Solvability of Some Nonlinear Integral Functional Equations

In this paper, we prove the local attractivity of solutions for a certain nonlinear Volterra type functional integral equations. We rely on a measure theoretic fixed point theorem of Dhage (2008) for nonlinear D-set-contraction in Banach spaces. Finally, we furnish an example to validate all the hypotheses of our main result and to ensure the existence and ultimate attractivity of solutions for a numerical nonlinear functional integral equation.

The Numerical Solution of Integral Equations on the Half-Line

We prove using hybrid fixed point theorem in Banach algebra an existence of solutions to fractional order quadratic functional integral equation in Also locally attractivity solutions for fractional order of nonlinear functional quadratic functional integral equations is proved by using Dhage’s Hybrid fixed point theorem.

In this paper, Darbo fixed-point theorem is employed as a mathematical tool to examine existence of the solution of some nonlinear functional stochastic integral equations which arise in many physical, chemical and biological problems. Throughout this paper, we consider (C[0, 1]; ∥.∥ ) as Banach space equipped with uniform norm.

Numerial Methods for Volterra Integral Equations with Singular Kernels

In this article, we establish some results for the existence of solution of nonlinear functional integral equations by using Darbo’s fixed point theorem in Banach algebra. As an application, we discuss some examples of nonlinear functional integral equations and existence of solutions.

Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn type

Nonlinear functional-integral equations contain the unknown function nonlinearly, occurring extensively in theory developed to a certain extent toward different applied problems and solutions thereof. Nonlinear science serves to reveal the nonlinear descriptions of widely different systems, has had a fundamental impact on complex dynamics. Accordingly, this study provides the basic definitions and results regarding Banach spaces with the proof as well as applications. Besides, proving the existence and uniqueness of the solutions by the Banach theorem is carried out. The existence results of the equations are generally obtained based on fundamental methods by which the fixed-point theorems are frequently applied. Subsequently, the definition of the measure pertaining to noncompactness on Banach space along with properties is presented. Hence, the aim is to study nonlinear functional-integral equations in the Banach space [Formula: see text] using the measure of noncompactness. In addition, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to [Formula: see text] is looked into. As the last stage, the definitions of global attractivity and asymptotical stability for equations on Banach space [Formula: see text] are given while proving the existence of solutions and global attractivity. Besides these, the existence and asymptotical stability are proven. Furthermore, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to [Formula: see text] for an example nonlinear integral equation has been demonstrated by one example premediated and the existence of blow-up point has been examined in one and more than one points regarding the nonlinear integral equation. The example nonlinear integral equation as addressed in this study has been examined in terms of its blow-up point and asymptotic stability with computational complexity for the first time. By doing so, it has been demonstrated that the existence of blow-up point has been the case in one and more than one points and besides this, the examination of its asymptotic stability has shown that it is asymptotically stable at least in one point, yet has no blow-up point pertaining to [Formula: see text]. This has enabled thorough analysis of computational complexity of the related nonlinear functional-integral equations within the example both addressing dimension of input in Banach space and that of dataset with [Formula: see text] dimension. This applicable approach assures the accurate transformation of the complex problems towards optimized solutions through the generation of advanced mathematical models in nonlinear dynamic settings characterized by complexity.

This work proposes a new defintion of the nonlinear functional Volterra integral equation ib 2D of the second kind with continuous kernel (NT-DFVIE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions are obtained by two powerful methods Adomian Decomposition Method (ADM) and Block by block method (BBM). The given numerical examples showed the efficiency and accuracy of the introduced methods.

This study introduces a novel implicit Jungck-type iterative scheme for solving nonlinear functional integral equations with double delays. By incorporating weak contractive conditions, the proposed scheme enhances computational efficiency and stability while achieving convergence equivalent to its explicit counterpart. Compared to existing Jungck-type explicit schemes, the implicit iteration exhibits superior stability and faster convergence.This work addresses key limitations in previous studies (Alam, Rohen in J. Appl. Math. Comput., https://doi.org/10.1007/s12190-024-02134-z, 2024; Copur et al. in Math. Comput. Simul. 215:476–497, 2024; Sharma et al. in Contemp. Math. 5:743–760, 2024) by:1. Establishing fixed-point results for a broader class of mappings with guaranteed existence and uniqueness of fixed point,2. Refining the stability framework for more reliable theoretical outcomes,3. Removing restrictive conditions on control sequences to improve convergence, and4. Deriving sharper estimates for data dependence, ensuring greater accuracy.Theoretical analysis within the weak w2-stability framework is supported by numerical experiments, including boundary value problems, confirming the scheme’s enhanced stability and efficiency. By refining and extending recent developments in fixed-point and coincidence-point theory, this study provides a robust framework for solving complex nonlinear problems.

An iterative solution of a particular non-linear Fredholm integral equation is considered. The method is based on the reduction of the equation for the residual at any stage of the iteration to a linear integral equation for the required perturbation, using Newton' s method. This equation is then solved by using a low order expansion in terms of Chebyshev polynomials. The numerical integrations required in the basic iterative process are carried out by the powerful Clenshaw-Curtis quadrature prescription. The non-linear integral equation considered arises from the unstable two-point boundary value problem for a tubular chemical reactor. The capacity of the present method in dealing with such numerical difficulties is illustrated in representative calculations. Comparisons are made with conventional solutions of the parent differential equation and also with attempts at direct iterative solution of the integral equation by standard Neumann-Liouville series and finally with the direct algebraic approach.

In this paper, we generalize the Meir-Keeler condensing operators via a concept of the class of operators $ O (f;.)$, that was given by Altun and Turkoglu [4], and apply this extension to obtain some tripled fixed point theorems. As an application of this extension, we analyze the existence of solution for a system of nonlinear functional integral equations of Volterra type. Finally, we present an example to show the effectiveness of our results. We use the technique of measure of noncompactness to obtain our results.

Under the idea of a measure of noncompactness, some fixed point results are proposed and a generalization of Darbo’s fixed point theorem is given in this manuscript. Furthermore, some novel quadruple fixed points results via a measure of noncompactness for a general class of functions are presented. Ultimately, the solutions to a system of non-linear functional integral equations by the fixed point results obtained are discussed, and non-trivial examples to illustrate the validity of our study are derived.

Existence of solutions of functional integral equations of convolution type using a new construction of a measure of noncompactness on [formula omitted

Many different problems in mathematics, physics, engineering can be expressed in the form of integral equations.Among these are diffraction problems, population growth, heat transfer, particle transport problems, electrical engineering, elasticity, control, elastic waves, diffusion problems, quantum mechanics, heat radiation, electrostatics and contact problems.Therefore, the solutions which are obtained by the mathematical methods play an important role in these fields.The most two basic types of integral equations are called Fredholm (FIEs) and Volterra (VIEs).In many instances, the ordinary and partial differential equations can be converted into Fredhom and Volterra integral equations that are solved more effectively.We aim through this research to present an improved Adomian decomposition method based on modified Bernstein polynomials (ADM-MBP) to solve nonlinear integral equations of the second kind.We introduced efficient method, constructed on modified Bernstein polynomials.The formulation is developed to solve nonlinear Fredholm and Volterra integral equations of second kind.This method is tested for some examples from nonlinear integral equations.Maple software was used to obtain the solutions of these examples.The results demonstrate reliability of the proposed method.Generally, the proposed method is very convenient to apply to find the solutions of Fredholm and Volterra integral equations of second kind.