Stieltjes Integral Research Articles

Ulam–Hyers Stability and Simulation of a Delayed Fractional Differential Equation with Riemann–Stieltjes Integral Boundary Conditions and Fractional Impulses

In this article, we delve into delayed fractional differential equations with Riemann–Stieltjes integral boundary conditions and fractional impulses. By using differential inequality techniques and some fixed-point theorems, some novel sufficient assessments for convenient verification have been devised to ensure the existence and uniqueness of solutions. We further employ the nonlinear analysis to reveal that this problem is Ulam–Hyers (UH) stable. Finally, some examples and numerical simulations are presented to illustrate the reliability and validity of our main results.

Positive Solutions to a System of Coupled Hadamard Fractional Boundary Value Problems

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main results, we employ several fixed-point theorems, namely the Banach contraction mapping principle, the Schauder fixed-point theorem, the Leggett–Williams fixed-point theorem, and the Guo–Krasnosel’skii fixed-point theorem.

Approximate controllability of second-order neutral measure driven systems with state-dependent delay

This article investigates the solvability and approximate controllability of a new class of measure driven control systems with state-dependent delay, which can deal with hybrid system without any restriction Zeno behaviour. By applying fixed point technology and Lebesgue Stieltjes integral, a new set of sufficient conditions is constructed that ensures the existence of mild solution and approximate controllability of the considered system. Lastly, an example is formulated to demonstrate the effectiveness of obtained results.

Fractional Calculus for Non-Discrete Signed Measures

In this paper, we suggest a first-ever construction of fractional integral and differential operators based on signed measures including a vector-valued case. The study focuses on constructing the fractional power of the Riemann–Stieltjes integral with a signed measure, using semigroup theory. The main result is a theorem that provides the exact form of a semigroup for the Riemann–Stieltjes integral with a measure having a countable number of extrema. This article provides examples of semigroups based on integral operators with signed measures and discusses the fractional powers of differential operators with partial derivatives.

A general form of Gronwall inequality with Stieltjes integrals

We present a new Gronwall inequality for Stieltjes integrals, which improves numerous existing results, and has a simple proof based on the quotient rule for Stieltjes integrals. As an application, we obtain uniqueness theorems for measure differential equations and nabla dynamic equations. Finally, we revisit the topic from the perspective of Stieltjes derivatives.

On a Coupled System of Nonlinear Generalized Fractional Differential Equations with Nonlocal Coupled Riemann–Stieltjes Boundary Conditions

On a Coupled System of Nonlinear Generalized Fractional Differential Equations with Nonlocal Coupled Riemann–Stieltjes Boundary Conditions

On Banach spaces of regulated functions of several variables. Analogue of the Riemann–Stieltjes integral

In the previous work of the authors, the concept of a regulated function of several variables $f\colon X\to\mathbb R$ was introduced, where $X\subseteq \mathbb R^n.$ The definition is based on the concept of a special partition of the set $X$ and the concept oscillation of the function $f$ on the elements of the partition. The space ${\rm G}(X)$ of such functions is Banach in the $\sup$-norm and is the closure of the space of step functions. In this paper, the space ${\rm G}^F(X)$ is defined and studied, which differs from ${\rm G}(X)$ in that here, in defining regulated functions of several variables, instead of special partitions, $F$-partitions are used: their elements are non-empty open sets measurable by the generalized Jordan measure (by the measure $m_{_{\!F}}$). (Symbol $F$ denotes the function generating the measure $m_{_{\!F}}.$) In the second part of the work, the concept of $F$-integrability of functions of several variables is defined. It is proved that if $X$ is the closure of a non-empty open bounded set $X_0\subseteq {\mathbb R}^n,$ measurable with respect to measure $m_{_{\!F}},$ and the function $f\colon X\to {\mathbb R}$ is integrable in the Riemann–Stieltjes sense with respect to the measure $m_{_{\!F}}$, then it is $F$-integrable. In this case, the values of the multiple integrals coincide. All functions from the space ${\rm G}^F(X)$ are $F$-integrable. The main properties of the Riemann–Stieltjes $F$-integral are proved.

On a (k;chi)-Hilfer fractional system with coupled nonlocal boundary conditions including various fractional derivatives and Riemann–Stieltjes integrals

In the present research, we investigate the existence and uniqueness of solutions for a system of (k; χ)-Hilfer fractional differential equations, subject to coupled nonlocal boundary conditions, which contain various fractional derivatives and Riemann–Stieltjes integrals. The uniqueness result relies on the Banach contraction mapping principle, while the existence results depend on the Leray–Schauder alternative and Krasnosel’skiĭ fixed point theorem. Examples are also constructed to illustrate the obtained results.

Existence of solutions for a fractional Riemann–Stieltjes integral boundary value problem

In this paper, we study a Riemann–Liouville-type fractional Riemann–Stieltjes integral boundary value problem under some conditions regarding the spectral radius of the relevant linear operator. The existence of nontrivial solutions is obtained using topological degree, and our results improve and generalize some results in the literature.

Mixed Hilfer and Caputo Fractional Riemann–Stieltjes Integro-Differential Equations with Non-Separated Boundary Conditions

In this paper, we investigate a sequential fractional boundary value problem which contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions. We establish the existence of a unique solution via Banach’s fixed point theorem, while by applying Leray–Schauder’s nonlinear alternative, we prove an existence result. Finally, examples are provided to demonstrate the results obtained.

