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In this paper, we study the existence of limits at infinity along almost every infinite curve for the upper and lower approximate limits of bounded variation functions on complete unbounded metric measure spaces. We prove that if the measure is doubling and supports a 1-Poincaré inequality, then for every bounded variation function f and for 1-a.e. infinite curve γ , for both the upper approximate limit f ∨ and the lower approximate limit f ∧ we have that lim t → + ∞ f ∨ (γ (t)) and lim t → + ∞ f ∧ (γ (t)) exist and are equal to the same finite value. We give examples showing that the conditions of the doubling property of the measure and a 1-Poincaré inequality are needed for the existence of limits. Furthermore, we establish a characterization for strictly positive 1-modulus of the family of all infinite curves in terms of bounded variation functions. These generalize results for Sobolev functions given in Koskela and Nguyen (J Funct Anal 285(11):110154, 2023). [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear abstract Cauchy problem with an m-accretive operator. If the initial value and the right-hand side lie in some interpolation sets with parameter α , then we obtain the rate of convergence O (| π | α / 2) for solutions of the corresponding implicit Euler schemes and for the Euler solution as the limit, we obtain Hölder continuity of exponent α . [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of the α -fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal functions. Further, we estimate the perturbation error between the given continuous function and its α -fractal function. Additionally, we define a new graph of a set-valued function different from the standard graph introduced in the literature and establish some bounds on the fractal dimension of the newly defined graph of some special classes of set-valued functions. Also, we explain the need to define this new graph with examples. In the sequel, we prove that this new graph of an α -fractal function is an attractor of an iterated function system. [ABSTRACT FROM AUTHOR]

We introduce discontinuous solutions to nonlinear impulsive control systems with state time delays in the dynamics and derive necessary optimality conditions in the form of a maximum principle for associated optimal control problems. In the case without delays, if the measure control is scalar valued, the corresponding discontinuous state trajectory, understood as a limit of classical state trajectories for absolutely continuous controls approximating the measure, is unique. For vector-valued measure controls, however, the limiting trajectory is not unique and a full description of the control must include additional "attached" controls affecting instantaneous state evolution at a discontinuity. For impulsive control systems with time delays we reveal a new phenomenon, namely, that the limiting state trajectory resulting from different approximations of a given measure control needs not to be unique, even in the scalar case. Correspondingly, our framework allows for additional attached controls, even though the measure control is scalar valued. [ABSTRACT FROM AUTHOR]

We prove BV continuity of a class of discrete maximal operators associated to a general function Φ , which cover the classical one-dimensional discrete uncentered Hardy–Littlewood maximal operator. The main result not only improves and generalizes some known ones, but also answers a question originally posed by Liu and Wu in 2019. [ABSTRACT FROM AUTHOR]

We study the problem of whether the centered Hardy–Littlewood maximal function of a singular function is absolutely continuous. For a parameter d ∈ (0 , 1) and a closed set E ⊂ [ 0 , 1 ] , let μ be a d-Ahlfors regular measure associated with E. We prove that for the cumulative distribution function f (x) = μ ([ 0 , x ]) its maximal function Mf is absolutely continuous. We then adapt our method to the multiparameter case and show that the same is true in the positive cone defined by these functions, i.e., for functions of the form f (x) = ∑ i = 1 n μ i ([ 0 , x ]) where { μ i } i = 1 n is any collection of d i -Ahlfors regular measures, d i ∈ (0 , 1) , associated with closed sets E i ⊂ [ 0 , 1 ] . This provides the first improvement of regularity for the classical centered maximal operator, and can be seen as a partial analogue of the result of Aldaz and Pérez Lázaro about the uncentered maximal operator. [ABSTRACT FROM AUTHOR]

In this paper we introduce a class of one dimensional non-centered minimal operator m ~ Φ associated to a function Φ , which covers the usual minimal operator. We establish the boundedness of m ~ Φ : BV (R) → BV (R) under a more restrictive condition on Φ . Here BV (R) is the set of functions of bounded variation defined on R . In the discrete setting, we prove the discrete minimal operator is bounded and continuous from BV (Z) to itself under a more restrictive condition on Φ . [ABSTRACT FROM AUTHOR]

We consider a control problem to maximize a profit from harvesting in agriculture or aquaculture, where the population is governed by size-structured population models with spatial diffusion. We show the existence of an optimal control of harvesting rate which is a measure with respect to size expressed by the distributional partial derivative of a function of bounded variation. [ABSTRACT FROM AUTHOR]

This paper intends to study the analytical properties of the Riemann–Liouville fractional integral and fractal dimensions of its graph on ℝ n . We show that the Riemann–Liouville fractional integral preserves some analytical properties such as boundedness, continuity and bounded variation in the Arzelá sense. We also deduce the upper bound of the box dimension and the Hausdorff dimension of the graph of the Riemann–Liouville fractional integral of Hölder continuous functions. Furthermore, we prove that the box dimension and the Hausdorff dimension of the graph of the Riemann–Liouville fractional integral of a function, which is continuous and of bounded variation in Arzelá sense, are n. [ABSTRACT FROM AUTHOR]

In the multidimensional Euclidean space, except the classical real Hardy space, there are numerous product ones. We associate with each of them a class of functions related to the known variations and new ones. Such a characterization is fulfilled by means of the integrability of the Fourier transform [ABSTRACT FROM AUTHOR]

In this paper, we have studied the pointwise convergence of the double Fourier-Legendre series of functions of the generalized bounded variation. In particular, we also have the convergence of double Fourier-Legendre series of functions of (p,q) -bounded variation. [ABSTRACT FROM AUTHOR]

This paper deals with the research of solutions of bounded variation (BV) to evolution inclusion coupled with a time and state dependent maximal monotone operator. Different problems are studied: existence of solutions, unicity of the solution, existence of periodic and bounded variation right continuous (BVRC) solutions. Second-order evolution inclusions and fractional (Caputo and Riemann–Liouville) differential inclusions are also considered. A result of the Skorohod problem driven by a time- and space-dependent operator under rough signal and a Volterra integral perturbation in the BRC setting is given. The paper finishes with some results for fractional differential inclusions under rough signals and Young integrals. Many of the given results are novel. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, the variation of functions has been defined, whose domain is a convex and compact set in the plane. Furthermore, in addition to presenting properties that satisfy this variation, the vector space formed by functions with finite variation is studied, demonstrating that it is a Banach space and its elements can be expressed as the difference of non-decreasing functions.

In this work, we prove the existence of a nontrivial solution to a quasilinear elliptic problem defined on the whole Euclidean space R N , N ≥ 2 , and involving a weighted 1-Laplacian operator. The nonlinear term has a singular behavior at the origin. This solution is obtained through an approximation technique, which consists in considering the problem with the 1-Laplacian operator as a limit of a family of problems with the p-Laplacian operators when p → 1 +. For that aim, a new version of Anzellotti’s L ∞ - divergence−measure pairing theory is established and new arguments are used. [ABSTRACT FROM AUTHOR]

We generalize the classical calculus rules satisfied by functions of bounded variation to the framework of RCD spaces. In the infinite dimensional setting, we are able to define an analogue of the distributional differential and, on finite dimensional spaces, we prove fine properties and suitable calculus rules, such as the Vol’pert chain rule for vector valued functions. [ABSTRACT FROM AUTHOR]

In this paper, the variation of functions has been defined, whose domain is a convex and compact set in the plane. Furthermore, in addition to presenting properties that satisfy this variation, the vector space formed by functions with finite variation is studied, demonstrating that it is a Banach space and its elements can be expressed as the difference of non-decreasing functions. [ABSTRACT FROM AUTHOR]

Convergence of classical Fourier series (trigonometric, Haar, Walsh, systems) of differentiable functions are trivial problems and they are well known. But general Fourier series, as it is known, even for the function does not converge. In such a case, if we want differentiable functions with respect to the general orthonormal system (ONS) to have convergent Fourier series, we must find the special conditions on the functions of system . This problem is studied in the present paper. It is established that the resulting conditions are best possible. Subsystems of general orthonormal systems are considered. [ABSTRACT FROM AUTHOR]

