The existence and numerical solution of functional integral equation via coupled fixed point theorem, measure of noncompactness and iterative algorithm Reza Arab, Mohsen Rabbani Measure of Noncompactness Fixed Point Theorems and Applications, 2024 Darbo's fixed point theorem [11] is a very important generalization of Schauder's fixed point theorem and includes the existence part of Banach's fixed point theorem. To generalize some fixed theorems and coupled fixed point theorems with the help of measure of noncompactness, we introduce notations, definitions, and preliminary facts which are used throughout this chapter. Denote by https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003436577/55cfbb76-b6b6-431d-9e38-10ed9379fd4a/content/math1_1.tif"/> the set of real numbers and put https://www.w3.org/1998/Math/MathML" display="inline"> R + = [ 0 , + ∞ ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003436577/55cfbb76-b6b6-431d-9e38-10ed9379fd4a/content/math1_2.tif"/> . Let https://www.w3.org/1998/Math/MathML" display="inline"> ( E , ∥ ⋅ ∥ ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003436577/55cfbb76-b6b6-431d-9e38-10ed9379fd4a/content/math1_3.tif"/> be a real Banach space with zero element 0. Let https://www.w3.org/1998/Math/MathML" display="inline"> B ― ( x , r ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003436577/55cfbb76-b6b6-431d-9e38-10ed9379fd4a/content/math1_4.tif"/> denote the closed ball centered at x with radius r. The symbol https://www.w3.org/1998/Math/MathML" display="inline"> B ― r https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003436577/55cfbb76-b6b6-431d-9e38-10ed9379fd4a/content/math1_5.tif"/> stands for the ball https://www.w3.org/1998/Math/MathML" display="inline"> B ― ( 0 , r ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003436577/55cfbb76-b6b6-431d-9e38-10ed9379fd4a/content/math1_6.tif"/> . For X, a nonempty subset of E, we denote by https://www.w3.org/1998/Math/MathML" display="inline"> X ― https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003436577/55cfbb76-b6b6-431d-9e38-10ed9379fd4a/content/math1_7.tif"/> and ConvX the closure and the closed convex hull of X, respectively. Moreover, let us denote by https://www.w3.org/1998/Math/MathML" display="inline"> M E https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003436577/55cfbb76-b6b6-431d-9e38-10ed9379fd4a/content/math1_8.tif"/> the family of nonempty bounded subsets of E and by https://www.w3.org/1998/Math/MathML" display="inline"> N E https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003436577/55cfbb76-b6b6-431d-9e38-10ed9379fd4a/content/math1_9.tif"/> its subfamily consisting of all relatively compact sets. We use the following definition of the measure of noncompactness given in [6].
Coupled fixed point theorem and measure of noncompactness for existence of solution of functional integral equations system and iterative algorithm to solve it Mohsen Rabbani, Reza Arab Measure of Noncompactness Fixed Point Theorems and Applications, 2024 Darbo's fixed point theorem [9] is a very important generalization of Schauder's fixed point theorem and Banach's fixed point theorem. We generalize some fixed theorems and coupled fixed point theorems using measure of noncompactness. As an application of obtained results we prove existence of solution of functional integral equations system as follows, 6.1 https://www.w3.org/1998/Math/MathML" display="block"> { x ( t ) = f ( t , x ( t ) , y ( t ) , ∫ 0 ∞ k ( t , s ) h ( s , x ( s ) , y ( s ) ) ) d s ) y ( t ) = f ( t , y ( t ) , x ( t ) , ∫ 0 ∞ k ( t , s ) h ( s , y ( s ) , x ( s ) ) ) d s ) , https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003436577/55cfbb76-b6b6-431d-9e38-10ed9379fd4a/content/math6_1.tif"/>
Common Fixed Point Theorem of Family of Contraction Maps and Its Applications in Integral Equations Thai Journal of Mathematics, 2023
EXISTENCE OF SOLUTION OF FUNCTIONAL VOLTERRA-FREDHOLM INTEGRAL EQUATIONS IN SPACE L∞((Formula presented)) AND SINC INTERPOLATION TO FIND SOLUTION Reza Arab, Mohsen Rabbani Journal of Integral Equations and Applications, 2022 A new measure of noncompactness (Mnc) and Darbo fixed point theorem are utilized on space L(R+) to prove the existence of solution for functional Volterra-Fredholm integral equations. An example is given to confirm the validity of results. Furthermore, we propound an iterative algorithm by Sinc interpolation to find the solution with an acceptable accuracy. In this algorithm, it does not need the problem is discretized to an algebraic system with unknown coefficients and we have an iterative processes to aproximate of solution with exponential convergence.
EXISTENCE OF SOLUTION OF FUNCTIONAL INTEGRAL EQUATIONS BY MEASURE OF NONCOMPACTNESS AND SINC INTERPOLATION TO FIND SOLUTION Mohsen Rabbani, Reza Arab, Bipan Hazarika, Nasser Aghazadeh Fixed Point Theory, 2022 In this article, we discuss the existence of solution for the functional integral equations in the Banach space BC(R + ) of real-valued continuous and bounded functions, using the method associated with the technique of measure of noncompactness and generalized Darbo fixed point theorem. We provide an example to illustrate our results, and we make an iterative algorithm by the Sinc interpolation to find solution to the above problem with acceptable accuracy.
Tripled fixed point results via a measure of noncompactness with applications Hemant Kumar Nashine, Reza Arab, Rabha W. Ibrahim Asian European Journal of Mathematics, 2021 In this paper, we create tripled fixed point outcomes via a subjective measure of noncompactness in the sense of Banas and Goebel. Furthermore, we introduce some applications of the measure of noncompactness notion to functional equations including nonlinear integral equations as well as local fractional integral equations.
Existence of Solution for Infinite System of Nonlinear Singular Integral Equations and Semi-Analytic Method to Solve it Bipan Hazarika, Mohsen Rabbani, Ravi P. Agarwal, Anupam Das, Reza Arab Iranian Journal of Science and Technology Transaction A Science, 2021 In this article, solvability of infinite system of nonlinear singular integral equations of two variables in Banach sequence spaces $$c_{0}$$ c 0 and $$\\ell _{1}$$ ℓ 1 is investigated. For this purpose, Hausdorff measure of noncompactness (in short, Hausdorff MNC) and Meir–Keeler condensing operators are employed. The guarantee of our results are given by some examples. Also to approach an approximation of semi-analytic solution, we introduce a coupled modified homotopy perturbation and Adomian decomposition method. Thus, an iterative algorithm is constructed to find the above solution. The numerical results show that the produced sequence to approximate the solution has a high accuracy.
On existence of solution of a class of quadratic-integral equations using contraction defined by simulation functions and measure of noncompactness Carpathian Journal of Mathematics, 2018
Construction of measures of noncompactness of Ck(Ω) and Ck0 and their application to functional integral-differential equations Bulletin of the Iranian Mathematical Society, 2017
On the existence of solutions for a class of systems of functional integral equations of volterra type in two variables Iranian Journal of Science and Technology Transaction A Science, 2015
An Iterative Algorithm to Approximate the Solution of Fractional Two-Dimensional Nonlinear Functional Integral Equations in a Banach Space S Darvishi Khezri, M Rabbani, R Arab, V Dadashi Computational Methods for Differential Equations , 2025 2025 Citations: 1
Common fixed point theorems in complete partial metric space R Arab, B Hazarika, T Abdeljawad, SP Kumari Afrika Matematika 36 (1), 3 , 2025 2025
Coupled fixed point theorem and measure of noncompactness for M Rabbani, R Arab Measure of Noncompactness, Fixed Point Theorems, and Applications, 85 , 2024 2024
of functional integral equation via coupled fixed point theorem, measure of noncompactness and iterative algorithm R Arab, M Rabbani Measure of Noncompactness, Fixed Point Theorems, and Applications, 1 , 2024 2024
Coupled fixed point theorem and measure of noncompactness for existence of solution of functional integral equations system and iterative algorithm to solve it M Rabbani, R Arab Measure of Noncompactness, Fixed Point Theorems, and Applications, 85-104 , 2024 2024
The existence and numerical solution of functional integral equation via coupled fixed point theorem, measure of noncompactness and iterative algorithm R Arab, M Rabbani Measure of Noncompactness, Fixed Point Theorems, and Applications, 1-23 , 2024 2024
Common fixed point theorem of family of contraction maps and its applications in integral equations R Arab, B Hazarika, M Imdad, A Das Thai Journal of Mathematics 21 (2) , 2023 2023 Citations: 2
On measure of noncompactness in Lebesgue and Sobolev spaces with an application to the functional integro-differential equation M Mursaleen, SMH Rizvi, R Arab, AS Haghighi, R Allahyari Aequationes mathematicae 97 (1), 199-217 , 2023 2023
Meir-Keeler Condensing Operator to Prove Existence of Solution for Infinite Systems of Differential Equations in the Banach Space and Numerical Method to Find the Solution M Rabbani, A Das, B Hazarika, R Arab Filomat 36 (10), 3217–3230 , 2022 2022
Existence of solution of functional Volterra-Fredholm integral equations in space L∞(R+) and sinc interpolation to find solution R Arab, M Rabbani Journal Of Integral Equations And Applications 34 (2), 151-164 , 2022 2022 Citations: 3
Existence of solution of functional integral equations by measure of noncompactness and sinc interpolation to find solution M Rabbani, R Arab, B Hazarika, N Aghazadeh Fixed Point Theory 23 (1), 331-334 , 2022 2022 Citations: 3
Generalization of Darbo-Type fixed point theorem and applications to integral equations HK Nashine, RW Ibrahim, R Arab, M Rabbani Advances in Metric Fixed Point Theory and Applications, 333-364 , 2021 2021 Citations: 2
A Solution of the System of Integral Equations in Product Spaces via Concept of Measures of Noncompactness HK Nashine, R Arab, RW Ibrahim Advances in Metric Fixed Point Theory and Applications, 133-155 , 2021 2021 Citations: 1
Best Proximity Point Results via Measure of Noncompactness and Application HK Nashine, R Arab, P R. Patle, DK Patel Numerical Functional Analysis and Optimization 42 (4), 430-442 , 2021 2021 Citations: 19
Existence of solutions of system of functional-integral equations using measure of noncompactness HK Nashine, R Arab, R Agarwal International Journal of Nonlinear Analysis and Applications 12 (1), 583-595 , 2021 2021 Citations: 13
Existence of Solution for Infinite System of Nonlinear Singular Integral Equations and Semi-Analytic Method to Solve it B Hazarik, M Rabbani, RP Agarwal, A Das, R Arab Iranian Journal of Science and Technology, Transactions A: Science 45, 235–245 , 2021 2021 Citations: 12
Tripled fixed point results via a measure of noncompactness with applications HK Nashine, R Arab, RW Ibrahim Asian-European Journal of Mathematics 14 (02), 2150008 , 2021 2021
COMMON FIXED POINT THEOREMS IN COMPLETE PARTIAL METRIC SPACE R Arab, B Hazarika, SP Kumari Journal of Applied and Fundamental Sciences 7 (2), 112-116 , 2021 2021
Measure of noncompactness of a new space of tempered sequences and its application on fractional differential equations M Rabbani, A Das, B Hazarika, R Arab Chaos, Solitons & Fractals 140, 110221 , 2020 2020 Citations: 61
Solvability of fractional dynamic systems utilizing measure of noncompactness HK Nashine, R W. Ibrahim, R Arab, M Rabbani Nonlinear Analysis: Modelling and Control 25 (4), 618–637 , 2020 2020 Citations: 10
MOST CITED SCHOLAR PUBLICATIONS
Application of simulation function and measure of noncompactness for solvability of nonlinear functional integral equations and introduction to an iteration algorithm to find … B Hazarika, HM Srivastava, R Arab, M Rabbani Applied Mathematics and Computation 360, 131-146 , 2019 2019 Citations: 67
Existence of solutions of infinite systems of integral equations in two variables via measure of noncompactness R Arab, R Allahyari, AS Haghighi Applied Mathematics and Computation 246, 283-291 , 2014 2014 Citations: 63
Measure of noncompactness of a new space of tempered sequences and its application on fractional differential equations M Rabbani, A Das, B Hazarika, R Arab Chaos, Solitons & Fractals 140, 110221 , 2020 2020 Citations: 61
Existence of solution for infinite system of nonlinear integral equations via measure of noncompactness and homotopy perturbation method to solve it B Hazarika, HM Srivastava, R Arab, M Rabbani Journal of Computational and Applied Mathematics 343, 341-352 , 2018 2018 Citations: 58
Existence of solution for two dimensional nonlinear fractional integral equation by measure of noncompactness and iterative algorithm to solve it M Rabbani, A Das, B Hazarika, R Arab Journal of Computational and Applied Mathematics 370 (112654), 1-12 , 2020 2020 Citations: 57
The Existence of Fixed Points via the Measure of Noncompactness and its Application to Functional-Integral Equations R Arab Mediterranean Journal of Mathematics 13 (2), 759–773 , 2016 2016 Citations: 55
Solvability of functional-integral equations (fractional order) using measure of noncompactness R Arab, H Kumar Nashine, NH Can, T Thanh Binh Advances in Difference Equations 12 (2020), 1-13 , 2020 2020 Citations: 52
Metric-like spaces to prove existence of solution for nonlinear quadratic integral equation and numerical method to solve it B Hazarika, E Karapinar, R Arab, M Rabbani Journal of Computational and Applied Mathematics 328, 302-313 , 2018 2018 Citations: 51
Fixed points of(ψ,φ, θ)-contractive mappings in partially ordered b-metric spaces and application to quadratic integral equations A Aghajani, R Arab Fixed Point Theory and Applications 2013 (1), 1-20 , 2013 2013 Citations: 46
Darbo type fixed and coupled fixed point results and its application to integral equation HK Nashine, R Arab, RP Agarwal, A Shole Haghighi Periodica Mathematica Hungarica 77 (1), 94–107 , 2018 2018 Citations: 44
Solvability of the infinite system of integral equations in two variables in the sequence spaces c0 and ℓ1 A Das, B Hazarika, R Arab, M Mursaleen Journal of Computational and Applied Mathematics 326, 183-192 , 2017 2017 Citations: 42
Applications of Measure of Noncompactness and Operator Type Contraction for Existence of Solution of Functional Integral Equations B Hazarika, R Arab, M Mursaleen Complex Analysis and Operator Theory 13 (8), 3837–385 , 2019 2019 Citations: 41
Applications of a fixed point theorem to the existence of solutions to the nonlinear functional integral equations in two variables A Das, B Hazarika, R Arab, M Mursaleen Rendiconti del Circolo Matematico di Palermo Series 2 68 (1), 139–152 , 2019 2019 Citations: 37
A generalization on weak contractions in partially ordered b -metric spaces and its application to quadratic integral equations R Allahyari, R Arab, A Shole Haghighi Journal of Inequalities and Applications 2014 (1), 355 , 2014 2014 Citations: 37
Solvability of nonlinear quadratic integral equation by using simulation type condensing operator and measure of noncompactness M Rabbani, R Arab, B Hazarika Applied Mathematics and Computation 349, 102-117 , 2019 2019 Citations: 36
Construction of a Measure of Noncompactness on BC (\ Omega) and its Application to Volterra Integral Equations R Arab, R Allahyari, AS Haghighi Mediterranean Journal of Mathematics 13 (3), 1197–1210 , 2016 2016 Citations: 30
Some fixed point theorems in generalized Darbo fixed point theorem and the existence of solutions for system of integral equations R Arab J. Korean Math. Soc. 52 (1), pp. 125–139 , 2015 2015 Citations: 30
Positive solutions of fractional integral equations by the technique of measure of noncompactness HK Nashine, R Arab, RP Agarwal, M De la Sen Journal of Inequalities and Applications 2017 (1), 225 , 2017 2017 Citations: 26
SOME GENERALIZATIONS OF DARBO FIXED POINT THEOREM AND ITS APPLICATION R Arab Miskolc Mathematical Notes 18 (2), 595-610 , 2017 2017 Citations: 25
Applications of measure of non-compactness and modified simulation function for solvability of nonlinear functional integral equations B Hazarika, R Arab, HK Nashine Filomat 33 (17), 5427-5439 , 2019 2019 Citations: 22
