Stieltjes Integral Research Articles

Viability for mixed stochastic differential equations driven by fractional Brownian motion and its application

ABSTRACT In this paper, we focus on a class of stochastic differential equations driven by the standard Brownian motion and fractional Brownian motion with Hurst parameter under a weaker conditions than Lipschitz one. In the sense of the pathwise Riemann–Stieltjes integral, we prove the convergence of solutions for the considered equations. By making use of some transformation techniques and approximation means, we obtain some sufficient conditions on the viability for the stochastic systems under investigation. Subsequently, by using some distance functions, we establish some necessary conditions and some sufficient conditions for the viability property with respect to a given non-empty closed set K, but K can be not smooth. As applications of the results, we explore the comparison theorems about the mixed stochastic differential equations driven by fractional Brownian motion under some non-Lipschitz conditions.

Positive Solutions for a System of Fractional Boundary Value Problems with r-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters

In this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fractional derivatives and positive parameters. We first change the unknown functions such that the new boundary conditions have no positive parameters, and then, by using the corresponding Green functions, we equivalently write this new problem as a system of nonlinear integral equations. By constructing an appropriate operator A, the solutions of the integral system are the fixed points of A. Following some assumptions regarding the nonlinearities of the system, we show (by applying the Schauder fixed-point theorem) that operator A has at least one fixed point, which is a positive solution of our problem, when the positive parameters belong to some intervals. Then, we present intervals for the parameters for which our problem has no positive solution.

Conformable derivative: A derivative associated to the Riemann-Stieltjes integral

Conformable derivative: A derivative associated to the Riemann-Stieltjes integral

Approximate Solution of the System of Linear Volterra–Stieltjes Integral Equations of the Second Kind

A numerical solution of the system of linear Volterra–Stieltjes integral equations of the second kind has been found and analyzed using the so-called generalized trapezoid rule. Conditions for estimating the error have also been determined and justified. A solution of an example obtained using the proposed method is given.

Uniqueness of the solution of one class of Volterra-Stieltjes linear integral equations of the third kind

In this paper, the question of uniqueness of the solution for one class of Volterra-Stieltjes linear integral equations of the third kind is investigated. The notion of derivative with respect to an increasing function was introduced by A. Asanov in 2001 and plays special role in the study. This notion is a generalization of the usual concept of a derivative function and is an inverse operator for one class of the Stieltjes integral. Basing on idea of such derivative, using the method of integral transformations and the method of non-negative quadratic forms, the uniqueness theorems for the solution of the considered class of integral equations are proved. Examples satisfying the conditions of uniqueness theorems are also constructed in the paper. It becomes clear from these examples that it is difficult to study Volterra-Stieltjes linear integral equations of the first and third kind without using the notion of derivative with respect to increasing function.

Existence Results for Coupled Nonlinear Sequential Fractional Differential Equations with Coupled Riemann–Stieltjes Integro-Multipoint Boundary Conditions

This paper is concerned with the existence of solutions for a fully coupled Riemann–Stieltjes, integro-multipoint, boundary value problem of Caputo-type sequential fractional differential equations. The given system is studied with the aid of the Leray–Schauder alternative and contraction mapping principle. A numerical example illustrating the abstract results is also presented.

Optimization of Preventive Maintenance and Replacement Strategies for Nonrenewing Two-Dimensional Warranty Products Experiencing Degradation and External Shocks

“Vehicle damage insurance” in China, “Protection Plus Mobile Elite” launched by Samsung, “iPhone Apple Care +” offered by Apple, and so on are special warranty services aiming at external shocks. These warranty terms, named accident insurance, exist widely in the marketplace. Considering the effect of external shocks on degradation, a two-dimensional preventive maintenance and replacement strategy for products, sold with a nonrenewing and two-dimensional rectangle warranty service, is proposed in this paper. Under this strategy, preventive maintenance actions are scheduled based on units of age or usage, which occurs first. There is a reduction in the intensity function after a preventive maintenance action. Each shock before the Nth shock causes a failure of the product or an increase in product failure rate. The product is replaced by a new one on the Nth shock if it survives the N − 1 shocks. From the view of manufacturers, the mean warranty servicing cost over the whole warranty region is obtained by using the renewal theory. Based on direct numerical Riemann–Stieltjes integration, an approximation algorithm of the cost is also given. The mean cost is minimized by the optimization of preventive maintenance interval and the number of shocks N. A numerical example is given to illustrate the feasibility of the proposed strategy. The effects of maintenance cost, the arrival rate of the shocks, and other model parameters on the optimal strategy are also investigated numerically.

Continuous dependence of Lyapunov matrices with respect to perturbations for linear delay systems

AbstractIn this article, we consider general linear systems whose right‐hand sides are represented by the Riemann–Stieltjes integral with kernels given by a matrix function of bounded variation. We establish some minimal requirements on the convergence of the kernels that ensure the convergence of Lyapunov matrices of the perturbed systems. Using functional analytic approach, we analyze the continuity of the dependence of Lyapunov matrices on the right‐hand sides. Results obtained in this paper improve the previously known results for multiple delay systems and distributed delay systems. One significant advantage of our approach is that no assumption on the exponential stability is being made.

Positive Solutions for a Second-Order Nonlinear Coupled System with Derivative Dependence Subject to Coupled Stieltjes Integral Boundary Conditions

Positive Solutions for a Second-Order Nonlinear Coupled System with Derivative Dependence Subject to Coupled Stieltjes Integral Boundary Conditions

Construction of solutions of integral equations with Stieltjes functionals and bifurcation parameters

The nonlinear Volterra integral equations with loads on the desired solution are studied. Loads are given using the Stieltjes integrals. The equations contain a parameter, for any value of which the equation has a trivial solution. The necessary and sufficient conditions on the values of the parameter are derived in the neighborhood where theequation has nontrivial real solutions.