We introduce a notion of non-local perimeter which is defined through an arbitrary positive Borel measure on R d which integrates the function 1 ∧ | x | . Such definition of non-local perimeter encompasses a wide range of perimeters which have been already studied in the literature, including fractional perimeters and anisotropic fractional perimeters. The main part of the article is devoted to the study of the asymptotic behaviour of non-local perimeters. As direct applications we recover well-known convergence results for fractional perimeters and anisotropic fractional perimeters. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we construct the Chlodowsky-type Szász operators defined via Boas–Buck-type polynomials. We prove some approximation properties and obtain the rate of the convergence for these operators. We also study the Voronovskaya-type theorem and weighted approximation.

This paper is an introduction to a sequence of nonlinear Baskakov-Durrmeyer operators $(NBD_{n})$ of the form \[ (NBD_{n})(f;x) =\int_{0}^\infty K_{n}(x,t,f(t))\,dt \] with $x\in [0,\infty)$ and $n\in\mathbb{N}$. While $K_{n}(x,t,u)$ provide convenient assumptions, these operators work on bounded functions, which are defined on all finite subintervals of $[0,\infty)$. This paper comprise some pointwise convergence results for these operators in certain functional spaces. As well as this study can be seen as a continuation of studies about nonlinear operators, it is the first study on nonlinear Baskakov-Durrmeyer or modified Baskakov operators, while there were more papers on linear part of the operators.

In this article, we defined a Nörlund type matrix 𝐴 = (𝑎𝑛𝑘) by 𝑎𝑛𝑘 = { 1 , 𝑘 = 𝑛 = 0 1 2 , 𝑛 − 1 ≤ 𝑘 ≤ 𝑛 0 , 𝑜𝑡ℎ𝑒𝑟𝑣𝑖𝑠𝑒 . Then we showed that the Nörlund type matrix 𝐴 = (𝑎𝑛𝑘) is a linear and bounded operator on the sequence spaces ℓ1 and 𝑏𝑣. Finally, we calculated the fine spectrum and its subdivisions of the operator 𝐴 = (𝑎𝑛𝑘) on the sequence spaces ℓ1 and 𝑏𝑣.

In this paper, we construct the Bézier variant of the operators constructed by Nasiruzzaman et al. (Iran J Sci Technol Trans A Sci 46(5):1495–1503, 2022). We use the notion of wavelets to construct Bézier type Kantorovich q-Baskakov wavelet operators. We calculate the moments and central moments and prove some approximation results for our new operators. [ABSTRACT FROM AUTHOR]

We prove that a rational trigonometric series with its coefficients c ⁢ (n) = o ⁢ (1) satisfying the condition ∑ n ∈ ℤ | Δ m ⁢ c ⁢ (n) | < ∞ , m ∈ ℕ , converges pointwise to some f ⁢ (x) for every x ∈ (0 , 2 ⁢ π) and also converges in L p ⁢ [ 0 , 2 ⁢ π) -metric to f for 0 < p < 1 m . This result is further extended to a double rational trigonometric series. [ABSTRACT FROM AUTHOR]

We prove an approximation result for a class of functionals G(u) = ∫Ω φ(x, Du) defined on BV (Ω) where φ(·, Du) ∈L¹ (Ω), Ω ⊂ RN bounded, φ(x, p) convex, radially symmetric and of the form φ(x, p) = { g(x, p) if |p| ≤ β ψ(x)|p| + k(x) if |p| > β. We show for each u ∈ BV (Ω) ∩ L p (Ω), 1 ≤ p < ∞, there exist uk ∈ W1,1 (Ω) ∩ C∞ (Ω) ∩ L p (Ω) so that G(uk) → G(u). Approximation theorems in BV are used to prove existence results for the strong solution to the time flow ut = div (∇pφ(x, Du)) inL¹ ((0, ∞); BV (Ω) ∩ L p (Ω)), typically with additional boundary condition or penalty term in u to ensure uniqueness. The functions in this work are not covered by previous approximation theorems since for fixed p we have φ(x, p) ∈L¹ (Ω) which do not in general hold for assumptions on φ in earlier work. [ABSTRACT FROM AUTHOR]

