Hammerstein Integral Equations Research Articles

Projection and modified projection methods for nonlinear Hammerstein integral equations on the real line using Hermite polynomials

Many physical problems represented as initial and boundary value problems are usually solved by transforming them into integral equations on the real line. Therefore, this paper proposes polynomially based projection and modified projection methods to solve Hammerstein integral equations on the real line with sufficiently smooth kernels. The approximating operator employed is either the orthogonal projection or an interpolatory projection using Hermite polynomials as basis functions. We analyse the convergence of the proposed approaches and its iterated version and we establish superconvergence results. Through different numerical tests, the effectiveness of the proposed methods is presented to demonstrate the given theoretical framework.

A hybrid Krasnosel’skiĭ-Schauder fixed point theorem for systems

We provide new results regarding the localization of the solutions of nonlinear operator systems. We make use of a combination of Krasnosel’skiĭ cone compression–expansion type methodologies and Schauder-type ones. In particular we establish a localization of the solution of the system within the product of a conical shell and of a closed convex set. By iterating this procedure we prove the existence of multiple solutions. We illustrate our theoretical results by applying them to the solvability of systems of Hammerstein integral equations. In the case of two specific boundary value problems and with given nonlinearities, we are also able to obtain a numerical solution, consistent with our theoretical results.

Discrete projection methods for Fredholm–Hammerstein integral equations using Kumar and Sloan technique

Discrete projection methods for Fredholm–Hammerstein integral equations using Kumar and Sloan technique

Mapped Gegenbauer functions for solving Hammerstein generalized integral equations with Green’s kernels on the whole line

This paper introduces a collocation method for numerically solving Hammerstein generalized integral equations on the whole line. The approach utilizes mapped Gegenbauer functions to transform the Hammerstein integral equation into a system of nonlinear algebraic equations. Additionally, convergence results are obtained using the standard L2-norm. Finally, the effectiveness of the proposed method is demonstrated through several numerical experiments, which also serve to illustrate our theoretical findings.

Asymptotically Stable Solutions of Infinite Systems of Quadratic Hammerstein Integral Equations

In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real axis with values in the classical Banach sequence space consisting of real bounded sequences. The main tool used in our investigations is the technique associated with the measures of noncompactness (MNCs) and a fixed point theorem of Darbo type. The applicability of our result is illustrated by a suitable example at the end of the paper.

AN ALGORITHM FOR APPROXIMATING SOLUTIONS OF SPLIT HAMMERSTEIN INTEGRAL EQUATIONS

In this paper, we introduce split Hammerstein integral equations of the form: u ∈ H_1 such that u + K_1F_1u = 0 and A(u) + K_2F_2(Au) = 0, where K_1, F_1 are maximal monotone maps defined on a real Hilbert space H_1, with D(K_1) = D(F_1) = H_1; K_2, F_2 are maximal monotone maps defined on a real Hilbert space H_2, with D(K_2) = D(F_2) = H_2 and A, a bounded linear map from H_1 to H_2. The sequence of the algorithm is proved to converge strongly to a solution of the split Hammerstein integral equation. As far as we know, this is the first algorithm whose sequence approximates a solution of a split Hammerstein integral equation in this direction. Finally, the theorem proved, improves and complements some important related recent results in the literature.

Approximated superconvergent methods for Volterra Hammerstein integral equations

In this article, we apply Kumar–Sloan based Galerkin and multi-Galerkin methods for solving the nonlinear Volterra–Hammerstein integral equations with both smooth and weakly singular kernels. For smooth kernels, we use piecewise polynomials basis functions based on uniform mesh, whereas for weakly singular kernels, we use piecewise polynomials basis functions based on graded mesh, to derive the superconvergence results. We obtain the improved superconvergence rates using Kumar–Sloan based Galerkin and multi-Galerkin methods without going to the iterated Galerkin and iterated multi-Galerkin methods. Infact without going to the iterated versions, we obtain the superconvergence rates in this method as in the case of iterated Galerkin and iterated multi-Galerkin methods. Numerical examples are given to verify the theoretical results.

Solving non-differentiable Hammerstein integral equations via first-order divided differences

Solving non-differentiable Hammerstein integral equations via first-order divided differences

Solving Integral Equations via Fixed Point Results Involving Rational-Type Inequalities

In this study, we establish unique and common fixed point results in the context of a complete complex-valued b-metric space using rational-type inequalities. The presented work generalizes some well-known results from the existing literature. Furthermore, to ensure the validity of the findings, we have included some examples and a section on the existence of solutions for the systems of Volterra–Hammerstein integral equations and Urysohn integral equations, respectively.

Discrete Galerkin and Iterated Discrete Galerkin Methods for Derivative-Dependent Fredholm–Hammerstein Integral Equations with Green’s Kernel

Discrete Galerkin and Iterated Discrete Galerkin Methods for Derivative-Dependent Fredholm–Hammerstein Integral Equations with Green’s Kernel

Superconvergence results for non-linear Hammerstein integral equations on unbounded domain

Superconvergence results for non-linear Hammerstein integral equations on unbounded domain

Spectral Methods For Hammerstein Integral Equations with Nonsmooth Kernels

In this paper, polynomial-based projection type and modified projection-type methods for approximating the solution of Hammerstein integral equations with a kernel of Green’s function type are proposed. The projection is either an orthogonal projection or an interpolatory projection using Legendre polynomial basis. The orders of convergence of these methods and those of superconvergence of the iterated modified projection-type methods are analyzed. A numerical example is given to illustrate the theoretical estimates.

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a General Discontinuous Kernel

In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, based on the Banach fixed point theorem. After applying the properties of a fractional integral, the Fr-NMIDE conformed to the Volterra–Hammerstein integral equation (V-HIE) of the second kind, with a general discontinuous kernel in position with the Hammerstein integral term and a continuous kernel in time to the Volterra term. Then, using a technique of the separating method, we obtained HIE, where its physical coefficients were variable in time. The Toeplitz matrix method (TMM) and its schemes were used to obtain a nonlinear algebraic system by studying the convergence of the system. The Maple 18 program was implemented to present the numerical results, along with corresponding errors.

A class of bijective Lotka–Volterra operators and its application

It is well known that any classical Lotka–Volterra (LV) operator (associated with quadratic stochastic operator) defined on the simplex is a homeomorphism. On the other hand, more general LV systems have important applications in the time evolution of conflicting species in biology. It is natural to study the bijectivity of such kind of LV operators. There is an example of a LV operator which is not injective. In this paper, we introduce a class of LV operators that are bijective. As an application of our result, the existence and uniqueness of solution of a class of Hammerstein integral equations is proved.

Superconvergent multi-Galerkin method for nonlinear Fredholm–Hammerstein integral equations

In this paper, we apply the multi-Galerkin method with Kumar–Sloan technique to improve the superconvergence rates and find the numerical solution of non-linear Fredholm–Hammerstein integral equations for both smooth as well as the weakly singular kernels. Considering piecewise polynomial as basis function of the approximating subspace, we derive the improved superconvergence rates for multi-Galerkin method based on Kumar–Sloan approximations which are exactly the same superconvergence rates for the approximate solution as in the case of iterated multi-Galerkin method. We have shown improved superconvergence results without the need for iterated multi-Galerkin method. Theoretical results are verified using numerical examples.

On the measure of noncompactness in $L_p(\mathbb{R}^+)$ and applications to a product of $n$-integral equations

In this article, we prove a new compactness criterion in the Lebesgue spaces $L_p({\mathbb{R}}^+), 1 \leq p < \infty$ and use such criteria to construct a measure of noncompactness in the mentioned spaces. The conjunction of that measure with the Hausdroff measure of noncompactness is proved on sets that are compact in finite measure. We apply such measure with a modified version of Darbo fixed point theorem in proving the existence of monotonic integrable solutions for a product of $n$-Hammerstein integral equations $n\geq 2$.

On the superposition operator in the space of functions of Hωq ([0,1

In this paper, we obtained a necessary and sufficient condition for the embedding H?q([0,1])? IBVqp([0,1]), where IBVq p denotes the set of functions of bounded q-integral p-variation. Additionally, the conditions for the composition and superposition operators were provided to map the space H?q([0,1]) into itself, by which these operators were bounded. Finally, we applied these results to examine the existence and uniqueness of solutions to Hammerstein integral equations in the space of H?q([0,1]).

Iterative Numerical Methods for a Fredholm–Hammerstein Integral Equation with Modified Argument

Iterative processes are a powerful tool for providing numerical methods for integral equations of the second kind. Integral equations with symmetric kernels are extensively used to model problems, e.g., optimization, electronic and optic problems. We analyze iterative methods for Fredholm–Hammerstein integral equations with modified argument. The approximation consists of two parts, a fixed point result and a quadrature formula. We derive a method that uses a Picard iterative process and the trapezium numerical integration formula, for which we prove convergence and give error estimates. Numerical experiments show the applicability of the method and the agreement with the theoretical results.

Numerical solution of two-dimensional Hammerstein integral equations via quadratic spline collocation

Numerical solution of two-dimensional Hammerstein integral equations via quadratic spline collocation

ON SOLVABILITY OF QUADRATIC HAMMERSTEIN INTEGRAL EQUATIONS IN HÖLDER SPACES

Using Schauder's fixed point theorem we consider the solvability of a quadratic Hammerstein integral equation in the space of functions satisfying a H\"{o}lder condition. An example is included to illustrate our results.