Stieltjes Integral Equations Research Articles

An Exact Solution to the Inverse Problem of Steady Free-Surface Flow over Topography

A simple exact solution is presented to the inverse problem in steady, two-dimensional idealised flow over topography that seeks the bottom profile given knowledge of the free-surface data. Attention is focused on the case when a uniform stream flows over a localised obstacle, although the solution is not restricted to this case. The inverse problem is formulated as a Stieltjes integral equation which is solved exactly using a Fourier transform. The solution requires the analytic continuation of two real functions representing the surface speed and the angle between the surface velocity vector and the horizontal. Some example surface profiles and their corresponding bottom topographies are discussed. Although the solution requires the prescription of the surface as a function of the velocity potential, it is shown to closely resemble the corresponding profile in physical space, even for quite large surface displacements, while significant discrepancy occurs at the bottom. Inference of the bottom profile from discrete surface data is accomplished by way of polynomial interpolation and rational approximation in the complex plane for the sample case of a hydraulic fall.

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.

The logistic equation in the context of Stieltjes differential and integral equations

In this paper, we introduce logistic equations with Stieltjes derivatives and provide explicit solution formulas. As an application, we present a population model which involves intraspecific competition, periods of hibernation, as well as seasonal reproductive cycles. We also deal with various forms of Stieltjes integral equations, and find the corresponding logistic equations. We show that our work extends earlier results for dynamic equations on time scales, which served as an inspiration for this paper.

Duality for Stieltjes differential and integral equations

We propose new definitions of adjoint equations to first-order linear Stieltjes differential and integral equations. We investigate the existence and uniqueness of their solutions, provide explicit solution formulas, and obtain corresponding versions of Lagrange's identity. We show that our results are compatible with known results for dynamic equations on time scales, which are included as a special case.

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.

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.

On the Solvability of a Class of Nonlinear Hammerstein—Stieltjes Integral Equations on the Whole Line

We consider a nonlinear integral equation on the whole line with a Hammerstein-Stieltjes integral operator whose pre-kernel is a continuous distribution function. Under certain conditions imposed on the nonlinearity, we prove constructive existence and uniqueness theorems for nonnegative monotone bounded solutions. Some qualitative properties of the constructed solution are also studied. In particular, the results proved in the paper contain a theorem of O. Diekmann as a special case.

Global existence and stability results for Hadamard--Volterra--Stieltjes integral equations

In this paper we prove the existence and stability of solutions of a class of Hadamard--Volterra--Stieltjes integral equations in the Banach space of continuous and bounded functions on unbounded interval. That result is proved under rather general hypotheses. The main tools used in our considerations are the concept of measure of noncompactness in conjunction with the Darbo and Monch fixed point theorems.

Uniformly consistently estimating the proportion of false null hypotheses via Lebesgue–Stieltjes integral equations

The proportion of false null hypotheses is a very important quantity in statistical modelling and inference based on the two-component mixture model and its extensions, and in control and estimation of the false discovery rate and false non-discovery rate. Most existing estimators of this proportion threshold p-values, deconvolve the mixture model under constraints on its components, or depend heavily on the location-shift property of distributions. Hence, they usually are not consistent, applicable to non-location-shift distributions, or applicable to discrete statistics or p-values. To eliminate these shortcomings, we construct uniformly consistent estimators of the proportion as solutions to Lebesgue–Stieltjes integral equations. In particular, we provide such estimators respectively for random variables whose distributions have Riemann–Lebesgue type characteristic functions, whose distributions form discrete natural exponential families with infinite supports, and whose distributions form natural exponential families with separable moment sequences. We provide the speed of convergence and uniform consistency class for each such estimator under independence. In addition, we provide two examples for which a consistent estimator of the proportion cannot be constructed using our techniques.

On the weak solutions of a coupled system of Volterra--Stieltjes integral equations

On the weak solutions of a coupled system of Volterra--Stieltjes integral equations

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

Nonlinear integral equations with respect to functions having bounded p-variation*

We prove an inequality for the p-variation, \( 1\leqslant p<1 \), of Picard iterates associated to a nonlinear refinement Riemann–Stieltjes integral equation with respect to a possibly discontinuous function of bounded p-variation.

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.

Computation of the eigenvalues of Fredholm–Stieltjes integral equations

The Rayleigh–Ritz and the inverse iteration methods are used in order to compute the eigenvalues of Fredholm–Stieltjes integral equations, i.e. Fredholm equations with respect to suitable Stieltjes-type measures. Some applications to the so-called ‘charged’ (in German ‘belastete’) integral equation, and particularly the problem of computing the eigenvalues of a string charged by a finite number of cursors are given.

Integral operators of Urysohn-Stieltjes type

We investigate some properties of nonlinear integral operators of Urysohn--Stieltjes type with kernels depending on two variables. Results concerning the continuity and compactness of these operators are obtained. The solvability of nonlinear Urysohn--Stieltjes integral equations is also studied.