Positive solutions of a fourth-order differential equation with integral boundary conditions by Seshadev Padhi, John R. Graef

“…We study the existence of positive solutions to the fourth-order two-point boundary value problem \begin{cases} u^{\prime\prime\prime\prime}(t) + f(t,u(t))=0, & 0 < t < 1, u^{\prime}(0) = u^\prime(1) = u^{\prime\prime}(0) =0, & u(0) = \alpha[u], \end{cases} where $\alpha[u]=\int^1_0u(t){\rm d}A(t)$ is a Riemann-Stieltjes integral with $A \geq0$ being a nondecreasing function of bounded variation and $f \in\mathcal{C}([0,1] \times\mathbb{R}_+, \mathbb{R}_+)$. …”
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Riemann–Liouville Fractional Newton’s Type Inequalities for Differentiable Convex Functions by Thanin Sitthiwirattham, Kamsing Nonlaopon, Muhammad Aamir Ali, Hüseyin Budak

“…Moreover, we prove some inequalities of Riemann–Liouville fractional Newton’s type for functions of bounded variation. It is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. …”
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Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem by Jochen Merker, Aleš Matas

“…As functions k∈BV(Ω)of bounded variation can be discontinuous, existence of strong solutions u of Aku=fcannot be guaranteed and existence of minimizers cannot be obtained by standard methods. …”
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On the behavior of entropy solutions for a fractional p-Laplacian problem as t tends to infinity by Mohammed Abdellaoui

“…We prove an asymptotic behavior result of entropy solutions to fractional parabolic problems whose simplest model is (P) ( ut(t, x) + (−∆)s pu(x) = µ in Q := (0, T) × Ω, u(0, x) = u0(x) in Ω, u(t, x) = 0 on Σ := (0, T) × ∂Ω, where, Ω is a bounded domain of RN (N ≥ 2), T > 0, (−∆)s pu is the fractional p-Laplace operator (ps < N, 0 < s < 1), p > 2 − s N , µ ∈ M+(Q) is a nonnegative measure with bounded variation over Q and u0 ∈ L1 (Ω) is a nonnegative function. …”
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General signature kernels by Cass, T, Lyons, T, Xu, X

“…Suppose that γ and σ are two continuous bounded variation paths which take values in a finite-dimensional inner product space V. …”

RIGHT INVERSES OF LEVY PROCESSES: THE EXCURSION MEASURE IN THE GENERAL CASE by Savov, M, Winkel, M

“…This description unifies known formulas in the case of a positive Gaussian coefficient and in the bounded variation case. While these known formulas relate to excursions away from a point starting negative continuously, and excursions started by a jump, the present description is in terms of excursions away from the supremum continued up to a return time. …”

Analysis of Correlation Bounds for Uniformly Expanding Maps on [0, 1] by Mohamed Abdelkader

“…As a principal tool of these studies, we use a coupling method for analyzing the coupling time of observables with bounded variation.…”
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On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system by Yujun Cui, Jingxian Sun

“….$$ where $\alpha[x], \beta[x]$ are bounded linear functionals on $C[0,1]$ given by $$\alpha[x]=\int_0^1x(t)dA(t), beta[x]=\int_0^1x(t)dB(t)$$ with $A,B$ functions of bounded variation with positive measures.…”
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Weighted signature kernels by Cass, T, Lyons, T, Xu, X

“…Suppose that γ and σ are two continuous bounded variation paths which take values in a finite-dimensional inner product space V. …”

Inverting the signature of a path by Xu, W

“…Finally, we prove a uniform estimate for the signatures of paths of bounded variations, and obtain a reconstruction theorem via that uniform estimate. …”

Overview of One-Dimensional Continuous Functions with Fractional Integral and Applications in Reinforcement Learning by Wang Jun, Cao Lei, Wang Bin, Gong Hongtao, Tang Wei

