Volterra-Stieltjes Integral Equation Research Articles

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.

Регуляризация и выбор параметра решений нелинейных интегральных уравнений Вольтерра-Стилтьеса третьего рода

The article considers nonlinear Volterra-Stieltjes integral equations of the third kind, and its solution by regularizing operator according to M. M. Lavrentev. A uniqueness theorem was proved, and a regularization parameter was chosen. The research uses the concept of a derivative with respect to an increasing function, the method of regularization according to M. M. Lavrentevs methods in functional analysis, methods of transformation of equations, methods of integral and differential equations. Proposed methods can be used to study the integral, integral-differential equations of the Volterra-Stieltjes type of high orders, as well as in the qualitative study of some applied processes in the field of physics, ecology, medicine, and the theory of control complex systems. They can be used in the further development of the theory of integral equations in classes of incorrect problems, in numerical solution of Volterra-Stieltjes integral equations of the third kind, and when solving specific applied problems that lead to equations of the third kind.

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.

Existence, uniqueness, variation-of-constant formula and controllability for linear dynamic equations with Perron Δ-integrals

In this paper, we investigate the existence and uniqueness of a solution for a linear Volterra-Stieltjes integral equation of the second kind, as well as for a homogeneous and a nonhomogeneous linear dynamic equations on time scales, whose integral forms contain Perron [Formula: see text]-integrals defined in Banach spaces. We also provide a variation-of-constant formula for a nonhomogeneous linear dynamic equations on time scales and we establish results on controllability for linear dynamic equations. Since we work in the framework of Perron [Formula: see text]-integrals, we can handle functions not only having many discontinuities, but also being highly oscillating. Our results require weaker conditions than those in the literature. We include some examples to illustrate our main results.

Volterra–Stieltjes integral equations and impulsive Volterra–Stieltjes integral equations

In this paper, we prove existence and uniqueness of solutions of Volterra– Stieltjes integral equations using the Henstock–Kurzweil integral. Also, we prove that these equations encompass impulsive Volterra–Stieltjes integral equations and prove the existence and uniqueness for these equations. Finally, we present some examples to illustrate our results.

Regularization and Choice of the Parameter for the Third Kind Nonlinear Volterra-Stieltjes Integral Equation Solutions

The article is considering the third kind of nonlinear Volterra-Stieltjes integral equations with the solution by Lavrentyev regularizing operator. A uniqueness theorem was proved, and a regularization parameter was chosen. This can be used in further development of the theory of the integral equations in non-standard problems, classes in the numerical solution of third kind Volterra-Stieltjes integral equations, and when solving specific problems that lead to equations of the third kind.

Combining Continuous and Discrete Phenomena for Feynman’s Operational Calculus in the Presence of a ( $${\varvec{C}}_{\mathbf{0 }}$$ C 0 ) Semigroup and Feynman–Kac Formulas with Lebesgue–Stieltjes Measures

This paper introduces the presence of a $$(C_{0})$$ semigroup of linear operators into the disentangling of functions of noncommuting operators in the setting where time-ordering measures with finitely supported discrete parts are allowed. Some examples are discussed. Furthermore, in this more general setting the main result of this paper, an evolution equation satisfied by disentangled exponential functions is obtained. A related generalized integral equation is also discussed. Via some detailed examples, Feynman–Kac formulas (Volterra–Stieltjes integral equations) with Lebesgue–Stieltjes measures are also derived and an associated differential equation is derived.

Solutions of a quadratic Volterra–Stieltjes integral equation in the class of functions converging at infinity

Solutions of a quadratic Volterra–Stieltjes integral equation in the class of functions converging at infinity

Solvability of a nonlinear Volterra–Stieltjes integral equation in the class of bounded and continuous functions of two variables

The aim of this paper is to investigate the solvability of a nonlinear Volterra–Stieltjes integral equation in two variables in the space \(BC(\mathbb {R_+} \times \mathbb {R_+},\mathbb {R})\). We will consider this equation in the space of real continuous and bounded functions on the set \(\mathbb {R_+} \times \mathbb {R_+}\). Applying the fixed point theorem of Darbo type and the technique of measure of noncompactness we will show that solutions of the mentioned equation exist. We will also give the characterization of these solutions. At the end of our considerations we will introduce some special cases of the studied equation and we will include an example illustrating our main result.

On Local Attractivity and Asymptotic Stability of Solutions of Nonlinear Volterra–Stieltjes Integral Equations in Two Variables

In this paper we will study the existence of solutions depending on two variables of a nonlinear integral equation of Volterra–Stieltjes type in two variables, in the space of real functions which are continuous and bounded on the set \mathbb{R}_{+}\times [0,M] . Moreover, we will give the characterization of those solutions. In our study we will utilize the Darbo type of fixed point theorem and apply a measure of noncompactness. In the last section we present particular cases of the considered equation.

