ABSTRACT We explore the well‐posedness and regularity of solutions for stochastic Volterra integral equations with general weakly singular kernels (SVIEGK). The kernels considered satisfy and . By employing a suitably weighted norm and Lebesgue's dominated convergence theorem, we establish the well‐posedness of solutions, including their existence, uniqueness, and continuous dependence on the initial value as well as on the parameters and . Additionally, for kernels with a specific structure, we extend the Hölder regularity results for stochastic Volterra integral equations.

Abstract This manuscript explores various characteristics of generalized fractal measures. We expand the concept of fractal integrals in relation to step functions and examine their numerous properties. Notably, since all step functions are classified as simple functions, we apply the aforementioned generalized measure to introduce Lebesgue-type integrals, referred to as FL -integrals. Additionally, we demonstrate that all F α {F^{\alpha}} -integrable functions are FL -integrals. Lastly, we address the bounded convergence theorem within the context of fractals.

In this paper, we study the concept of statistical convergence with respect to the power series method in measure, which is a new modification of the power series method. Also, using this convergence, we obtain the Lebesgue bounded convergence theorem and the fundamental theorems of measure theory. Finally, we prove Korovkin’s theorem as an application using this notion of convergence.

Under a wide concept of generalized integrals, classical results like the Lebesgue dominated convergence theorem and the rearrangement Riemann theorem are presented and proven. At the end of the paper, an application of the rearrangement theorem provides us with more information about the connection between generalized and improper integrals.

Abstract In this study, the fundamental theory of Kurzweil integral for fuzzy-number-valued functions with two variables will be discussed. Firstly, we provide the concept and fundamental properties of this integral. Then, we prove the characterize theorem of this kind of integral by using support functions. In addition, we obtained the convergence theorem based on weak equi-integrability and dominated convergence theorem for fuzzy Kurzweil integral. Finally, we obtained some properties of primitives of fuzzy Kurzweil integrable functions by weak generalized absolutely continuity. Furthermore, we obtain the relationship between the fuzzy Kurzweil integral and the fuzzy Henstock integral based on the connection of fuzzy Kurzweil integral and the fuzzy Henstock integral in the sense of partial derivatives.

In this paper, a new kind of Meyer-K?nig-Zeller type operators will be constructed by their relationship with Baskakov type operators. Firstly, the Baskakov type operators preserving the function e-?x(? > 0) will be introduced by the dominated convergence theorem. Secondly, some approximation properties of this kind of new Baskakov operators will be discussed. Finally, with the help of a trans-formation from [0, 1) to [0, ?), we will give the definition and the approximation theorem of the new Meyer-K?nig-Zeller type operators.

Pth moment stability and approximate controllability of neutral differential equations (NDEs) with delayed impulse and time-varying delays driven by fractional Brownian motion (fBm) with H ? (0, 1/2) are studied. Unlike the fBm with Hurst parameter H ? (1/2, 1) in most of the literature, the fBm with Hurst parameter H ? (0, 1/2) in this work. The delayed impulse here means time delay in impulse, and the delay is no more than the minimum value of impulsive intervals. Based on Lyapunov stability theory and analytic semigroup theory, combined with stochastic analysis methods and impulse integral inequalities, some conditions guaranteeing the pth moment stability of the mild solution are established. Afterward, approximate controllability conditions of the system are acquired by the Lebesgue-dominated convergence theorem. At last, the validity of secured results is verified by an example.

This paper presents an enhanced and rigorously analyzed methodology for deriving implied riskfree rates through the consistency condition between distributional and CAPM-based derivatives of beta. The approach equates two distinct constructions: (i) a distributional derivative obtained through dominated convergence theorems applied to joint density integrands, and (ii) the analytical derivative of the CAPM expression. This consistency yields a quadratic equation in the risk-free rate whose roots provide model-implied candidates. This enhanced version strengthens mathematical foundations with complete proofs, expands econometric discussion incorporating post-2020 literature on density estimation and option-implied rates, provides detailed statistical inference procedures, and comprehensively analyzes practical limitations and implementation challenges. It demonstrates that while theoretically sound, the methodology faces significant empirical hurdles including density derivative estimation noise, measurement error amplification, and model specification sensitivity, positioning it primarily as a diagnostic rather than estimation tool.