In this paper, our object of investigation is the endpoint regularity of the following general maximal operator, M ~ Φ f (x) = sup r , s ≥ 0 r + s > 0 Φ (r + s) ∫ x - r x + s | f (y) | d y , and minimal operator, m ~ Φ f (x) = inf r , s ≥ 0 r + s > 0 Φ (r + s) ∫ x - r x + s | f (y) | d y , where Φ (t) : (0 , ∞) → (0 , ∞) is a non-increasing continuous function and satisfies B q : = sup t > 0 t Φ (t) q < ∞ for some q ≥ 1 . We prove that if f : R → R is a function of bounded variation, then max { Var q (M ~ Φ f) , Var q (m ~ Φ f) } ≤ (8 B q) 1 / q Var (f). Here, Var q (f) denotes the q-variation of f and Var q (f) = Var (f) when q = 1 . Similar results are proved for the discrete versions of the above operators. [ABSTRACT FROM AUTHOR]

This work is concerned with operators of the type A = -c A acting in domains fi := fi0 x (0,H) C Rd x Rc The diffusion coefficient c >0 depends on one coordinate y e (0, H) and is bounded but may be discontinuous. This corresponds to the physical model of "layered media," appearing in acoustics, elasticity, optical fibers. Dirichlet boundary conditions are assumed. In general, for each s > 0, the set of eigenfunctions is divided into a disjoint union of three subsets: Fng (non-guided), Fg (guided) and Fres (residual). The residual set shrinks as s ! 0: The customary physical terminology of guided/non-guided is often replaced in the mathematical literature by concentrating/non-concentrating solutions, respectively. For guided waves, the assumption of "layered media" enables us to obtain rigorous estimates of their exponential decay away from concentration zones. The case of non-guided waves has attracted less attention in the literature. While it is not so closely connected to physical models, it leads to some very interesting questions concerning oscillatory solutions and their asymptotic properties. Classical asymptotic methods are available for c (y) e C2 but a lesser degree of regularity excludes such methods. The associated eigenfunctions (in Fng) are oscillatory. However, this fact by itself does not exclude the possibility of "flattening out" of the solution between two consecutive zeros, leading to concentration in the complementary segment. Here we show it cannot happen when c(y) is of bounded variation, by proving a "minimal amplitude hypothesis." However the validity of such results when c(y) is not of bounded variation (even if it is continuous) remains an open problem. [ABSTRACT FROM AUTHOR]

We introduce two discrete multilinear maximal and minimal operators associated to the function Φ , which cover the classical discrete multilinear maximal and minimal operators and their fractional variants. Under a more restrictive condition on Φ , we establish some new variation boundedness and continuity of the above operators, which is new and interesting even in the linear case. It should be pointed out that the main results we obtain not only extend and improve some known ones in the maximal setting, but also supplement some new results in the minimal setting. [ABSTRACT FROM AUTHOR]

This paper deals with the research of solutions of bounded variation (BV) to evolution inclusion coupled with a time and state dependent maximal monotone operator. Different problems are studied: existence of solutions, unicity of the solution, existence of periodic and bounded variation right continuous (BVRC) solutions. Second-order evolution inclusions and fractional (Caputo and Riemann–Liouville) differential inclusions are also considered. A result of the Skorohod problem driven by a time- and space-dependent operator under rough signal and a Volterra integral perturbation in the BRC setting is given. The paper finishes with some results for fractional differential inclusions under rough signals and Young integrals. Many of the given results are novel.