Solvability of a Volterra–Stieltjes integral equation in the class of functions having limits at infinity

Solvability of a Volterra–Stieltjes integral equation in the class of functions having limits at infinity

On Monotonic and Nonnegative Solutions of a Nonlinear Volterra-Stieltjes Integral Equation

We study the existence of monotonic and nonnegative solutions of a nonlinear quadratic Volterra-Stieltjes integral equation in the space of real functions being continuous on a bounded interval. The main tools used in our considerations are the technique of measures of noncompactness in connection with the theory of functions of bounded variation and the theory of Riemann-Stieltjes integral. The obtained results can be easily applied to the class of fractional integral equations and Volterra-Chandrasekhar integral equations, among others.

A Unified Approach to Some Classes of Nonlinear Integral Equations

We are going to discuss some important classes of nonlinear integral equations such as integral equations of Volterra-Chandrasekhar type, quadratic integral equations of fractional orders, nonlinear integral equations of Volterra-Wiener-Hopf type, and nonlinear integral equations of Erdélyi-Kober type. Those integral equations play very significant role in applications to the description of numerous real world events. Our aim is to show that the mentioned integral equations can be treated from the view point of nonlinear Volterra-Stieltjes integral equations. The Riemann-Stieltjes integral appearing in those integral equations is generated by a function of two variables. The choice of a suitable generating function enables us to obtain various kinds of integral equations. Some results concerning nonlinear Volterra-Stieltjes integral equations in several variables will be also discussed.

Solvability of Fractional Integral Equations on an Unbounded Interval through the Theory of Volterra-Stieltjes Integral Equations

In this paper we study the existence of solutions of a nonlinear quadratic Volterra-Stieltjes integral equation in the space of real functions being continuous and bounded on the interval of nonnegative numbers. Moreover, we also investigate the solvability of the equation in question in the classes of functions being asymptotically stable or having limits at infinity, for example. The main tool used in our considerations is the technique of measures of noncompactness constructed in a special way. It is shown that results obtained in the paper are applicable to the class of fractional integral equations and Volterra-Chandrasekhar integral equations, among others.

The technique of Volterra–Stieltjes integral equations in the application to infinite systems of nonlinear integral equations of fractional orders

Our study in this paper concerns the infinite systems of nonlinear integral equations of Volterra–Stieltjes type. We show that some assumptions imposed on terms of the mentioned systems guarantee the solvability of those systems in the space C(I,c0) with I=[0,1] and c0 being the classical sequence space. As a particular and very important case of the investigated systems of nonlinear Volterra–Stieltjes integral equations we obtain infinite systems of nonlinear integral equations of fractional orders. Thus, the approach applied in the paper enables us to obtain rather general results concerning the existence of solutions of infinite systems of nonlinear fractional integral equations.

Existence of solutions for a class of nonlinear Volterra singular integral equations

In this paper we prove some results concerning the existence of solutions for a large class of nonlinear Volterra singular integral equations in the space C [ 0 , 1 ] consisting of real functions defined and continuous on the interval [ 0 , 1 ] . The main tool used in the proof is the concept of a measure of noncompactness. We also present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results. Moreover, we compare our theory with the approach depending on the use of the theory of Volterra–Stieltjes integral equations. We also show that the results of the paper are applicable in the study of the so-called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems.

Existence results for Volterra-Stieltjes quadratic integral equations on an unbounded interval

We prove a few existence results for a nonlinear quadratic Volterra-Stieltjes integral equation on an unbounded interval. Our proof depends on suitable combination of the technique of measures of noncompactness and the Schauder fixed point principle. Such an approach permits us to obtain existence theorems under rather general assumptions.

Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes

Let X be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieltjes integral equation techniques we investigate Hölder continuity and differentiability of first passage time distributions of X with respect to continuous lower and upper moving boundaries. Under mild assumptions on the transition function of X we prove the existence of a continuous first passage time density to one-sided differentiable moving boundaries and derive a new integral equation for this density. We apply our results to Brownian motion and its nonrandom Markovian transforms, in particular to the Ornstein-Uhlenbeck process.

Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes

Let X be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieltjes integral equation techniques we investigate Hölder continuity and differentiability of first passage time distributions of X with respect to continuous lower and upper moving boundaries. Under mild assumptions on the transition function of X we prove the existence of a continuous first passage time density to one-sided differentiable moving boundaries and derive a new integral equation for this density. We apply our results to Brownian motion and its nonrandom Markovian transforms, in particular to the Ornstein-Uhlenbeck process.

Discrete and Continuous Liouville–Green–Olver Approximations: a Unified Treatment via Volterra–Stieltjes Integral Equations

A unified treatment of the Liouville–Green–Olver approximation theory for linear second-order differential and difference equations is presented. This is based on reduction to Volterra–Stieltjes integral equations with respect to complex measures. The present approach embodies and improves several previous results. Moreover, error bounds are obtained for recessive solutions of certain difference equations, for which only qualitative results were known. The theory can be applied, for instance, to the asymptotics of certain families of orthogonal polynomials.