Inequalities for Riemann - Stieltjes integral

Two new inequalities for Riemann--Stieltjes integral are introduced for functions of bounded $p$-variation and H\"{o}lder continuous integrators.

Two-time-scale stochastic differential delay equations driven by multiplicative fractional Brownian noise: Averaging principle

The main goal of this article is to study an averaging principle for a class of two-time-scale stochastic differential delay equations in which the slow-varying process includes a multiplicative fractional Brownian noise with Hurst parameter H∈(12,1) and the fast-varying process is a rapidly-changing diffusion. We would like to emphasize that the approach proposed in this paper is based on the fact that a stochastic integral with respect to fractional Brownian motion with Hurst parameter in (12,1) can be defined as a generalized Stieltjes integral. In particular, to prove a limit theorem for the averaging principle, we will introduce a sequence of stopping times to control the size of multiplicative fractional Brownian noise. Then, inspired by the Khasminskii's approach, an averaging principle is developed in the sense of convergence in the p-th moment uniformly in time.

Probability Models with Discrete and Continuous Parts

In mathematical statistics courses, students learn that the quadratic function is minimized when x is the mean of the random variable X, and that the graphs of this function for any two distributions of X are simply translates of each other. We focus on the problem of minimizing the function defined by in the context of mixtures of probability distributions of the discrete, absolutely continuous, and singular continuous types. This problem is important, for example, in Bayesian statistics, when one attempts to compute the decision function, which minimizes the expected risk with respect to an absolute error loss function. Although the literature considers this problem, it does so only under restrictive conditions on the distribution of the random variable X, by, for example, assuming that the corresponding cumulative distribution function is discrete or absolutely continuous. By using Riemann-Stieltjes integration, we prove a theorem, which solves this minimization problem under completely general conditions on the distribution of X. We also illustrate our result by presenting examples involving mixtures of distributions of the discrete and absolutely continuous types, and for the Cantor distribution, in which case the cumulative distribution function is singular continuous. Finally, we prove a theorem that evaluates the function y(x) when X has the Cantor distribution.

Unified Approach to the Existence of Solutions for a Coupled System of Fractional Differential‐Integral Equations with Infinite Points and Riemann–Stieltjes Integral Boundary Conditions

In this article, by using the Schauder fixed point theorem, we first study the existence of solutions for a new coupled system of Caputo fractional differential equations with multipoint boundary value conditions under the assumption that the nonlinear term satisfies two types of the Carathéodory conditions. Using this result, we provide the existence of solutions of the system with infinite points and Riemann–Stieltjes integral boundary conditions, respectively, instead of doing it directly. Finally, we give three examples to illustrate the feasibility of main results.

Positive solutions to three classes of non-local fourth-order problems with derivative-dependent nonlinearities

In the article, we investigate three classes of fourth-order boundary value problems with dependence on all derivatives in nonlinearities under the boundary conditions involving Stieltjes integrals. A Gronwall-type inequality is employed to get an a priori bound on the third-order derivative term, and the theory of fixed-point index is used on suitable open sets to obtain the existence of positive solutions. The nonlinearities have quadratic growth in the third-order derivative term. Previous results in the literature are not applicable in our case, as shown by our examples.

Fixed-time and state-dependent time discontinuities in the theory of Stieltjes differential equations

In the present paper, we are concerned with a very general problem, namely the Stieltjes differential Cauchy problem involving state-dependent discontinuities. Given that the theory of Stieltjes differential equations covers the framework of impulsive problems with fixed-time impulses, in the present work we generalize this setting by allowing the occurrence of fixed-time impulses, as well as the occurrence of state-dependent impulses. Along with an existence result obtained under an overarching set of assumptions involving Stieltjes integrals, it is showed that a least and a greatest solution can be found.

Existence and Uniqueness for Coupled Systems of Hilfer Type Sequential Fractional Differential Equations Involving Riemann–Stieltjes Integral Multistrip Boundary Conditions

In this paper, we study a coupled system of Hilfer type sequential fractional differential equations supplemented with Riemann–Stieltjes integral multistrip boundary conditions. The standard tools of the fixed point theory are employed to prove the existence and uniqueness results for the considered problem. Examples are constructed for the illustration of the obtained results.

Positive Solutions of a Singular Fractional Boundary Value Problem with r-Laplacian Operators

We investigate the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators and nonnegative singular nonlinearities depending on fractional integrals, supplemented with nonlocal uncoupled boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. In the proof of our main results we apply the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type.

A new Simpson’s 1/3-type quadrature scheme with geometric mean derivative for the Riemann-Stieltjes integral

The main purpose of this research is to develop and improve the Simpson’s 1/3-type quadrature scheme numerically utilizing the geometric mean derivative for the Riemann- Stieltjes integral. The proposed scheme of Simpson’s 1/3-type is described in basic form and also in composite form. The performance of the proposed scheme is compared with existing schemes by experimental results using MATLAB. It has been noted in numerical results that the performance of new proposed scheme is more efficient against the existing schemes in terms of errors, computational cost, and average CPU time.

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

Abstract Volterra integro-differential equations with kernels of integral operators representable by Stieltjes integrals are investigated. The presented results are based on the approach related to the study of one-parameter semigroups for linear evolution equations. We present the method of reduction of the original initial-value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation in an extended function space. The existence of the contractive C0-semigroup is proved. An estimate for the exponential decay of the semigroup is obtained.