Boundary value problems for the Schrödinger equation involving the Henstock-Kurzweil integral

Salvador Sánchez-Perales; Francisco J. Mendoza-Torres

Displaying similar documents to “Boundary value problems for the Schrödinger equation involving the Henstock-Kurzweil integral”

Henstock-Kurzweil integral on $BV$ sets

Jan Malý, Washek Frank Pfeffer (2016)

Mathematica Bohemica

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The generalized Riemann integral of Pfeffer (1991) is defined on all bounded $BV$ subsets of $ℝ^{n}$ , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $σ$ -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $BV$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect...

Variational Henstock integrability of Banach space valued functions

Luisa Di Piazza, Valeria Marraffa, Kazimierz Musiał (2016)

Mathematica Bohemica

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We study the integrability of Banach space valued strongly measurable functions defined on $[0, 1]$ . In the case of functions $f$ given by $\sum_{n = 1}^{\infty} x_{n} χ_{E_{n}}$ , where $x_{n}$ are points of a Banach space and the sets $E_{n}$ are Lebesgue measurable and pairwise disjoint subsets of $[0, 1]$ , there are well known characterizations for Bochner and Pettis integrability of $f$ . The function $f$ is Bochner integrable if and only if the series $\sum_{n = 1}^{\infty} x_{n} | E_{n} |$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability...

Elementary construction of Hölder functions such that the Kurzweil-Stieltjes integral does not exist

Martin Rmoutil (2025)

Czechoslovak Mathematical Journal

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For any $α, β > 0$ with $α + β < 1$ we provide a simple construction of an $α$ -Hölde function $f : [0, 1] \to ℝ$ and a $β$ -Hölder function $g : [0, 1] \to ℝ$ such that the integral $\int_{0}^{1} f d g$ fails to exist even in the Kurzweil-Stieltjes sense.

$H^{p}$ spaces associated with Schrödinger operators with potentials from reverse Hölder classes

Jacek Dziubański, Jacek Zienkiewicz (2003)

Colloquium Mathematicae

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Let A = -Δ + V be a Schrödinger operator on $ℝ^{d}$ , d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of $H_{A}^{p}$ if the maximal function $s u p_{t > 0} | T_{t} f (x) |$ belongs to $L^{p} (ℝ^{d})$ , where ${T_{t}}_{t > 0}$ is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space $H_{A}^{p}$ admits a special atomic decomposition.

Compact operators and integral equations in the $ℋ𝒦$ space

Varayu Boonpogkrong (2022)

Czechoslovak Mathematical Journal

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The space $ℋ𝒦$ of Henstock-Kurzweil integrable functions on $[a, b]$ is the uncountable union of Fréchet spaces $ℋ𝒦 (X)$ . In this paper, on each Fréchet space $ℋ𝒦 (X)$ , an $F$ -norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $ℋ𝒦 (X)$ space. It is known that every control-convergent sequence in the $ℋ𝒦$ space always belongs to a $ℋ𝒦 (X)$ space for some $X$ . We illustrate how...

On coincidence of Pettis and McShane integrability

Marián J. Fabián (2015)

Czechoslovak Mathematical Journal

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R. Deville and J. Rodríguez proved that, for every Hilbert generated space $X$ , every Pettis integrable function $f : [0, 1] \to X$ is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space $X$ and a scalarly null (hence Pettis integrable) function from $[0, 1]$ into $X$ , which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from $[0, 1]$ (mostly) into $C (K)$ spaces. We focus in more detail on...

The topology of the space of $ℋ𝒦$ integrable functions in $ℝ^{n}$

Varayu Boonpogkrong (2025)

Czechoslovak Mathematical Journal

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It is known that there is no natural Banach norm on the space $ℋ𝒦$ of $n$ -dimensional Henstock-Kurzweil integrable functions on $[a, b]$ . We show that the $ℋ𝒦$ space is the uncountable union of Fréchet spaces $ℋ𝒦 (X)$ . On each $ℋ𝒦 (X)$ space, an $F$ -norm ${∥ \cdot ∥}^{X}$ is defined. A ${∥ \cdot ∥}^{X}$ -convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$ -norm is also defined for a ${∥ \cdot ∥}^{X}$ -continuous linear operator. Hence, many important results in functional analysis hold for the $ℋ𝒦 (X)$ space. It is well-known that every...

Unconditional uniqueness of higher order nonlinear Schrödinger equations

Friedrich Klaus, Peer Kunstmann, Nikolaos Pattakos (2021)

Czechoslovak Mathematical Journal

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We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data $u_{0} \in X$ , where $X \in {M_{2, q}^{s} (ℝ), H^{σ} (𝕋), H^{s_{1}} (ℝ) + H^{s_{2}} (𝕋)}$ and $q \in [1, 2]$ , $s \geq 0$ , or $σ \geq 0$ , or $s_{2} \geq s_{1} \geq 0$ . Moreover, if $M_{2, q}^{s} (ℝ) ↪ L^{3} (ℝ)$ , or if $σ \geq \frac{1}{6}$ , or if $s_{1} \geq \frac{1}{6}$ and $s_{2} > \frac{1}{2}$ we show that the Cauchy problem is unconditionally wellposed in $X$ . Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ...

Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

Rodolfo Collegari, Márcia Federson, Miguel Frasson (2018)

Czechoslovak Mathematical Journal

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We present a variation-of-constants formula for functional differential equations of the form $\dot{y} = ℒ (t) y_{t} + f (y_{t}, t), y_{t_{0}} = ϕ,$ where $ℒ$ is a bounded linear operator and $ϕ$ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application $t \mapsto f (y_{t}, t)$ is Kurzweil integrable with $t$ in an interval of $ℝ$ , for each regulated function $y$ . This means that $t \mapsto f (y_{t}, t)$ may admit not only many discontinuities, but it can also be highly oscillating and yet, we are...

On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals

Abraham Racca, Emmanuel Cabral (2016)

Mathematica Bohemica

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Equiintegrability in a compact interval $E$ may be defined as a uniform integrability property that involves both the integrand $f_{n}$ and the corresponding primitive $F_{n}$ . The pointwise convergence of the integrands $f_{n}$ to some $f$ and the equiintegrability of the functions $f_{n}$ together imply that $f$ is also integrable with primitive $F$ and that the primitives $F_{n}$ converge uniformly to $F$ . In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers...