Nonlinear Integral Equations Research Articles

The $ AA $-iterative algorithm in hyperbolic spaces with applications to integral equations on time scales

<p>We explored the $ AA $-iterative algorithm within the hyperbolic spaces (HSs), aiming to unveil a stability outcome for contraction maps and convergence outcomes for generalized $ (\alpha, \beta) $-nonexpansive ($ G\alpha \beta N $) maps in such spaces. Through this algorithm, we derived compelling outcomes for both strong and $ \Delta $-convergence and weak $ w^2 $-stability. Furthermore, we provided an illustrative example of $ G\alpha \beta N $ maps and conducted a comparative analysis of convergence rates against alternative iterative methods. Additionally, we demonstrated the practical relevance of our findings by applying them to solve the linear Fredholm integral equations (FIEs) and nonlinear Fredholm-Hammerstein integral equations (FHIEs) on time scales.</p>

On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems

<abstract><p>The main aim of this paper is to study the Cauchy problem for nonlinear differential equations of fractional order containing the weighted Riemann-Liouville fractional derivative of a function with respect to another function. The equivalence of this problem and a nonlinear Volterra-type integral equation of the second kind have been presented. In addition, the existence and uniqueness of the solution to the considered Cauchy problem are proved using Banach's fixed point theorem and the method of successive approximations. Finally, we obtain a new estimate of the weighted Riemann-Liouville fractional derivative of a function with respect to functions at their extreme points. With the assistance of the estimate obtained, we develop the comparison theorems of fractional differential inequalities, strict as well as nonstrict, involving weighted Riemann-Liouville differential operators of a function with respect to functions of order $ \delta $, $ 0 < \delta < 1 $.</p></abstract>

Rezultati fiksne tačke za β - F-slaba mapiranja kontrakcije u potpunim S-metričkim prostorima

Introduction/purpose: This paper introduces the concept of β - F-weak contraction by using the concepts of F-weak contraction and a - pscontraction. Methods: The use of the β - F-weak contraction proves some fixed points theorems in the framework of S-metric spaces. Results: The obtained results on fixed points in S-metric spaces generalize some known results in the literature. Conclusions: The β - F-weak contraction generalizes some important contraction types and examines the existence of a fixed point in S-metric spaces. The results are used to solve a non-linear Fredholm integral equation.

Improved composite methods using radial basis functions for solving nonlinear Volterra-Fredholm integral equations

Improved composite methods using radial basis functions for solving nonlinear Volterra-Fredholm integral equations

A new approach of overlapping generation model via fixed point technique

<abstract><p>This article explored a specific variant of the overlapping generation model using a nonlinear Fredholm integral equation. We considered assumptions related to the Ćirić operator, offering a new perspective compared to existing research. To solve the equation, we employed the Galerkin method, which approximates it as a finite system of equations. By combining these approaches, we conducted a comprehensive analysis of the model, providing insights into its dynamics and potential applications.</p></abstract>

A study of the multipoint boundary value problem for nonlinear differential equations of arbitrary fractional order based on the fixed point theorem for composite nonlinear operators

Abstract The existence and distinctiveness of solutions, as well as the iterative approximations of the distinctive answer to the initial boundary value issue of nonlinear differential equations, may be solved with the help of the theory of nonlinear operators, which offers a strong theoretical guarantee and a fundamental tool. In this work, we examine the fixed point theory for composites nonlinear operators, which may offer a fresh approach to solving linear differential equations having arbitrary order fractional multipoint boundary value problems. The existing result of the boundary value issue’s solution is confirmed by the fixed point theorem for composites nonlinear operators, and the four-point boundary value problem for fractional-order linear equations of the Riemann-Liouville type is examined. A group of fractional-order nonlinear differential equations with deviating quantities are examined to examine the presence of an unusual positive solution for this boundary value problem. The existence of this unique positive solution is determined by the use of both the mixed monotone operator and the combined nonlinear operator fixed point theorem. Ultimately, the nonlinear Bagley-Torvik equation with a fractional order variable coefficient four-point boundary value problem is converted into a Fredholm-Hammerstein integral the formula of the second type with weakly singular or continuous kernels, and the fixed point principle is used to demonstrate the uniqueness of the solution in the space of continuous functions. Approximate solutions are provided for the second class of nonlinear Fredholm-Hammerstein integral equations with weakly singular kernels. The outcomes demonstrate the applicability and validity of the fixed point theorem to composite nonlinear operators, offering a fresh approach to solving the multipoint boundary value issue for nonlinear differential equations of any fractional order.

