On a Functional Integral Equation

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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Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations

Mathematica Slovaca ◽

10.1515/ms-2017-0082 ◽

2018 ◽

Vol 68 (1) ◽

pp. 77-88

Author(s):

Marcin Borkowski ◽

Daria Bugajewska

Keyword(s):

Integral Equation ◽

Differential Equations ◽

Integral Equations ◽

Bounded Variation ◽

Volterra Integral Equation ◽

Functions Of Bounded Variation ◽

Hammerstein Integral Equation ◽

Linear Differential ◽

Unbounded Intervals ◽

Differential And Integral Equations

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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Representation Stability of Power Sets and Square Free Polynomials

Canadian Journal of Mathematics ◽

10.4153/cjm-2014-029-2 ◽

2015 ◽

Vol 67 (5) ◽

pp. 1024-1045

Author(s):

Samia Ashraf ◽

Haniya Azam ◽

Barbu Berceanu

Keyword(s):

Symmetric Group ◽

Braid Group ◽

Point Of View ◽

Pure Braid Group ◽

Representation Stability ◽

Stability Point ◽

Power Set ◽

The Stability

AbstractThe symmetric group 𝓢n acts on the power set 𝓟(n) and also on the set of square free polynomials in n variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

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An application of Newton's method to differential and integral equations

The ANZIAM Journal ◽

10.1017/s1446181100012001 ◽

2001 ◽

Vol 42 (3) ◽

pp. 372-386 ◽

Cited By ~ 1

Author(s):

J. M. Gutiérrez ◽

M. A. Hernández

Keyword(s):

Integral Equations ◽

Error Estimates ◽

Newton’S Method ◽

Existence And Uniqueness ◽

Newton's Method ◽

Uniqueness Of A Solution ◽

Differential And Integral Equations

AbstractNewton's method is applied to an operator that satisfies stronger conditions than those of Kantorovich. Convergence and error estimates are compared in the two situations. As an application, we obtain information on the existence and uniqueness of a solution for differential and integral equations.

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The Integral Expansions of Arbitrary Functions connected with Integral Equations

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100002917 ◽

1924 ◽

Vol 22 (2) ◽

pp. 169-185 ◽

Cited By ~ 1

Author(s):

J. Hyslop

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Point Of View ◽

General Treatment ◽

Integral Equation Theory

The following paper aims at a more general treatment than has hitherto been given, of the integral expansions of arbitrary functions, from the point of view of integral equation theory.

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Existence, Uniqueness, and Approximation for Solutions of a Functional-integral Equation in Lp Spaces

TEMA (São Carlos) ◽

10.5540/tema.2019.020.03.403 ◽

2019 ◽

Vol 20 (3) ◽

pp. 403

Author(s):

Suzete M Afonso ◽

Juarez S Azevedo ◽

Mariana P. G. Da Silva ◽

Adson M Rocha

Keyword(s):

Integral Equation ◽

Fixed Point ◽

Fixed Point Theorem ◽

Existence And Uniqueness ◽

Functional Integral ◽

Successive Approximation Method ◽

Banach Fixed Point Theorem ◽

Numerical Examples ◽

Lp Spaces ◽

Chebyshev Quadrature

In this work we consider the general functional-integral equation: \begin{equation*}y(t) = f\left(t, \int_{a}^{b} k(t,s)g(s,y(s))ds\right), \qquad t\in [a,b],\end{equation*}and give conditions that guarantee existence and uniqueness of solution in $L^p([a,b])$, with {$1<p<\infty$}.We use Banach Fixed Point Theorem and employ the successive approximation method and Chebyshev quadrature for approximating the values of integrals. Finally, to illustrate the results of this work, we provide some numerical examples.

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Solving a Class of Integral Equations via Contractive Mapping with Rational Type

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v13i4.3831 ◽

2020 ◽

Vol 13 (4) ◽

pp. 995-1015

Author(s):

Abdullah Abdullah ◽

Muhammad Sarwar ◽

Zead Mustafa ◽

Mohammed M.M. Jaradat

Keyword(s):

Integral Equation ◽

Fixed Point ◽

Integral Equations ◽

Existence And Uniqueness ◽

Metric Spaces ◽

Coupled Fixed Point ◽

Contractive Mapping ◽

Coupled Fixed Point Theorem ◽

Set Up ◽

Rational Type

In this paper, using rational type contractive conditions, the existence and uniqueness of common coupled fixed point theorem in the set up of Gb-metric spaces is studied. The derived result cover and generalize some well-known comparable results in the existing literature. Then we use the derived results to prove the existence and uniqueness solution for some classes of integral equations. Further more, an example of such type of integral equation is presented.

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QUANTUM INTEGRAL EQUATIONS OF VOLTERRA TYPE WITH GENERALIZED OPERATOR-VALUED KERNELS

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025700000315 ◽

2000 ◽

Vol 03 (04) ◽

pp. 505-517 ◽

Cited By ~ 2

Author(s):

ZHIYUAN HUANG ◽

CAISHI WANG ◽

XIANGJUN WANG

Keyword(s):

Integral Equation ◽

Explicit Expression ◽

Integral Equations ◽

Existence And Uniqueness ◽

Continuous Dependence ◽

Uniqueness Of Solutions ◽

Volterra Type ◽

Quantum Integral ◽

Generalized Operator

Quantum integral equation of Volterra type with generalized operator-valued kernel is introduced. Existence and uniqueness of solutions are established, explicit expression of the solution is given, the continuity, continuous dependence on free terms and other properties of the solution are proved.

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Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein–Kantorovich Operators

Mathematics ◽

10.3390/math9111193 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1193

Author(s):

Suzan Cival Buranay ◽

Mehmet Ali Özarslan ◽

Sara Safarzadeh Falahhesar

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Unknown Function ◽

Input Data ◽

Linear Equations ◽

Numerical Approach ◽

Volterra Integral Equations ◽

Test Problems ◽

The Stability ◽

Kantorovich Operators

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations.

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Solvability of functional-integral equations (fractional order) using measure of noncompactness

Advances in Difference Equations ◽

10.1186/s13662-019-2487-4 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 2

Author(s):

Reza Arab ◽

Hemant Kumar Nashine ◽

N. H. Can ◽

Tran Thanh Binh

Keyword(s):

Banach Space ◽

Integral Equation ◽

Fixed Point ◽

Banach Spaces ◽

Integral Equations ◽

Fractional Order ◽

Measure Of Noncompactness ◽

Functional Integral ◽

Arbitrary Measure ◽

Functional Integral Equations

AbstractWe investigate the solutions of functional-integral equation of fractional order in the setting of a measure of noncompactness on real-valued bounded and continuous Banach space. We introduce a new μ-set contraction operator and derive generalized Darbo fixed point results using an arbitrary measure of noncompactness in Banach spaces. An illustration is given in support of the solution of a functional-integral equation of fractional order.

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The Existence and Uniqueness of the Solution of a Nonlinear Fredholm–Volterra Integral Equation with Modified Argument via Geraghty Contractions

Mathematics ◽

10.3390/math9010029 ◽

2020 ◽

Vol 9 (1) ◽

pp. 29

Author(s):

Maria Dobriţoiu

Keyword(s):

Integral Equation ◽

Fixed Point ◽

Integral Equations ◽

Volterra Integral Equation ◽

Existence And Uniqueness ◽

Metric Spaces ◽

Nonlinear Integral Equations ◽

Uniqueness Of The Solution

Using some of the extended fixed point results for Geraghty contractions in b-metric spaces given by Faraji, Savić and Radenović and their idea to apply these results to nonlinear integral equations, in this paper we present some existence and uniqueness conditions for the solution of a nonlinear Fredholm–Volterra integral equation with a modified argument.

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