Solutions of linear Fredholm integral equations with the three-step iteration method

Year 2024, Volume: 42 Issue: 1, 245 - 251, 27.02.2024

Lale Cona Kadir Şengül

Abstract

In this article, the solution of the second type of nonhomogeneous linear Fredholm integral equations is investigated using a three-step iteration algorithm. It has been shown that the sequences obtained from this algorithm converge to the solution of the mentioned equations. Morever, data dependency is obtained for the second type of nonhomogeneous linear Fred-holm integral equations. This result is supported by an example.

Keywords

Banach Fixed Point Theorem, Data Dependency, Fredholm Integral Equations, Three-Step Iteration Method

References

• REFERENCES
• [1] Abbas M, Nazir T. A new faster iteration process applied to constrained minimization and feasibility problems. Mat Vesnik 2014;66:223234.
• [2] Adebisi AF, Ojurongbe TA, Okunlola KA, Peter OJ. Application of Chebyshev polynomial basis function on the solution of Volterra integro-differential equations using Galerkin method. Math Comput Sci 2021;2:4151.
• [3] Adebisi AF, Okunola KA, Raji MT, Adedeji JA, Peter OJ. Galerkin and perturbed collocation methods for solving a class of linear fractional integro-differential equations. Aligarh Bull Math 2020;40:4557.
• [4] Agarwal RP, O'Regan D, Sahu DR. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J Nonlinear Convex Anal 2007;8:6179.
• [5] Akbulut A. Sabit nokta teoremlerinin Cauchy problemine ve integral denklemlere uygulanması [Master's thesis]. Ankara: Gazi Üniversitesi; 2007.
• [6] Atalan Y. Yeni bir iterasyon yöntemi için hemen-hemen büzülme dönüşümleri altında bazı sabit nokta teoremleri. Marmara Fen Bilim Derg 2018;30:276285. [Turkish] [CrossRef]
• [7] Atalan Y. İteratif yaklaşım altında bir fonksiyonel-integral denklem sınıfının çözümünün incelenmesi. J Inst Sci Technol 2019;9:16221632. [Turkish]
• [8] Atalan Y, Maldar S. A new note on convergence and data dependence concept for a Volterra integral equation by fixed point iterative algorithm. J Math Ext 2023;17:118.
• [9] Banach S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam Math 1922;3:133181. [CrossRef]
• [10] Berinde V, Takens F. Iterative approximation of fixed points. Berlin: Springer; 2007. [CrossRef]
• [11] Brouwer LEJ. Über abbildung von mannigfaltigkeiten. Math Ann 1911;71:97115. [CrossRef]
• [12] Chugh R, Kumar V, Kumar S. Strong convergence of a new three-step iterative scheme in Banach spaces. Am J Comput Math 2012;2:345357. [CrossRef]
• [13] Cona L, Şengül K. On data dependency and solutions of nonlinear Fredholm integral equations with the three-step iteration method. Ikonion J Math 2023;2:5364. [CrossRef]
• [14] Cona L, Tuna T. Solutions of linear Fredholm integral equations with the Picard's three-step iteration method. In: Proceedings of the 11th International Summit Scientific Research Congress; 2023 Dec 1517; Gaziantep, Turkey.
• [15] Cona L, Tuna T. Data dependence with Picard's three-step iteration method for linear Fredholm integral equations. In: Proceedings of the 11th International Summit Scientific Research Congress; 2023 Dec 1517; Gaziantep, Turkey.
• [16] Cona L. Solutions of some fractional order integro-differential equation by the three-step iteration method. In: Proceedings of the Bilsel International Ahlat Scientific Researches Congress; 2023 Dec 910; Bitlis, Turkey.
• [17] Cona L. Data dependency for some fractional order integro-differential equation by the three-step iteration method. In: Proceedings of the Bilsel International Ahlat Scientific Researches Congress; 2023 Dec 910; Bitlis, Turkey.