SOME NEW TYPES OF GRONWALL-BELLMAN INEQUALITY ON FRACTAL SET

Gronwall–Bellman-type inequalities provide a very effective way to investigate the qualitative and quantitative properties of solutions of nonlinear integral and differential equations. In recent years, local fractional calculus has attracted the attention of many researchers. In this paper, based on the basic knowledge of local fractional calculus and the method of proving inequality on the set of real numbers, some new Gronwall–Bellman inequalities are discussed and established on the fractal space. Our results are a powerful supplement to the existing literature.

A Qualitative Study of Solutions of Nonlinear Volterra Integral Equation in the Plane, from the Perspective of Shiba Bounded Variation

In this article a nonlinear Volterra integral equation is studied in the space of functions of the bounded variation in the Shiba sense on the plane. A series of conditions is established for the functions and the kernel involved in this equation which guarantee the global existence and uniqueness of the solution in this space.

Fixed point and endpoint theorems of multivalued mappings in convex $ b $-metric spaces with an application

<abstract><p>In this paper, we investigated several new fixed points theorems for multivalued mappings in the framework $ b $-metric spaces. We first generalized $ S $-iterative schemes for multivalued mappings to above spaces by means of a convex structure and then we developed the existence of fixed points and approximate endpoints of the multivalued contraction mappings using iteration techniques. Furthermore, we introduced the modified $ S $-iteration process for approximating a common endpoint of a multivalued $ \alpha_{s} $-nonexpansive mapping and a multivalued mapping satisfying conditon $ (E^{'}) $. We also showed that this new iteration process converges faster than the $ S $-iteration process in the sense of Berinde. Some convergence results for this iterative procedure to a common endpoint under some certain additional hypotheses were proved. As an application, we applied the $ S $-iteration process in finding the solution to a class of nonlinear quadratic integral equations.</p></abstract>

A study of (2+1)-dimensional variable coefficients equation: Its oceanic solitons and localized wave solutions

In this work, shallow ocean-wave soliton, breather, and lump wave solutions, as well as the characteristics of interaction between the soliton and lump wave in a multi-dimensional nonlinear integrable equation with time-variable coefficients, are investigated. The Painlevé analysis is used to verify the integrability of this model. Based on the bilinear form of this model, we use the simplified Hirota's method obtained from the perturbation approach and various auxiliary functions to construct the aforementioned solutions. Besides, the interaction between the soliton and lump wave solutions is also examined. In addition, by imposing specific constraint conditions on the N-soliton solutions, we further derive higher-order breather solutions. To show the physical characteristics of this model, several graphical representations of the discovered solutions are established. These graphs show that the time-variable coefficients result in a variety of novel dynamic behaviors that differ significantly from those for integrable equations with constant coefficients. The acquired results are useful for the study of shallow water waves in fluid dynamics, marine engineering, nonlinear sciences, and ocean physics.

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

Best proximity point of $ \alpha $-$ \eta $-type generalized $ F $-proximal contractions in modular metric spaces

<abstract><p>The purpose of this paper is to present a study of $ \alpha $-$ \eta $-type generalized $ F $-proximal contraction mappings in the framework of modular metric spaces and to prove some best proximity point theorems for these types of mappings. Some examples are given to show the validity of our results. We also apply our results to establish the existence of solutions for a certain type of non-linear integral equation.</p></abstract>