“…Therefore, the existing conclusions will be systematically sorted out according to the bounded variation, unbounded variation and h<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>o</mi><mo>¨</mo></mover></semantics></math></inline-formula>lder continuity. …”
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On Stability Criteria Induced by the Resolvent Kernel for a Fractional Neutral Linear System with Distributed Delays by Ekaterina Madamlieva, Marian Milev, Tsvetana Stoyanova

“…In the case when the initial functions are with bounded variation, it is proven that this IP has a unique solution. …”
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The Riemann-Lebesgue Integral of Interval-Valued Multifunctions by Danilo Costarelli, Anca Croitoru, Alina Gavriluţ, Alina Iosif, Anna Rita Sambucini

“…Some classic properties of the <inline-formula><math display="inline"><semantics><mrow><mi>R</mi><mi>L</mi></mrow></semantics></math></inline-formula> integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the purpose of illustration; an example is given in case of fractal image coding for image compression, and for edge detection algorithm. …”
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Distributional, differential and integral problems: equivalence and existence results by Giselle Monteiro, Bianca Satco

“…We prove that, for regulated functions $g$, \eqref{1a} and \eqref{3} are equivalent if $f$ satisfies a bounded variation assumption. The relation between problems \eqref{2} and \eqref{3} is described for very general $f$, though, more restrictive assumptions over the function $g$ are required. …”
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Variable-Order Fractional Linear Systems with Distributed Delays—Existence, Uniqueness and Integral Representation of the Solutions by Hristo Kiskinov, Mariyan Milev, Milena Petkova, Andrey Zahariev

“…We prove that the initial problem for the variable-order fractional system with a discontinuous initial function possesses a unique continuous solution on the half-axis when the function describing the variable order of differentiation is locally bounded, Lebesgue integrable and has an appropriate decomposition similar to the Lebesgue decomposition of functions with bounded variation. The obtained results lead to the existence and uniqueness of a fundamental matrix for the studied variable-order fractional homogeneous system. …”
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Impulsive Control of Systems with Network Structure Describing Spread of Political Influence by M. V. Staritsyn, N. S. Maltugueva, N. I. Pogodaev, S.P. Sorokin

“…Next, we discuss an impulsive trajectory extension of the prototypic model in an appropriate coarse topology of the space of right continuous functions with bounded variation. For an equivalent classical problem (obtained by a discontinuous time reparameterization of the extended system) we present a detalization of the Maximum Principle. …”
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The Range of a Module Measure Defined on an Effect Algebra by Francisco Javier García-Pacheco

“…Among other results, we prove that if <i>L</i> is an sRDP <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-effect algebra with a natural basis and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>:</mo><mi>L</mi><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> is a countably additive measure, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> has bounded variation.…”
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The adaptive patched cubature filter and its implementation by Lee, W, Lyons, T

“…The algorithm can be performed by integrating along a number of carefully selected bounded variation paths. The iterated application of the KLV method has a tendency for the number of particles to increase. …”

Stability of conical shocks in the three-dimensional steady supersonic isothermal flows past lipschitz perturbed cones by Chen, G-QG, Kuang, J, Zhang, Y

“…We first formulate the shock stability problem as an initial boundary value problem with the leading conical shock-front as a free boundary, and then establish the existence and structural/asymptotic stability of global entropy solutions of bounded variation (BV) of the problem. To achieve this, we first develop a modified Glimm scheme to construct approximate solutions via self-similar solutions as building blocks in order to incorporate with the geometric source term. …”

Completion of the Kernel of the Differentiation Operator by Anatoly N. Morozov

“….\) Consequently, the space of the functions of the bounded variation \(H_1 ^ 1 \) is contained in each \( Y_p ^ 1 ,\) and the differentiation operator on \(H_1^1\) satisfies the relation \(\Lambda f = f^{\; '}.\)Also, we come to the conclusion that every function from the added part of the kernel can be logically named singular.…”
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