In this paper, we investigate noncoercive monotone convex sweeping processes with velocity constraints, a topic not previously investigated. Using some fundamental results on Bochner integration, the Tikhonov regularization method, a solution existence result for coercive convex sweeping processes with velocity constraints and a useful fact on measurable single-valued mappings, we prove the solution existence and properties of the solution set of noncoercive convex sweeping processes with velocity constraints under suitable conditions. These results represent a significant contribution, addressing a specific aspect of an open question posed in our recent work [Adly et al., Convex and nonconvex sweeping processes with velocity constraints: well-posedness and insights. Appl Math Optim. 2023;88(Paper No. 45)]. Additionally, we resolve two other open questions from the same paper concerning the behaviour of the regularized trajectories, relying on the Dominated Convergence Theorem for Bochner integration.

The strong fuzzy variational Henstock multiple integral on n-dimensional fuzzy number space

Measurements of Bose–Einstein correlations played a crucial role in the discovery and the subsequent detailed exploration of the Quark-Gluon-Plasma (QGP) created in high-energy collisions of heavy nuclei. Such measurements gave rise to femtoscopy, a flourishing sub-field of high-energy physics, and there are important new directions to explore and discoveries to be made in the near future. One of these important current topics is the precise investigation of the shape of the correlation functions utilizing Lévy-stable sources. In this paper, we utilize Lebesgue’s dominated convergence theorem and Fubini’s theorem to present a novel method of calculating the shape of the two-particle correlation functions, including the Coulomb final-state interaction. This method relies on an exact calculation of a large part of the necessary integrals of the Coulomb wave function and can be utilized to calculate the correlation function for any source function with an easily accessible Fourier transform. In this way, it is eminently applicable to Lévy-stable source functions. In this particular case, we find that the new method is more robust and allows the investigation of a wider parameter range than the previously utilized techniques. We present an easily applicable software package that is ready to use in experimental studies.

The Dominated Convergence Theorem, Fatou's Lemma, and the Monotone Convergence Theorem are collectively referred to as the continuity theorems of Lebesgue integration. These ubiquitous results are stated and proved in the literature for sequences of measurable functions. It is well known that these results do not hold for arbitrary nets of measurable functions. These results, as well as the Levy Continuity Theorem, Riesz's characterization of $\mathcal{L}_p$ convergence in terms of almost everywhere convergence and convergence of norms, and a stronger version of Scheffe's Theorem, are derived for nets indexed by countably accessible directed sets.

The interchange of the ‘limit of an integral’ with the ‘integral of a limit’ for sequenc- es of functions is crucial in relevant applications, such as Fourier series for decom- posing periodic functions into sinusoidal components, and Fubini’s theorem for changing the order of integration of multivariable functions. This expository paper reviews three classical results in real analysis for cases where the limit of an integral of a sequence of functions equals the integral of the limiting function: (1) Mono- tone Convergence Theorem, (2) Uniform Convergence Theorem, and the broad- est result, (3) Dominated Convergence Theorem. While proofs of (2) are typically studied in undergraduate analysis, the proofs of (1) and (3) are usually reserved for graduate-level measure theory, where they are taught in a more general context. The purpose of this paper is to summarize and adapt W. A. J. Luxembourg’s un- dergraduate-friendly proof [7] of (3) Arzel`a’s Dominated Convergence Theorem, to demonstrate the nontrivial direction of (1) Monotone Convergence Theorem for Riemann Integrals. Our aim is to demystify the hidden logic involved in these well-established theorems, making them more accessible for undergraduate analysis.