Double composed metric-like spaces via some fixed point theorems

<p>The manuscript introduces the concept of a double-composed metric-like space, which is an extension of the notion of a double-composed metric space. In this new space, the self-distance is not necessarily zero, but if the distance metric equals zero, it must be for identical points of distance. Furthermore, this paper presents several results related to this novel concept in the literature, with a particular focus on Hardy–Rogers type contractions. It establishes fixed point theorems accompanied by some illustrative examples that elucidate the findings. Finally, this research provides an application to nonlinear integral equation to substantiate our theorems.</p>

CONTACT OF AN ANISOTROPIC LAYER WITH A RIGID PERIODICALLY CONVEX SURFACE

By using Fourier integral transforms applied to integral equations, exact solutions are constructed for spatial problems of full contact of a transversely isotropic layer with a rigid doubly periodic sinusoidal surface. One layer face is subjected to sliding or rigid support, and isotropy planes are parallel to the layer faces. It is established that the full contact over the whole layer face is always possible except the degenerate case of isotropic incompressible layer with one face rigidly fixed. It is indicated that the full contact is also realizable for the case of isotropy planes perpendicular to the layer faces when one face is subjected to sliding support. The problems of partial contact are also considered for the transversely isotropic layer with a rigid surface relief of which is described by a doubly periodic function in the form of a system of paraboloids of revolution. On the base of the theory of generalized functions, the problems are reduced to integral equations kernels of which do not contain integrals. The complication of the problems of partial contact is that the contact domain is a priori unknown that leads us to additional inequalities and nonlinearities. With the help of the B.A. Galanov method, the system of the integral equation and inequality can be reduced to only one nonlinear integral equation for which the modified Newton numerical method can be applied. The exact solutions for the full contact are usable to check calculations. Contact domains and contact pressures are determined. Mechanical characteristics are calculated for the percolation process of initially discrete contact spots in the strengthening contact. Minimal settlements of the paraboloids of revolution are calculated to begin or complete percolation for an anisotropic material.

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.

Solving hybrid functional-fractional equations originating in biological population dynamics with an effect on infectious diseases

<abstract><p>This paper study was designed to establish solutions for mixed functional fractional integral equations that involve the Riemann-Liouville fractional operator and the Erdélyi-Kober fractional operator to describe biological population dynamics in Banach space. The results rely on the measure of non-compactness and theoretical concepts from fractional calculus. Darbo's fixed-point theorem for Banach spaces has been utilized. Moreover, the solvability of a specific non-linear integral equation that models the spread of infectious diseases with a seasonally varying periodic contraction rate has been explored by using the Banach contraction principle. Finally, two numerical examples demonstrate the practical application of these findings in the realm of fractional integral equation theory.</p></abstract>

SOLUTION OF INTEGRAL EQUATION INVOLVINGINTERPOLATIVE ENRICHED CYCLIC KANNANCONTRACTION MAPPINGS

The purpose of this paper is to introduce the class of interpolative enriched cyclic Kannan contraction mappings defined on an Banach space and to prove the existence and uniqueness of fixed point of the such mappings. An example is presented to support the concept introduced herein. Moreover, an application of the main result to solve nonlinear integral equations is also given. Our result extend and generalize various results in the existing literature.

NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEM FOR A SECOND ORDER IMPULSIVE SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS WITH MIXED MAXIMA

A two-point nonlinear boundary value problem for a second order system of ordinary integro-differential equations with impulsive effects and mixed maxima is investigated. By applying some transformations is obtained a system of nonlinear functional integral equations. The existence and uniqueness of the solution of the nonperiodic two-point boundary value problem are reduced to the one valued solvability of the system of nonlinear functional integral equations in Banach space . The method of successive approximations in combination with the method of compressing mapping is used in the proof of one-valued solvability of nonlinear functional integral equations.

Correction to: Lagrange interpolation polynomials for solving nonlinear stochastic integral equations

Correction to: Lagrange interpolation polynomials for solving nonlinear stochastic integral equations

Resurgent aspects of applied exponential asymptotics

AbstractIn many physical problems, it is important to capture exponentially small effects that lie beyond‐all‐orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans‐series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans‐series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans‐series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial‐over‐power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial‐over‐power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences.