System Of Nonlinear Integral Equations Research Articles

The Smirnov Property for Weighted Lebesgue Spaces

We establish lower norm bounds for multivariate functions within weighted Lebesgue spaces, characterised by a summation of functions whose components solve a system of nonlinear integral equations. This problem originates in portfolio selection theory, where these equations allow one to identify mean-variance optimal portfolios, composed of standard European options on several underlying assets. We elaborate on the Smirnov property—an integrability condition for the weights that guarantees the uniqueness of solutions to the system. Sufficient conditions on weights to satisfy this property are provided, and counterexamples are constructed, where either the Smirnov property does not hold or the uniqueness of solutions fails.

Tripled Fixed Point Methodologies for Finding a Solution to a System of Nonlinear Integral Equations on Orthogonal Metric Spaces

Tripled Fixed Point Methodologies for Finding a Solution to a System of Nonlinear Integral Equations on Orthogonal Metric Spaces

On infinite systems of nonlinear integral equations in two variables in Banach Space $BC(\mathbb{R_+}\times \mathbb{R_+},c_0$)

In this paper, the solvability of an infinite system of integral equations of the Volterra-Hammerstein type in Banach space $BC(\mathbb{R_+}\times \mathbb{R_+},c_0$) is examined. Technique associated with the measure of noncompactness plays the most important role in adopted analysis and authors present an example to validate the applicability of the result.

ENHANCING CONVERGENCE AND ACCURACY IN SOLVING SYSTEM OF NONLINEAR INTEGRAL EQUATIONS USING MULTILAYER NEURAL NETWORKS

Abstract. This paper presents a novel approach for solving nonlinear integral equations of Fredholm and Volterra types using a multi-layer neural network (MLNN) trained through unrestricted optimization. The learning process involves minimizing the integral equation error function using the Nelder-Mead simplex method (NMSM), enabling efficient convergence in both Fredholm and Volterra equations. Our method ensures high accuracy despite potentially time-consuming computations for complex problems, maintaining convergence and approximation precision. Furthermore, the proposed approach demonstrates robustness against high-dimensional and variant equations. Numerical experiments employing the NMSM showcase its superior performance compared to existing methods. This research offers a promising avenue for solving a system of nonlinear integral equations efficiently and accurately, with broad applicability across diverse domains.

On qualitative properties of the solution of a boundary value problem for a system of nonlinear integral equations

On qualitative properties of the solution of a boundary value problem for a system of nonlinear integral equations

On qualitative properties of the solution of a boundary value problem for a system of nonlinear integral equations

Исследуется одна краевая задача для системы нелинейных интегральных уравнений на полуоси, матричное ядро которой имеет единичный спектральный радиус. эта краевая задача имеет приложения в различных областях физики и биологии. в частности, такие задачи возникают в динамической теории $p$-адических струн для скалярного поля тахионов, в математической теории распространения эпидемий, в кинетической теории газов и в теории переноса излучения. обсуждаются вопросы существования, отсутствия и единственности нетривиального решения этой краевой задачи. доказано, что краевая задача с нулевыми краевыми условиями на бесконечности имеет только тривиальное решение в классе неотрицательных и ограниченных функций. также доказано, что если хотя бы одно из краевых значений на бесконечности положительно, то эта задача имеет нетривиальное решение, которое является выпуклым, неотрицательным, ограниченным и непрерывным. приведены примеры матричных ядер и нелинейностей, удовлетворяющих всем условиям доказанных теорем.

On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line

We consider a special system of integral equations of convolution type with a monotone convex nonlinearity naturally arising when searching for stationary or limit states in various dynamic models of applied nature, for example, in models of the spread of epidemics, and prove theorems stating the existence or absence of a nontrivial bounded solution with limits at +@ depending on the values of these limits and on the structure of the matrix kernel of the system. We also study the uniqueness of such a solution assuming that it exists. Specific examples of systems whose parameters satisfy the conditions stated in our theorems are given.

On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line

On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line

Coupled fixed point results on orthogonal metric spaces with application to nonlinear integral equations

In this article, we prove some well-known coupled fixed point theorems in 0-complete metric spaces. Also, we present some corollaries related to our study. In addition to this, we give an example showing that our results successfully obtain the existence and uniqueness of the coupled fixed point for 0-complete metric spaces, but the results are not valid for complete metric spaces. Finally, we apply our results to examine the existence and uniqueness of a solution of the system of nonlinear integral equations.

On the solvability of a nonlinear optimization problem with boundary vector control of oscillatory processes

In the paper, the solvability of the nonlinear boundary optimization problem has been investigated for the oscillation processes, described by the integro-differential equation in partial derivatives with Fredholm integral operator. It has been established that the components of the boundary vector control are defined as a solution to a system of nonlinear integral equations of a specific form, and the equations of this system have the property of equal relations. An algorithm for constructing a solution to the problem of nonlinear optimization has been developed.

On the Solvability of a System of Nonlinear Integral Equations with the Hammerstein–Stieltjes Operator on the Half-Line

We study a system of nonlinear integral equations of the Hammerstein–Stieltjes type whose pre-kernels are continuous distribution functions. A constructive existence theorem for a nontrivial, nonnegative, and bounded solution of the system is proved. The integral asymptotics of the constructed solution is studied. Examples of systems for which all the conditions of the theorem are satisfied are given.

Fixed Point Theorems in Generalized Banach Algebras and an Application to Infinite Systems in 𝒞([0, 1], c 0) × 𝒞([0, 1], c 0)

The purpose of this paper is to extend Boyd and Wong’s fixed point theorem in complete generalized metric spaces and to apply this result under the so-called G-weak topology in generalized Banach algebras. Some tools will be introduced and involved in our work as the generalized measures of weak non-compactness and the sequential condition Our results will extend many known theorems in the literature. Also, we will give an application for an infinite system of nonlinear integral equations defined on the generalized Banach algebra where c 0 is the space of all real sequences converging to zero.

Predicting the last zero before an exponential time of a spectrally negative Lévy process

AbstractGiven a spectrally negative Lévy process, we predict, in an $L_1$ sense, the last passage time of the process below zero before an independent exponential time. This optimal prediction problem generalises [2], where the infinite-horizon problem is solved. Using a similar argument as that in [24], we show that this optimal prediction problem is equivalent to solving an optimal prediction problem in a finite-horizon setting. Surprisingly (unlike the infinite-horizon problem), an optimal stopping time is based on a curve that is killed at the moment the mean of the exponential time is reached. That is, an optimal stopping time is the first time the process crosses above a non-negative, continuous, and non-increasing curve depending on time. This curve and the value function are characterised as a solution of a system of nonlinear integral equations which can be understood as a generalisation of the free boundary equations (see e.g. [21, Chapter IV.14.1]) in the presence of jumps. As an example, we numerically calculate this curve in the Brownian motion case and for a compound Poisson process with exponential-sized jumps perturbed by a Brownian motion.

On non-trivial solvability of one system of non-linear integral equations on the real axis

Рассматривается система сингулярных интегральных уравнений с монотонной и выпуклой нелинейностью на всей числовой прямой. Данная система имеет приложения во многих направлениях естествознания. В частности, такие системы встречаются в теории $p$-адических открыто-замкнутых струн, в математической теории пространственно-временного распространения эпидемии в рамках известной модели Дикмана-Капера, в кинетической теории газов, в теории переноса излучения. Доказывается теорема существования нетривиального и ограниченного решения. Исследуется также асимптотическое поведение построенного решения на $\pm\infty$. Приводятся конкретные примеры нелинейностей и ядерных функций, имеющих прикладной характер. Библиография: 18 наименований.

On non-trivial solvability of one system of non-linear integral equations on the real axis

A system of singular integral equations with monotonic and convex non-linearity on the entire real line is considered. System of this form have applications in many areas of natural science. In particular, such systems arise in the theory of $p$-adic open-closed strings, in the mathematical theory of spatial-temporal epidemic spread within the framework of the well known Diekmann-Kaper model, in the kinetic theory of gases, in the radiative transfer theory. An existence theorem for a non-trivial and bounded solution is proved. The asymptotic behaviour of the constructed solution at $\pm\infty$ is also studied. Specific examples of non-linearities and kernel functions having an applied character are given.

Stability Analysis of the Solution to a System of Nonlinear Integral Equations Arising in a Logistic Dynamics Model

Stability Analysis of the Solution to a System of Nonlinear Integral Equations Arising in a Logistic Dynamics Model

On Solvability for Some Classes of System of Non-Linear Integral Equations in Two Dimensions via Measure of Non-Compactness

In this paper, we present some results of coupled fixed points for the system of non-linear integral equations in Banach space. Our results enlarge the results of newer papers. Additionally, we prove the applicability of those results to the solvability of the system of non-linear integral equations. Finally, we give an example to validate the applicability of our results.

Application of a Generalized Fixed Point Principle to the Study of a System of Nonlinear Integral Equations Arising in the Population Dynamics Model

Application of a Generalized Fixed Point Principle to the Study of a System of Nonlinear Integral Equations Arising in the Population Dynamics Model

On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel

On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel\Abstracteng{In this article the problems of the unique classical solvability and theconstruction of the solution of a nonlinear boundary value problem for a fifth orderpartial integro-differential equations with degenerate kernel are studied. Dirichletboundary conditions are specified with respect to the spatial variable. So, the Fourierseries method, based on the separation of variables is used. A countable system of~thesecond order ordinary integro-differential equations with degenerate kernel is obtained.The method of degenerate kernel is applied to this countable system of ordinaryintegro-differential equations. A system of~countable systems of algebraic equations isderived. Then the countable system of nonlinear Fredholm integral equations is obtained.Iteration process of solving this integral equation is constructed. Sufficient coefficientconditions of the unique solvability of the countable system of nonlinear integralequations are established for the regular values of parameter. In proof of uniquesolvability of the obtained countable system of nonlinear integral equations the method ofsuccessive approximations in combination with the contraction mapping method is used.In the proof of the convergence of Fourier series the Cauchy--Schwarz and Besselinequalities are applied. The smoothness of solution of the boundary value problemis also proved.

Existence of Solutions for Nonlinear Integral Equations in Tempered Sequence Spaces via Generalized Darbo-Type Theorem

Two concepts—one of Darbo-type theorem and the other of Banach sequence spaces—play a very important and active role in ongoing research on existence problems. We first demonstrate the generalized Darbo-type fixed point theorems involving the concept of continuous functions. Keeping one of these theorems into our account, we study the existence of solutions of system of nonlinear integral equations in the setting of tempered sequence space. Moreover, a very interesting and illustrative example is designed to visualize our findings.