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Journal articles on the topic 'The Hammerstein integral equation'

Author: Grafiati

Published: 4 May 2022

Last updated: 18 July 2025

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1

Marian, Daniela, Sorina Anamaria Ciplea, and Nicolaie Lungu. "On a Functional Integral Equation." Symmetry 13, no. 8 (2021): 1321. http://dx.doi.org/10.3390/sym13081321.

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In this paper, we establish some results for a Volterra–Hammerstein integral equation with modified arguments: existence and uniqueness, integral inequalities, monotony and Ulam-Hyers-Rassias stability. We emphasize that many problems from the domain of symmetry are modeled by differential and integral equations and those are approached in the stability point of view. In the literature, Fredholm, Volterra and Hammerstein integrals equations with symmetric kernels are studied. Our results can be applied as particular cases to these equations.

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2

Borkowski, Marcin, and Daria Bugajewska. "Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations." Mathematica Slovaca 68, no. 1 (2018): 77–88. http://dx.doi.org/10.1515/ms-2017-0082.

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Abstract In this paper we are going to apply the Henstock-Kurzweil integrals defined on an unbounded intervals to differential and integral equations defined on such intervals. To deal with linear differential equations we examine convolution involving functions integrable in Henstock-Kurzweil sense. In the case of nonlinear Hammerstein integral equation as well as Volterra integral equation we look for solutions in the space of functions of bounded variation in the sense of Jordan.

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3

Tunç, Osman, Cemil Tunç, and Jen-Chih Yao. "New Results on Ulam Stabilities of Nonlinear Integral Equations." Mathematics 12, no. 5 (2024): 682. http://dx.doi.org/10.3390/math12050682.

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This article deals with the study of Hyers–Ulam stability (HU stability) and Hyers–Ulam–Rassias stability (HUR stability) for two classes of nonlinear Volterra integral equations (VIEqs), which are Hammerstein-type integral and Hammerstein-type functional integral equations, respectively. In this article, both the HU stability and HUR stability are obtained for the first integral equation and the HUR stability is obtained for the second integral equation. Among the used techniques, we present fixed point arguments and the Gronwall lemma as a basic tool. Two supporting examples are also provided to demonstrate the applications and effectiveness of the results.

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4

Sahu, P. K., A. K. Ranjan, and S. Saha Ray. "B-spline Wavelet Method for Solving Fredholm Hammerstein Integral Equation Arising from Chemical Reactor Theory." Nonlinear Engineering 7, no. 3 (2018): 163–69. http://dx.doi.org/10.1515/nleng-2017-0116.

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Abstract Mathematical model for an adiabatic tubular chemical reactor which processes an irreversible exothermic chemical reaction has been considered. For steady state solution for an adiabatic tubular chemical reactor, the model can be reduced to ordinary differential equation with a parameter in the boundary conditions. Again the ordinary differential equation has been converted into a Hammerstein integral equation which can be solved numerically. B-spline wavelet method has been developed to approximate the solution of Hammerstein integral equation. This method reduces the integral equation to a system of algebraic equations. The numerical results obtained by the present method have been compared with the available results.

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5

Miranda, José Carlos Simon de. "A mixed Hammerstein integral equation." São Paulo Journal of Mathematical Sciences 2, no. 2 (2008): 145. http://dx.doi.org/10.11606/issn.2316-9028.v2i2p145-160.

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6

Tunç, Osman, and Cemil Tunç. "On Ulam Stabilities of Delay Hammerstein Integral Equation." Symmetry 15, no. 9 (2023): 1736. http://dx.doi.org/10.3390/sym15091736.

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In this paper, we consider a Hammerstein integral equation (Hammerstein IE) in two variables with two variables of time delays. The aim of this paper is to investigate the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of the considered IE via Banach’s fixed point theorem (Banach’s FPT) and the Bielecki metric. The proofs of the new outcomes of this paper are based on these two basic tools. As the new contributions of the present study, here, for the first time, we develop the outcomes that can be found in the earlier literature on the Hammerstein IE, including variable time delays. The present study also includes complementary outcomes for the symmetry of Hammerstein IEs. Finally, a concrete example is given at the end of this study for illustrations.

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7

Amer, Yasser. "Hermite collocation method for solving Hammerstein integral equations." JOURNAL OF ADVANCES IN MATHEMATICS 14, no. 1 (2018): 7413–23. http://dx.doi.org/10.24297/jam.v14i1.6716.

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In this paper, we are presenting Hermite collocation method to solve numer- ically the Fredholm-Volterra-Hammerstein integral equations. We have clearly presented a theory to …nd ordinary derivatives. This method is based on replace- ment of the unknown function by truncated series of well known Hermite expan-sion of functions. The proposed method converts the equation to matrix equation which corresponding to system of algebraic equations with Hermite coe¢ cients. Thus, by solving the matrix equation, Hermite coe¢ cients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

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8

Kant, Kapil, Rakesh Kumar, and B. V. Rathish Kumar. "Convergence Analysis for Linear Fredholm and Nonlinear Fredholm Hammerstein Integral Equations." Journal of Basic & Applied Sciences 18 (December 29, 2022): 158–65. http://dx.doi.org/10.29169/1927-5129.2022.18.16.

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In this article, we consider the linear Fredholm integral equations and Fredholm-Hammerstein’s integral equations. We propose the Legendre polynomial based degenerate kernel method to solve linear Fredholm and Fredholm-Hammerstein integral equations. We discuss the convergence and error analysis of the proposed method and also obtain the superconvergence results for iterated degenerate kernel method.

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9

Pachpatte, B. G. "On a generalized Hammerstein-type integral equation." Journal of Mathematical Analysis and Applications 106, no. 1 (1985): 85–90. http://dx.doi.org/10.1016/0022-247x(85)90132-5.

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10

RASHIDINIA, J., and ALI PARSA. "SEMI-ORTHOGONAL SPLINE SCALING FUNCTIONS FOR SOLVING HAMMERSTEIN INTEGRAL EQUATIONS." International Journal of Wavelets, Multiresolution and Information Processing 09, no. 03 (2011): 427–43. http://dx.doi.org/10.1142/s0219691311004134.

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We developed a new numerical procedure based on the quadratic semi-orthogonal B-spline scaling functions for solving a class of nonlinear integral equations of the Hammerstein-type. Properties of the B-spline wavelet method are utilized to reduce the Hammerstein equations to some algebraic equations. The advantage of our method is that the dimension of the arising algebraic equation is 10 × 10. The operational matrix of semi-orthogonal B-spline scaling functions is sparse which is easily applicable. Error estimation of the presented method is analyzed and proved. To demonstrate the validity and applicability of the technique the method applied to some illustrative examples and the maximum absolute error in the solutions are compared with the results in existing methods.20,25,27,29

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11

Navascués, María A. "Hammerstein Nonlinear Integral Equations and Iterative Methods for the Computation of Common Fixed Points." Axioms 14, no. 3 (2025): 214. https://doi.org/10.3390/axioms14030214.

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In the first part of this article, a special type of Hammerstein nonlinear integral equation is studied. A theorem of the existence of solutions is given in the framework of L2-spaces. Afterwards, an iterative method for the resolution of this kind of equations is considered, and the convergence of this algorithm towards a solution of the equation is proved. The rest of the paper considers two modifications of the algorithm. The first one is devoted to the sought of common fixed points of a family of nearly asymptotically nonexpansive mappings. The second variant focuses on the search of common fixed points of a finite number of nonexpansive operators. The characteristics of convergence of these methods are studied in the context of uniformly convex Banach spaces. The iterative scheme is applied to approach the common solution of three nonlinear integral equations of Hammerstein type.

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12

Jan, A. R., M. A. Abdou, and M. Basseem. "A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a Quadrature Nystrom Method." Fractal and Fractional 7, no. 9 (2023): 656. http://dx.doi.org/10.3390/fractalfract7090656.

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In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space L2Ω×C0,T, T<1. The FrNMIoDE conformed to the Volterra-Hammerstein integral equation (V-HIE) of the second kind, after applying the characteristics of a fractional integral, with a general discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time to the Volterra integral (VI) term. Then, using a separation technique methodology, we developed HIE, whose physical coefficients were time-variable. By examining the system’s convergence, the product Nystrom technique (PNT) and associated schemes were employed to create a nonlinear algebraic system (NAS).

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13

Szufla, Stanislaw. "On the HAMMERSTEIN Integral Equation in BANACH Spaces." Mathematische Nachrichten 124, no. 1 (1985): 7–14. http://dx.doi.org/10.1002/mana.19851240102.

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14

Casey, Kendall F. "Periodic traveling-wave solutions to the Whitham equation." Mathematical Modelling of Natural Phenomena 13, no. 2 (2018): 16. http://dx.doi.org/10.1051/mmnp/2018026.

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We investigate periodic traveling-wave solutions to the Whitham equation. It is shown that for solutions of this type, the Whitham equation can be expressed as a nonlinear integral equation of Hammerstein form. Solutions to this integral equation are then obtained by iteration. Representative numerical results are presented to illustrate the waveshapes and the nonlinear dispersion characteristics of the solutions thus obtained.

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15

Micula, Sanda. "A numerical method for two-dimensional Hammerstein integral equations." Studia Universitatis Babes-Bolyai Matematica 66, no. 2 (2021): 267–77. http://dx.doi.org/10.24193/subbmath.2021.2.03.

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"In this paper we investigate a collocation method for the approximate solution of Hammerstein integral equations in two dimensions. As in [8], col- location is applied to a reformulation of the equation in a new unknown, thus reducing the computational cost and simplifying the implementation. We start with a special type of piecewise linear interpolation over triangles for a refor- mulation of the equation. This leads to a numerical integration scheme that can then be extended to any bounded domain in R2, which is used in collocation. We analyze and prove the convergence of the method and give error estimates. As the quadrature formula has a higher degree of precision than expected with linear interpolation, the resulting collocation method is superconvergent, thus requiring fewer iterations for a desired accuracy. We show the applicability of the proposed scheme on numerical examples and discuss future research ideas in this area."

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16

Li, Wanjun. "Eigenvalue of Coupled Systems for Hammerstein Integral Equation with Two Parameters." ISRN Mathematical Analysis 2011 (October 11, 2011): 1–11. http://dx.doi.org/10.5402/2011/989401.

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17

Alhazmi, Sharifah E., and Mohamed A. Abdou. "A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a General Discontinuous Kernel." Fractal and Fractional 7, no. 2 (2023): 173. http://dx.doi.org/10.3390/fractalfract7020173.

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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.

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18

Zhang, Jian, Jinjiao Hou, Jing Niu, Ruifeng Xie, and Xuefei Dai. "A high order approach for nonlinear Volterra-Hammerstein integral equations." AIMS Mathematics 7, no. 1 (2021): 1460–69. http://dx.doi.org/10.3934/math.2022086.

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<abstract><p>Here a scheme for solving the nonlinear integral equation of Volterra-Hammerstein type is given. We combine the related theories of homotopy perturbation method (HPM) with the simplified reproducing kernel method (SRKM). The nonlinear system can be transformed into linear equations by utilizing HPM. Based on the SRKM, we can solve these linear equations. Furthermore, we discuss convergence and error analysis of the HPM-SRKM. Finally, the feasibility of this method is verified by numerical examples.</p></abstract>

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19

Abduragimov, G. E. "On the existence and uniqueness of a positive solution to a boundary value problem with integral boundary conditions for one nonlinear ordinary differential equation of the second order." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 12 (January 4, 2025): 12–19. https://doi.org/10.26907/0021-3446-2024-12-12-19.

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The article considers a boundary value problem with integral boundary conditions for one nonlinear second-order functional differential equation. Using Green’s function, the boundary value problem is reduced to the equivalent nonlinear Hammerstein integral equation. Next, having identified the necessary properties of Green’s function, we prove that the Hammerstein operator contracts the corresponding cone. The last circumstance, by virtue of the well-known Krasnoselsky theorem, guarantees the existence of at least one positive solution to the boundary value problem. Using a priori estimates and the principle of compressed mappings, sufficient conditions for the uniqueness of a positive solution were obtained. At the end of the article, there is a non-trivial example illustrating the results obtained here.

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20

Banaś, Józef. "Integrable solutions of Hammerstein and Urysohn integral equations." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 46, no. 1 (1989): 61–68. http://dx.doi.org/10.1017/s1446788700030378.

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AbstractIn this paper we prove theorems on the existence of integrable and monotonic solutions of Hammserstein and Urysohn integral equations. The basic tool used in the proof is the fixed point principle for contractions with respect to the so-called measure of weak noncompactness.

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21

Darwish, Mohamed Abdalla, Mohamed M. A. Metwali, and Donal O'Regan. "ON SOLVABILITY OF QUADRATIC HAMMERSTEIN INTEGRAL EQUATIONS IN HÖLDER SPACES." Matematički Vesnik 74, no. 4 (2022): 242–48. http://dx.doi.org/10.57016/mv-nuyr4938.

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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.

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22

Minhós, Feliz, and Robert de Sousa. "Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line." Mathematics 8, no. 1 (2020): 111. http://dx.doi.org/10.3390/math8010111.

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In this work, we consider a generalized coupled system of integral equations of Hammerstein-type with, eventually, discontinuous nonlinearities. The main existence tool is Schauder’s fixed point theorem in the space of bounded and continuous functions with bounded and continuous derivatives on R , combined with the equiconvergence at ± ∞ to recover the compactness of the correspondent operators. To the best of our knowledge, it is the first time where coupled Hammerstein-type integral equations in real line are considered with nonlinearities depending on several derivatives of both variables and, moreover, the derivatives can be of different order on each variable and each equation. On the other hand, we emphasize that the kernel functions can change sign and their derivatives in order to the first variable may be discontinuous. The last section contains an application to a model to study the deflection of a coupled system of infinite beams.

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23

Al-Bugami, A. M. "Singular Hammerstein-Volterra Integral Equation and Its Numerical Processing." Journal of Applied Mathematics and Physics 09, no. 02 (2021): 379–90. http://dx.doi.org/10.4236/jamp.2021.92026.

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24

Liu, Xi-Lan. "On a nonlinear Hammerstein integral equation with a parameter." Nonlinear Analysis: Theory, Methods & Applications 70, no. 11 (2009): 3887–93. http://dx.doi.org/10.1016/j.na.2008.07.038.

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25

Jeribi, Aref, Bilel Krichen, and Bilel Mefteh. "Existence of Solutions of a Nonlinear Hammerstein Integral Equation." Numerical Functional Analysis and Optimization 35, no. 10 (2014): 1328–39. http://dx.doi.org/10.1080/01630563.2014.884582.

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26

Sidorov, N. A., and L. R. D. Dreglea Sidorov. "On the Solution of Hammerstein Integral Equations with Loads and Bifurcation Parameters." Bulletin of Irkutsk State University. Series Mathematics 43 (2022): 78–90. http://dx.doi.org/10.26516/1997-7670.2023.43.78.

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The Hammerstein integral equation with loads on the desired solution is considered. The equation contains a parameter for any value of which the equation has a trivial solution. Necessary and sufficient conditions are obtained for the coefficients of the equation and those values of the parameter (bifurcation points) in its neighborhood the equation has a nontrivial real solutions. The leading terms of the asymptotics of such branches of solutions are constructed. Examples are given illustrating the proven existence theorems

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27

Fimin, Nikolay Nikolaevich, and Valery Mihailovich Chechetkin. "Large–scale cosmological structures and Hammerstein–type equation for potential." Keldysh Institute Preprints, no. 79-e (2024): 1–18. https://doi.org/10.20948/prepr-2024-79-e.

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The criteria for the formation of stationary pseudo-periodic structures in a system of gravitating particles, described by the Vlasov–Poisson system of equations. Conditions studied branching solutions of a nonlinear integral equation for a generalized gravitational potential, leading to the emergence of coherent complex states of relative equilibrium in non-stationary systems of massive particles.

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28

Mollapourasl, Reza, and Joseph Siebor. "Numerical Solution of Nonlinear Quadratic Integral Equation of Hammerstein Type Based on Fixed-Point Scheme." Mathematics 13, no. 9 (2025): 1413. https://doi.org/10.3390/math13091413.

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Existence of the solution for the nonlinear quadratic integral equation of the Hammerstein type in the Banach space BC(R+) has been proved by using the technique of measure of noncompactness and fixed-point theorem. In this article, we obtain an approximate solution for the quadratic integral equation by using the Sinc method and the fixed-point technique. Moreover, the convergence of the numerical scheme for the solution of the integral equation is demonstrated by a theorem, and numerical experiments are presented to show the accuracy of the numerical scheme and guarantee the analytical results.

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29

Jones, Kenneth L., and Yunkai Chen. "Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation." International Journal of Mathematics and Mathematical Sciences 22, no. 3 (1999): 643–48. http://dx.doi.org/10.1155/s0161171299226439.

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The generalized forced Boussinesq equation,utt−uxx+[f(u)]xx+uxxxx=h0, and its periodic traveling wave solutions are considered. Using the transformz=x−ωt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein type integral equation is then established by using the Green's function method. This integral equation generates compact operators in a Banach space of real-valued continuous periodic functions with a given period2T. The Schauder's fixed point theorem is then used to prove the existence of solutions to the integral equation. Therefore, the existence of nonconstant periodic traveling wave solutions to the generalized forced Boussinesq equation is established.

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30

Balachandran, K., and S. Ilamaran. "A note on integrable solutions of Hammerstein integral equations." Proceedings of the Indian Academy of Sciences - Section A 105, no. 1 (1995): 99–103. http://dx.doi.org/10.1007/bf02840593.

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31

Karamolegos, Antonios, and Dimitrios Kravvaritis. "Nonlinear random operator equations and inequalities in Banach spaces." International Journal of Mathematics and Mathematical Sciences 15, no. 1 (1992): 111–18. http://dx.doi.org/10.1155/s0161171292000139.

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In this paper we give some new existence theorems for nonlinear random equations and inequalities involving operators of monotone type in Banach spaces. A random Hammerstein integral equation is also studied. In order to obtain random solutions we use some results from the existing deterministic theory as well as from the theory of measurable multifunctions and, in particular, the measurable selection theorems of Kuratowski/Ryll-Nardzewski and of Saint-Beuve.

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32

El-Bary, A. A. "Sobolev’s Method for Hammerstein Integral Equations." Mathematical and Computational Applications 11, no. 2 (2006): 91–94. http://dx.doi.org/10.3390/mca11020091.

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33

Tersian, Stepan A., and Petr P. Zabrejko. "Hammerstein Integral Equations with Nontrivial Solutions." Results in Mathematics 19, no. 1-2 (1991): 179–88. http://dx.doi.org/10.1007/bf03322425.

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34

ROBINSON, P. D., and P. K. YUEN. "Bivariational Methods for Hammerstein Integral Equations." IMA Journal of Applied Mathematics 39, no. 2 (1987): 177–88. http://dx.doi.org/10.1093/imamat/39.2.177.

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35

ARGYROS, I. K., J. A. EZQUERRO, M. A. HERNANDEZ, S. HILOUT, N. ROMERO, and A. I. VELASCO. "Expanding the applicability of secant-like methods for solving nonlinear equations." Carpathian Journal of Mathematics 31, no. 1 (2015): 11–30. http://dx.doi.org/10.37193/cjm.2015.01.02.

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We use the method of recurrent functions to provide a new semilocal convergence analysis for secant-like methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our sufficient convergence criteria are weaker than in earlier studies such as [18, 19, 20, 21, 25, 26]. Therefore, the new approach has a larger convergence domain and uses the same constants. A numerical example involving a nonlinear integral equation of mixed Hammerstein type is given to illustrate the advantages of the new approach. Another example of nonlinear integral equations is presented to show that the old convergence criteria are not satisfied but the new convergence are satisfied.

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36

Cordero, Alicia, Eva G. Villalba, Juan R. Torregrosa, and Paula Triguero-Navarro. "Convergence and Stability of a Parametric Class of Iterative Schemes for Solving Nonlinear Systems." Mathematics 9, no. 1 (2021): 86. http://dx.doi.org/10.3390/math9010086.

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A new parametric class of iterative schemes for solving nonlinear systems is designed. The third- or fourth-order convergence, depending on the values of the parameter being proven. The analysis of the dynamical behavior of this class in the context of scalar nonlinear equations is presented. This study gives us important information about the stability and reliability of the members of the family. The numerical results obtained by applying different elements of the family for solving the Hammerstein integral equation and the Fisher’s equation confirm the theoretical results.

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37

Raad, Sameeha. "On Numerical Treatments to Solve a Volterra - Hammerstein Integral Equation." British Journal of Mathematics & Computer Science 14, no. 6 (2016): 1–15. http://dx.doi.org/10.9734/bjmcs/2016/23821.

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38

Banaś, J., J. Rocha Martin, and K. Sadarangani. "On solutions of a quadratic integral equation of Hammerstein type." Mathematical and Computer Modelling 43, no. 1-2 (2006): 97–104. http://dx.doi.org/10.1016/j.mcm.2005.04.017.

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39

Abdou, M. A., M. M. El-Borai, and M. M. El-Kojok. "Toeplitz matrix method and nonlinear integral equation of Hammerstein type." Journal of Computational and Applied Mathematics 223, no. 2 (2009): 765–76. http://dx.doi.org/10.1016/j.cam.2008.02.012.

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40

Ngoc, Le Thi Phuong, and Nguyen Thanh Long. "On a nonlinear volterra-hammerstein integral equation in two variables." Acta Mathematica Scientia 33, no. 2 (2013): 484–94. http://dx.doi.org/10.1016/s0252-9602(13)60013-2.

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41

Khachatryan, A. Kh, and Kh A. Khachatryan. "Existence and uniqueness theorem for a Hammerstein nonlinear integral equation." Opuscula Mathematica 31, no. 3 (2011): 393. http://dx.doi.org/10.7494/opmath.2011.31.3.393.

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42

Liu, Xi-Lan, and Jian-Hua Wu. "Positive solutions for a Hammerstein integral equation with a parameter." Applied Mathematics Letters 22, no. 4 (2009): 490–94. http://dx.doi.org/10.1016/j.aml.2008.06.022.

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43

BUONG, NGUYEN. "OPTIMAL CONTROL FOR SYSTEMS GOVERNED BY DISCONTINUOUS NONLINEARITY." Tamkang Journal of Mathematics 30, no. 4 (1999): 289–94. http://dx.doi.org/10.5556/j.tkjm.30.1999.4234.

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   The aim of this paper is to present an existence theorem of optimal control for systems descrided by the operator equation of Hammerstein type $x + K F(u, x) = 0$ with the discontinuous monotone nonlinear operator $F$ in $x$. Then, the theoretical result is applied to investigate an optimal control problem for system, where the state is written in the form of nonlinear integral equations in $L_p(\omega)$.   

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44

Abas, Abas Wisam Mahdi. "The calculation of the solution of multidimensional integral equations with methods Monte Carlo and quasi-Monte Carlo." T-Comm 15, no. 10 (2021): 55–63. http://dx.doi.org/10.36724/2072-8735-2021-15-10-55-63.

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The article considers an approach based on the random cubature method for solving both single and multidimensional singular integral equations, Volterra and Fredholm equations of the 1st kind, for ill-posed problems in the theory of integral equations, etc. A variant of the quasi-Monte Carlo method is studied. The integral in an integral equation is approximated using the traditional Monte Carlo method for calculating integrals. Multidimensional interpolation is applied on an arbitrary set of points. Examples of applying the method to a one-dimensional integral equation with a smooth kernel using both random and low-dispersed pseudo-random nodes are considered. A multidimensional linear integral equation with a polynomial kernel and a multidimensional nonlinear problem – the Hammerstein integral equation – are solved using the Newton method. The existence of several solutions is shown. Multidimensional integral equations of the first kind and their solution using regularization are considered. The Monte Carlo and quasi-Monte Carlo methods have not been used to solve such problems in the studied literature. The Lavrentiev regularization method was used, as well as random and pseudo-random nodes obtained using the Halton sequence. The problem of eigenvalues is solved. It is established that one of the best methods considered is the Leverrier-Faddeev method. The results of solving the problem for a different number of quadrature nodes are presented in the table. An approach based on parametric regularization of the core, an interpolation-projection method, and averaged adaptive densities are studied. The considered methods can be successfully applied in solving spatial boundary value problems for areas of complex shape. These approaches allow us to expand the range of problems in the theory of integral equations solved by Monte Carlo and quasi-Monte Carlo methods, since there are no restrictions on the value of the norm of the integral operator. A series of examples demonstrating the effectiveness of the method under study is considered.

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45

Singh, Sukhjit, Eulalia Martínez, Abhimanyu Kumar, and D. K. Gupta. "Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations." Mathematics 8, no. 3 (2020): 384. http://dx.doi.org/10.3390/math8030384.

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In this work, we performed an study about the domain of existence and uniqueness for an efficient fifth order iterative method for solving nonlinear problems treated in their infinite dimensional form. The hypotheses for the operator and starting guess are weaker than in the previous studies. We assume omega continuity condition on second order Fréchet derivative. This fact it is motivated by showing different problems where the nonlinear operators that define the equation do not verify Lipschitz and Hölder condition; however, these operators verify the omega condition established. Then, the semilocal convergence balls are obtained and the R-order of convergence and error bounds can be obtained by following thee main theorem. Finally, we perform a numerical experience by solving a nonlinear Hammerstein integral equations in order to show the applicability of the theoretical results by obtaining the existence and uniqueness balls.

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Alharbi, Faizah M. "Stability Analysis of the Solution for the Mixed Integral Equation with Symmetric Kernel in Position and Time with Its Applications." Symmetry 16, no. 8 (2024): 1048. http://dx.doi.org/10.3390/sym16081048.

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Under certain assumptions, the existence of a unique solution of mixed integral equation (MIE) of the second type with a symmetric kernel is discussed, in is the position domain of integration and T is the time. The convergence error and the stability error are considered. Then, after using the separation technique, the MIE transforms into a system of Hammerstein integral equations (SHIEs) with time-varying coefficients. The nonlinear algebraic system (NAS) is obtained after using the degenerate method. New and special cases are derived from this work. Moreover, numerical results are computed using MATLAB R2023a software.

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Micula, Sanda. "Iterative Numerical Methods for a Fredholm–Hammerstein Integral Equation with Modified Argument." Symmetry 15, no. 1 (2022): 66. http://dx.doi.org/10.3390/sym15010066.

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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.

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Chidume, C. E., and N. Djitté. "Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type." ISRN Mathematical Analysis 2012 (March 25, 2012): 1–12. http://dx.doi.org/10.5402/2012/169751.

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Suppose that H is a real Hilbert space and F,K:H→H are bounded monotone maps with D(K)=D(F)=H. Let u* denote a solution of the Hammerstein equation u+KFu=0. An explicit iteration process is shown to converge strongly to u*. No invertibility or continuity assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our result is a significant improvement on the Galerkin method of Brézis and Browder.

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González, Daniel. "A variant of the Newton-Kantorovich theorem for nonlinear integral equations of mixed Hammerstein type." Applied Mathematics and Computation 218, no. 18 (2012): 9536–46. https://doi.org/10.1016/j.amc.2012.03.049.

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<em>We study nonlinear integral equations of mixed Hammerstein type using Newton’s method </em><em>as follows. We investigate the theoretical significance of Newton’s method to draw conclusions </em><em>about the existence and uniqueness of solutions of these equations. After that, we </em><em>approximate the solutions of a particular nonlinear integral equation by Newton’s method. </em><em>For this, we use the majorant principle, which is based on the concept of majorizing </em><em>sequence given by Kantorovich, and milder convergence conditions than those of Kantorovich. </em><em>Actually, we prove a semilocal convergence theorem which is applicable to situations </em><em>where Kantorovich’s theorem is not.</em>

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50

Debashis Dey and Mantu Saha. "Application of random fixed point theorems in solving nonlinear stochastic integral equation of the Hammerstein type." Malaya Journal of Matematik 1, no. 02 (2013): 54–59. http://dx.doi.org/10.26637/mjm102/007.

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In the present paper, we apply random analogue Kannan fixed point theorem [10] to analyze the existence of a solution of a nonlinear stochastic integral equation of the Hammerstein type of the form$$x(t ; \omega)=h(t ; \omega)+\int_S k(t, s ; \omega) f(s, x(s ; \omega)) d \mu(s)$$where $t \in S$, a $\sigma$-finite measure space with certain properties, $\omega \in \Omega$, the supporting set of a probability measure space $(\Omega, \beta, \mu)$ and the integral is a Bochner integral.

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