This paper deals with construction and studying wavelet type generalized Bézier operators by using the compactly supported Daubechies wavelets of the given function $ f $. The basis used in this construction are the wavelet expansion of the function $ f $ instead of its rational sampling values $ f\left(\frac{k}{n}\right) $. By using the Chanturiya modulus of variation we estimate the rate of pointwise convergence of $ \left(WB_{n, \alpha }f\right) (x)\ $at those $ x>0 $ at which the one-sided limits $ f(x+) $, $ f(x-) $ exist.Clearly our wavelet type operators contain at least classical version, Durrmeyer and the Kantorovich form of the generalized Bézier operators and hence our results extend some of the previous results on generalized B ézier operators, such as [1], [11] and [12]. [ABSTRACT FROM AUTHOR]

We propose certain Durrmeyer-type operators for Apostol-Genocchi polynomials in this research. We explore these operators' approximation attributes and measure the rate of convergence. In addition, we present a direct approximation theorem based on first and second-order modulus of continuity, local approximation findings for Lipschitz class functions and a direct theorem based on the typical modulus of continuity. Finally, we showed a graph illustrating the convergence of the suggested operators and an error table. [ABSTRACT FROM AUTHOR]

In this paper, we construct the Chlodowsky-type Szász operators defined via Boas–Buck-type polynomials. We prove some approximation properties and obtain the rate of the convergence for these operators. We also study the Voronovskaya-type theorem and weighted approximation. [ABSTRACT FROM AUTHOR]

Cartoon-texture decomposition of images is a hot research topic in image processing, its main goal is to decompose a given image into cartoon and texture parts measured by different norms. In the existing partial decomposition model, the H−1 functional is often used to measure the oscillation function, which is obtained by the Hodge decomposition. Because a divergence-free vector function is neglected in the calculation of the Hodge decomposition, some directional information will be lost in the texture part, which will affect the decomposition results. In this paper, a new cartoon-texture decomposition model with divergence-free vector field constraint is established by analyzing the theory of the vector field. Mathematically, the new model makes the texture more accurate and complete. In order to solve this model, we transform the constrained optimization problem into an unconstrained problem by using the augmented Lagrange method, and then solve it with the alternating direction method. The proposed cartoon-texture decomposition model contains the divergence-free vector field information, so it improves the decomposition effect obviously, and the experimental results confirm the validity of the proposed model. [ABSTRACT FROM AUTHOR]

Recently, the characterization of the compactness in the space BV([0, 1]) of functions of bounded Jordan variation was given. Here, certain generalizations of this result are given for the spaces of functions of bounded Waterman Μ-variation, Young Φ-variation and integral variation. It appears that, on the compact sets, the norm is uniformly approximated by certain seminorms induced by the selection of finitely many intervals in [0, 1]. [ABSTRACT FROM AUTHOR]

This paper intends to estimate the box dimension of the Weyl–Marchaud fractional derivative (Weyl–M derivative) for various choices of continuous functions on a compact subset of ℝ. We show that the Weyl–M derivative of order γ of a continuous function satisfying Hölder condition of order μ also satisfies Hölder condition of order μ − γ and the upper box dimension of the Weyl–M derivative increases at most linearly with the order γ. Moreover, the upper box dimension of the Weyl–M derivative of a continuous function satisfying the Lipschitz condition is not more than the sum of the box dimension of the function itself and order γ. Furthermore, we prove that the box dimension of the Weyl–M derivative of a certain continuous function which is of bounded variation is one. [ABSTRACT FROM AUTHOR]

We study the Dirichlet problem for least gradient functions for domains in metric spaces equipped with a doubling measure and supporting a (1,1)-Poincaré inequality when the boundary of the domain satisfies a positive mean curvature condition. In this setting, it was shown by Malý, Lahti, Shanmugalingam, and Speight (2019) that solutions exist for continuous boundary data. We extend these results, showing existence of solutions for boundary data that are approximable from above and below by continuous functions. We also show that for each f ∈ L¹ (∂Ω)there is a least gradient function in Ω whose trace agrees with f at all points of continuity of f, and so we obtain existence of solutions for boundary data which are continuous almost everywhere. This is in contrast to a result of Spradlin and Tamasan (2014), who constructed an L1 -function on the unit circle which has no least gradient solution in the unit disk in R². Modifying the example of Spradlin and Tamasan, we show that the space of solvable L¹ -functions on the unit circle is non-linear, even though the unit disk satisfies the positive mean curvature condition. [ABSTRACT FROM AUTHOR]