Abstract This article presents a proof of the bounded convergence theorem for Riemann integrals. An effort has been made to keep the exposition concise and self-contained.

This paper is concerned with a nonlocal dispersal susceptible–infected–recovered (SIR) epidemic model adopted with the mass action infection mechanism. We mainly study the existence and non‐existence of traveling waves connecting the infection‐free equilibrium state and the endemic equilibrium state. The main difficulty lies in the fact that the semiflow generated here does not admit the order‐preserving property. Meanwhile, this new model brings some new challenges due to the unboundedness of the nonlinear term. We overcome these difficulties to obtain the boundedness of traveling waves with the speed by some analysis techniques firstly and then prove the existence of traveling waves by employing Lyapunov–LaSalle theorem and Lebesgue dominated convergence theorem. By utilizing an approximating method, we study the existence of traveling waves with the critical wave speed . Our results on this new model may provide some implications on disease modeling and controls.

Abstract This paper is concerned with a version of the Lebesgue dominated convergence theorem (DCT) which has been stated for the Kurzweil–Stieltjes integral of real functions. Our objective in this work is to analyze the extension of this result to include vector functions with values in Banach spaces. We establish that the mentioned convergence theorem for the Kurzweil–Stieltjes integral can be formulated in weaker versions for reflexive and separable Banach spaces, and spaces having the Schur property, nonetheless it is not verified in the general case.

"In this paper, necessary and sufficient conditions are establish of the solutions to second-order delay differential equations of the form \begin{equation} \Big(r(t)\big(x'(t)\big)^\gamma\Big)' +\sum_{i=1}^m q_i(t)f_i\big(x(\sigma_i(t))\big)=0 \text{ for } t \geq t_0,\notag \end{equation} We consider two cases when $f_i(u)/u^\beta$ is non-increasing for $\beta<\gamma$, and non-decreasing for $\beta>\gamma$ where $\beta$ and $\gamma$ are the quotient of two positive odd integers. Our main tool is Lebesgue's Dominated Convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem."

Many interesting physical theories have analytic classical actions. We show how Feynman’s path integral may be defined non-perturbatively, for such theories, without a Wick rotation to imaginary time. We start by introducing a class of smooth regulators which render interference integrals absolutely convergent and thus unambiguous. The analyticity of the regulators allows us to use Cauchy’s theorem to deform the integration domain onto a set of relevant, complex “thimbles” (or generalized steepest descent contours) each associated with a classical saddle. The regulator can then be removed to obtain an exact, non-perturbative representation. We show why the usual method of gradient flow, used to identify relevant saddles and steepest descent “thimbles” for finite-dimensional oscillatory integrals, fails in the infinite-dimensional case. For the troublesome high frequency modes, we replace it with a method we call “eigenflow” which we employ to identify the infinite-dimensional, complex “eigenthimble” over which the real time path integral is absolutely convergent. We then bound the path integral over high frequency modes by the corresponding Wiener measure for a free particle. Using the dominated convergence theorem we infer that the interacting path integral defines a good measure. While the real time path integral is more intricate than its Euclidean counterpart, it is superior in several respects. It seems particularly well-suited to theories such as quantum gravity where the classical theory is well developed but the Euclidean path integral does not exist.

In this paper, we introduce new conditions to prove that the existence and boundedness of the solution by convergent sequences and convergent series. The theorem of Krasnoselskii, Lebesgue’s dominated convergence theorem and fixed point theorem are used to get some sufficient conditions for the existence of solutions. Furthermore, we get sufficient conditions to guarantee the oscillatory property for all solutions in this class of equations. An illustrative example is included as an application to the main results.

Using the Newton-Leibniz formula for the Lebesgue integral and Lebesgue’s dominated convergence theorem, this Note offers a sufficient condition for the equality of mixed partial derivatives, improving Peano’s result.