Generalized Chain Rules and Applications to Stieltjes Differential and Integral Equations

We present new generalizations of the chain rule, which involve Stieltjes derivatives and integrals. The results are subsequently used to obtain the power rules for two generalizations of the exponential function, and to investigate Bernoulli-type equations with Stieltjes derivatives and integrals.

On the S-asymptotically ω-periodic mild solutions for multi-term time fractional measure integro-differential equations

This paper deals with a class of nonlocal problem for multi-term time-fractional measure integro-differential evolution equations of mixed type in Hilbert spaces. Firstly, we introduce the concept of S-asymptotically ω-periodic mild solution, on the premise of by utilizing -resolvent family and measure functional (Henstock–Lebesgue–Stieltjes integral). Secondly, combining regulated functions and fixed point theorem, the existence S-asymptotically ω-periodic mild solution for a classes of measure-driven differential equation is discussed. Finally, as samples of applications, we consider the nonlocal problem to a class of time-fractional measure partial differential equation with nonlocal condition at the end of this paper.

Essential norm of Riemann–Stieltjes operator on weighted Bergman spaces with doubling weights

Abstract Let ω be a doubling weight and 0 < p ≤ q < ∞ {0<p\leq q<\infty} . The essential norm of Riemann–Stieltjes operator T g {T_{g}} from the weighted Bergman space A ω p {A^{p}_{\omega}} to A ω q {A^{q}_{\omega}} was investigated in the unit ball of ℂ n {\mathbb{C}^{n}} .

Semigroups of Operators Generated by Integro-Differential Equations with Kernels Representable by Stieltjes Integrals

Semigroups of Operators Generated by Integro-Differential Equations with Kernels Representable by Stieltjes Integrals

Монотонизация модифицированной схемы с эрмитовой интерполяцией для численного решения неоднородного уравнения переноса с поглощением

A hybrid scheme for the numerical solving of a transport equation is implemented. The high order scheme is a modified CIP (Cubic Interpolation Polynomial) scheme with Hermitian interpolation. The third order of approximation of the CIP scheme is achieved by including in the list of unknowns not only the nodal values of the function, but also the nodal values of its derivatives. In the modification under consideration the Euler–Maclaurin formula is used to calculate derivatives. The low order scheme is the characteristic scheme of the first order of approximation. Local, layerwise and global monotonization are considered, the hybridization is performed, respectively, after a cell, a time layer or the whole grid processing. It is shown that the best results are obtained by a scheme with local monotonization. The convergence orders of the hybrid scheme on tests of different smoothness of exact solution do not differ significantly from the convergence orders of the CIP scheme. It is proposed to calculate an integral along a characteristic using the Simpson formula for the Stieltjes integral in the case of the large optical thickness; it provides a significant reduction in the numerical solution errors.

THE SCALING-LAW FLOWS: AN ATTEMPT AT SCALING-LAW VECTOR CALCULUS

In this paper, the scaling-law vector calculus, which is connected between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The scaling-law Gauss–Ostrogradsky-like, Stokes-like and Green-like theorems, and Green-like identities are considered in sense of the scaling-law vector calculus. The strong and weak conjectures for the scaling-law flows are obtained in detail. The obtained result is a potentially mathematical tool proposed to develop an important way of approaching this challenge for the scaling-law flows.

Systems of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Boundary Conditions

We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply various fixed point theorems, notably including the Banach contraction mapping principle, the Krasnosel’skii fixed point theorem applied to the sum of two operators, the Schaefer fixed point theorem, and the Leray–Schauder nonlinear alternative.

Approximate controllability of time‐varying measure differential problem of second order with state‐dependent delay and noninstantaneous impulses

It is comprehended that the systems without any limitation on their Zeno action are enthralled in a vast class of hybrid systems. This article is influenced by a new category of nonautonomous second‐order measure differential problems with state‐dependent delay (SDD) and noninstantaneous impulses (NIIs). Some new sufficient postulates are created that guarantee the solvability and approximate controllability. We employ the fixed point strategy and theory of Lebesgue–Stieltjes integral in the space of piecewise regulated functions. The measure of noncompactness is applied to establish the existence of a solution. Moreover, the measured differential equations generalize the ordinary impulsive differential equations. Thus, our findings are more prevalent than that encountered in the literature. At last, an example is comprised that exhibits the significance of the developed theory.

Existence of Solutions for the One-Dimensional Fourth-Order p-Laplacian Impulsive Differential Equation Involving Nonlinear Stieltjes Integral Boundary Conditions

Existence of Solutions for the One-Dimensional Fourth-Order p-Laplacian Impulsive Differential Equation Involving Nonlinear Stieltjes Integral Boundary Conditions

Positive Solutions for a System of Hadamard Fractional Boundary Value Problems on an Infinite Interval

Our investigation is devoted to examining the existence, uniqueness, and multiplicity of positive solutions for a system of Hadamard fractional differential equations. This system is defined on an infinite interval and is subject to coupled nonlocal boundary conditions. These boundary conditions encompass both Hadamard fractional derivatives and Riemann–Stieltjes integrals, and the nonlinearities within the system are non-negative functions that may not be bounded. To establish the main results, we rely on the utilization of mathematical theorems such as the Schauder fixed-point theorem, the Banach contraction mapping principle, and the Avery–Peterson fixed-point theorem.