A fixed-point type result for some non-differentiable Fredholm integral equations
Abstract
In this paper, we present a new fixed-point result to draw conclusions about the existence and uniqueness of the solution for a nonlinear Fredholm integral equation of the second kind with non-differentiable Nemytskii operator. To do this, we will transform the problem of locating a fixed point for an integral operator into the problem of locating a solution of an integral equation. Thus, assuming conditions on the Nemytskii operator, we will obtain a global convergence domain for the solution of the considered integral equation, taking for this a uniparametric family of derivativefree iterative processes with quadratic convergence. This result provides us a new fixed-point result for the integral operator considered.
Keyword : fixed point theorem, global convergence, Fredholm integral equations, derivative-free iterative processes
How to Cite
Hernández-Verón, M. A., Singh, S., Martínez, E., & Yadav, N. (2024). A fixed-point type result for some non-differentiable Fredholm integral equations. Mathematical Modelling and Analysis, 29(1), 161–177. https://doi.org/10.3846/mma.2024.18338
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Saberi-Nadjafi and M. Heidari. Solving nonlinear integral equations in the Urysohn form by Newton–Kantorovich–quadrature method. Computers & Mathematics with Applications, 60(7):2058–2065, 2010. https://doi.org/10.1016/j.camwa.2010.07.046
S.M. Shakhno. Nonlinear majorants for investigation of methods of linear interpolation for the solution of nonlinear equations. In ECCOMAS 2004-European Congress on Computational Methods in Applied Sciences and Engineering, 2004.
A.-M. Wazwaz. A First Course in Integral Equations. World Scientific Publishing Company, 2015. https://doi.org/10.1142/9570
I.K. Argyros and S. George. Improved convergence analysis for the Kurchatov method. Nonlinear Functional Analysis and Applications, 22(1):41–58, 2017.
K. Atkinson and J. Flores. The discrete collocation method for nonlinear integral equations. IMA journal of numerical analysis, 13(2):195–213, 1993. https://doi.org/10.1093/imanum/13.2.195
F. Awawdeh, A. Adawi and S. Al-Shara. A numerical method for solving nonlinear integral equations. International Mathematical Forum, 4(17):805–817, 2009.
V. Berinde and M. Pacurar. Iterative approximation of fixed points of almost contractions. Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2007), pp. 387–392, 2007. https://doi.org/10.1109/SYNASC.2007.49
Y. Cherruault, G. Saccomandi and B. Some. New results for convergence of Adomian’s method applied to integral equations. Mathematical and Computer Modelling, 16(2):85–93, 1992. https://doi.org/10.1016/0895-7177(92)90009-A
J.A. Ezquerro, D. González and M.A. Hernández. A variant of the Newton– Kantorovich theorem for nonlinear integral equations of mixed Hammerstein type. Applied Mathematics and Computation, 218(18):9536–9546, 2012. https://doi.org/10.1016/j.amc.2012.03.049
J.A. Ezquerro and M.Á. Hernández-Verón. How to obtain global convergence domains via Newton’s method for nonlinear integral equations. Mathematics, 7(6):553, 2019. https://doi.org/10.3390/math7060553
M. Ghasemi, M. Tavassoli Kajani and E. Babolian. Numerical solutions of the nonlinear Volterra–Fredholm integral equations by using homotopy perturbation method. Applied Mathematics and Computation, 188(1):446–449, 2007. https://doi.org/10.1016/j.amc.2006.10.015
M. Grau-Sánchez, M. Noguera and S. Amat. On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods. Journal of Computational and Applied Mathematics, 237(1):363– 372, 2013. https://doi.org/10.1016/j.cam.2012.06.005
M. Grau-Sánchez, M. Noguera and J.M. Gutiérrez. On some computational orders of convergence. Applied Mathematics Letters, 23(4):472–478, 2010. https://doi.org/10.1016/j.aml.2009.12.006
M.A. Hernández and M.J. Rubio. A uniparametric family of iterative processes for solving nondifferentiable equations. Journal of mathematical analysis and applications, 275(2):821–834, 2002. https://doi.org/10.1016/S0022-247X(02)00432-8
M.A. Hernández and M.A. Salanova. A Newton-like iterative process for the numerical solution of Fredholm nonlinear integral equations. The Journal of Integral Equations and Applications, 17(1):1–17, 2005. https://doi.org/10.1216/jiea/1181075309
M.A. Hernández-Verón, N. Yadav, Á. Magreñán, E. Martínez and S. Singh. An improvement of the Kurchatov method by means of a parametric modification. Mathematical Methods in the Applied Sciences, 45(11):6844–6860, 2022. https://doi.org/10.1002/mma.8209
R. Hongmin and W. Qingbiao. The convergence ball of the secant method under Hölder continuous divided differences. Journal of Computational and Applied Mathematics, 194(2):284–293, 2006. https://doi.org/10.1016/j.cam.2005.07.008
F. Mirzaee and S. Bimesl. Application of Euler matrix method for solving linear and a class of nonlinear Fredholm integro-differential equations. Mediterranean Journal of Mathematics, 11(3):999–1018, 2014. https://doi.org/10.1007/s00009-014-0391-4
F. Mirzaee and E. Hadadiyan. Using operational matrix for solving nonlinear class of mixed Volterra–Fredholm integral equations. Mathematical Methods in the Applied Sciences, 40(10):3433–3444, 2017. https://doi.org/10.1002/mma.4237
F. Mirzaee and N. Samadyar. On the numerical solution of stochastic quadratic integral equations via operational matrix method. Mathematical Methods in the Applied Sciences, 41(12):4465–4479, 2018. https://doi.org/10.1002/mma.4907
Y. Ordokhani and M. Razzaghi. Solution of nonlinear Volterra–Fredholm– Hammerstein integral equations via a collocation method and rationalized Haar functions. Applied Mathematics Letters, 21(1):4–9, 2008. https://doi.org/10.1016/j.aml.2007.02.007
D. Porter and D.S.G. Stirling. Integral equations: a practical treatment, from spectral theory to applications. Cambridge University Press, 1990. https://doi.org/10.1017/CBO9781139172028
J. Rashidinia and M. Zarebnia. New approach for numerical solution of Hammerstein integral equations. Applied mathematics and computation, 185(1):147–154, 2007. https://doi.org/10.1016/j.amc.2006.07.017
J. Saberi-Nadjafi and M. Heidari. Solving nonlinear integral equations in the Urysohn form by Newton–Kantorovich–quadrature method. Computers & Mathematics with Applications, 60(7):2058–2065, 2010. https://doi.org/10.1016/j.camwa.2010.07.046
S.M. Shakhno. Nonlinear majorants for investigation of methods of linear interpolation for the solution of nonlinear equations. In ECCOMAS 2004-European Congress on Computational Methods in Applied Sciences and Engineering, 2004.
A.-M. Wazwaz. A First Course in Integral Equations. World Scientific Publishing Company, 2015. https://doi.org/10.1142/9570