A remark of J.J. Moreau in his seminal paper on sweeping processes with closed convex-valued moving set C : [ 0 , T ] → P (R d) having finite retraction: e (C (s) , C (t)) ≤ g (t) - g (s) , ∀ 0 ≤ s ≤ t ≤ T , where g : [ 0 , T ] → R is a right-continuous nondecreasing function, suggests a very natural involvement of the Stieltjes derivative with respect to g in the study of evolution equations governed by the sweeping process. We therefore focus on such problems with periodic boundary value conditions and multivalued perturbation - u g ′ (t) ∈ N C (t) (u (t)) + F (t , u (t)) , μ g - a. e. t ∈ [ 0 , T ] u (0) = u (T) including the Stieltjes derivative u g ′ and the Stieltjes measure μ g associated to g, N C (t) (u (t)) denoting the normal cone of C(t) at the point u(t). An application of Kakutani–Ky Fan’s fixed point theorem provides us, under Carathéodory assumptions on F, with the existence of right-continuous solutions of bounded variation. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for any bounded set of finite perimeter \Omega \subset \mathbb {R}^n, we can choose smooth sets E_k \Subset \Omega such that E_k \rightarrow \Omega in L^1 and \begin{align*} \limsup _{i \rightarrow \infty } P(E_i) \le P(\Omega)+C_1(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*} In the above \Omega ^1 is the measure-theoretic interior of \Omega, P(\cdot) denotes the perimeter functional on sets, and C_1(n) is a dimensional constant. Conversely, we prove that for any sets E_k \Subset \Omega satisfying E_k \rightarrow \Omega in L^1, there exists a dimensional constant C_2(n) such that the following inequality holds: \begin{align*} \liminf _{k \rightarrow \infty } P(E_k) \ge P(\Omega)+ C_2(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*} In particular, these results imply that for a bounded set \Omega of finite perimeter, \begin{align*} \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1)=0 \end{align*} holds if and only if there exists a sequence of smooth sets E_k such that E_k \Subset \Omega, E_k \rightarrow \Omega in L^1 and P(E_k) \rightarrow P(\Omega). [ABSTRACT FROM AUTHOR]

Let S : [ 0 , 1 ] → [ 0 , 1 ] be a piecewise increasing mapping satisfying some generalized convexity condition, so that it possesses an invariant density that is a decreasing function. We show that this invariant density can be computed by a family of Markov finite approximations that preserve the monotonicity of integrable functions. We also construct a quadratic spline Markov method and demonstrate its merits numerically. [ABSTRACT FROM AUTHOR]

We consider a nonlinear profit maximization problem in the Lotka–McKendrick model of age-structured harvested population describing farmed populations in agriculture and aquaculture. The control functions are time- and age-dependent harvesting rate and time dependent supply of newborns. We establish the existence of optimal controls with measure-valued harvesting rate by using distributional partial derivatives of functions of bounded variation through the equivalent integrated form to the original problem. [ABSTRACT FROM AUTHOR]

The motive of this article is to introduce the Bézier variant of a sequence of summation-integral type operators involving inverse Pólya-Eggenberger distribution and Păltănea operators [17]. For these operators, we estimate the approximation behaviour including first and second-order modulus of smoothness. Lastly, we establish the rate of convergence with a class of functions of derivatives of bounded variation. [ABSTRACT FROM AUTHOR]

The author's recent attempt to generalize the problem of L 1 convergence of trigonometric series to the non-periodic case is finalized here. It can now be said that any known condition which guarantees integrability of the Fourier transform of a function of bounded variation also leads to the L 1 convergence of its partial integrals provided that a universal necessary and sufficient condition is added. [ABSTRACT FROM AUTHOR]