seng; Lee, Peng Yee On the divergence theorem on manifolds. J. Math. Anal. Appl. 397 (2013), no. 1, 182–190.

An Existence Result for Fractional Kirchhoff-Type Equations

The aim of this paper is to study a class of nonlocal fractional Laplacian equations of Kirchhoff-type. More precisely, by using an appropriate analytical context on fractional Sobolev spaces, we establish the existence of one non-trivial weak solution for nonlocal fractional problems exploiting suitable variational methods.

Generalized Henstock integrals in the theory of series with respect to multiplicative system

Integration by parts for the Lr Henstock-Kurzweil integral

Musial and Sagher [4] described a Henstock-Kurzweil type integral that integrates Lr-derivatives. In this article, we develop a product rule for the Lr-derivative and then an integration by parts formula.

On the problem of recovering the coefficients of series with respect to characters of zero-dimensional groups

It is proved that if a series with respect to the characters of abelian compact zero-dimensional group is convergent everywhere except, possibly, a countable number of points, then the coefficients of this series can be recovered from its sum by generalized Fourier formulas in which a Henstock-Kurzweil type integral is used.

Inversion formulae for the integral transform on a locally compact zero-dimensional group

Abstract Generalized inversion formulae for multiplicative integral transform with a kernel defined by characters of a locally compact zero-dimensional abelian group are obtained using a Kurzweil-Henstock type integral.

Perron type integral on compact zero-dimensional Abelian groups

Perron and Henstock type integrals defined directly on a compact zero-dimensional Abelian group are studied. It is proved that the considered Perron type integral defined by continuous majorants and minorants is equivalent to the integral defined in the same way, but without assumption on continuity of majorants and minorants.

On the possible values of upper and lower derivatives with respect to differentiation bases of product structure.

A solution of the Guzmán's problem on possible values of upper and lower derivatives is given for the class of translation invariant and product type differentiation bases formed by ndimensional intervals. Namely, the bases from the mentioned class are characterized, for which integral means of a summable function can boundedly diverge only on a set of zero measure

Denjoy and P-path integrals on compact groups in an inversion formula for multiplicative transforms

Abstract Denjoy and P-path Kurzweil-Henstock type integrals are defined on compact subsets of some locally compact zero-dimensional abelian groups. Those integrals are applied to obtain an inversion formula for the multiplicative integral transform.