• [18] Gürsoy F. A Picard-S iterative method for approximating fixed point of weak-contraction mappings. Filomat 2016;30:28292845. [CrossRef]
• [19] Halpern B. Fixed points of nonexpanding maps. Bull Am Math Soc 1967;73:957961. [CrossRef]
• [20] Hussain N, Chugh R, Kumar V, Rafiq A. On the rate of convergence of Kirk-type iterative schemes. J Appl Math 2012;2012:526503. [CrossRef]
• [21] Ishikawa S. Fixed points by a new iteration method. Proc Am Math Soc 1974;44:147150. [CrossRef]
• [22] Ishola CY, Taiwo OA, Adebisi AF, Peter OJ. Numerical solution of two-dimensional Fredholm integro-differential equations by Chebyshev integral operational matrix method. J Appl Math Comput Mech 2022;21:2940. [CrossRef]
• [23] Karahan İ, Özdemir M. A general iterative method for approximation of fixed points and their applications. Adv Fixed Point Theory 2013;3:510526. [CrossRef]
• [24] Karakaya V, Atalan Y, Doğan K, Bouzara NEH. Some fixed point results for a new three steps iteration process in Banach spaces. Fixed Point Theory 2017;18:625640. [CrossRef]
• [25] Khan SH. A Picard-Mann hybrid iterative process. Fixed Point Theory Appl 2013;2013:69.
 [CrossRef]
• [26] Kirk WA. On successive approximations for nonexpansive mappings in Banach spaces. Glasgow Math J 1971;12:69. [CrossRef]
• [27] Krasnosel’skii MA. Two comments on the method of successive approximations. Usp Mat Nauk 1955;10:123127.
• [28] Mann WR. Mean value methods in iteration. Proc Am Math Soc 1953;4:506510. [CrossRef]
• [29] Noor MA. New approximation schemes for general variational inequalities. J Math Anal Appl 2000;251:217229. [CrossRef]
• [30] Olatinwo MO. Some stability results for two hybrid fixed point iterative algorithms in normed linear space. Mat Vesnik 2009;61:247256.
• [31] Oyedepo T, Adebisi AF, Tayo RM, Adedeji JA, Ayinde MA, Peter OJ. Perturbed least squares technique for solving Volterra fractional integro-differential equations based on constructed orthogonal polynomials. J Math Comput Sci 2020;11:203218.
• [32] Oyedepo T, Uwaheren OA, Okperhie EP. Solution of fractional integro-differential equation using modified homotopy perturbation technique and constructed orthogonal polynomials as basis functions. ATBU J Sci Technol Educ. 2019;7:157164.
• [33] Peter OJ. Transmission dynamics of fractional order Brucellosis model using Caputo-Fabrizio operator. Int J Differ Equ 2020;2020:2791380. [CrossRef]
• [34] Peter OJ, Adebisi AF, Oguntolu FA, Bitrus S, Akpan CE. Multi-step homotopy analysis method for solving Malaria model. Malays J Appl Sc 2018;3:3445.
• [35] Phuengrattana W, Suantai S. On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J Comput Appl Math 2011;235:30063014. [CrossRef]
• [36] Picard E. Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives. J Math Pures Appl 1890;6:145210.
• [37] Rhoades BE, Soltuz SM. The equivalence between Mann–Ishikawa iterations and multistep iteration. Nonlinear Anal Theory Methods Appl. 2004;58:219228. [CrossRef]
• [38] Schaefer H. Über die methode sukzessiver approximationen. Jahresber Dtsch Math-Ver 1957;59:131140.
• [39] Şoltuz ŞM, Grosan T. Data dependence for Ishikawa iteration when dealing with contractive-like operators. Fixed Point Theory Appl 2008;2008:17. [CrossRef]
• [40] Thianwan S. Common fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space. J Comput Appl Math 2009;224:688695.[CrossRef]
• [41] Uwaheren OA, Adebisi AF, Olotu OT, Etuk MO, Peter OJ. Legendre Galerkin method for solving fractional integro-differential equations of Fredholm type. Aligarh Bull Math 2021;40:1527.
• [42] Weng X. Fixed point iteration for local strictly pseudo-contractive mapping. Proc Am Math Soc 1991;113:727731. [CrossRef]