A fixed-point type result for some non-differentiable Fredholm integral equations. (English) Zbl 1542.45001
Summary: 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.
MSC:
| 45B05 | Fredholm integral equations |
| 45L05 | Theoretical approximation of solutions to integral equations |
| 47H10 | Fixed-point theorems |
| 47N20 | Applications of operator theory to differential and integral equations |
| 65R20 | Numerical methods for integral equations |
| 65J15 | Numerical solutions to equations with nonlinear operators |
Keywords:
fixed point theorem; global convergence; Fredholm integral equations; derivative-free iterative processesReferences:
| [1] | I.K. Argyros. On the secant method. Publicationes Mathematicae Debrecen, 43(3-4):223-238, 1993. https://doi.org/10.5486/PMD.1993.1215. · Zbl 0796.65075 · doi:10.5486/PMD.1993.1215 |
| [2] | I.K. Argyros and S. George. Improved convergence analysis for the Kurchatov method. Nonlinear Functional Analysis and Applications, 22(1):41-58, 2017. · Zbl 1368.65078 |
| [3] | K. Atkinson and J. Flores. The discrete collocation method for nonlinear in-tegral equations. IMA Journal of Numerical Analysis, 13(2):195-213, 1993. https://doi.org/10.1093/imanum/13.2.195. · Zbl 0771.65090 · doi:10.1093/imanum/13.2.195 |
| [4] | F. Awawdeh, A. Adawi and S. Al-Shara. A numerical method for solving nonlin-ear integral equations. International Mathematical Forum, 4(17):805-817, 2009. · Zbl 1203.65282 |
| [5] | V. Berinde and M. Pacurar. Iterative approximation of fixed points of al-most contractions. Ninth International Symposium on Symbolic and Nu-meric Algorithms for Scientific Computing (SYNASC 2007), pp. 387-392, 2007. https://doi.org/10.1109/SYNASC.2007.49. · doi:10.1109/SYNASC.2007.49 |
| [6] | 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. · Zbl 0756.65083 · doi:10.1016/0895-7177(92)90009-A |
| [7] | J.A. Ezquerro, D. González and M.A. Hernández. A variant of the Newton-Kantorovich theorem for nonlinear integral equations of mixed Hammer-stein type. Applied Mathematics and Computation, 218(18):9536-9546, 2012. https://doi.org/10.1016/j.amc.2012.03.049. · Zbl 1245.65179 · doi:10.1016/j.amc.2012.03.049 |
| [8] | 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. · doi:10.3390/math7060553 |
| [9] | M. Ghasemi, M. Tavassoli Kajani and E. Babolian. Numerical solutions of the nonlinear Volterra-Fredholm integral equations by using homotopy perturba-tion method. Applied Mathematics and Computation, 188(1):446-449, 2007. https://doi.org/10.1016/j.amc.2006.10.015. · Zbl 1114.65367 · doi:10.1016/j.amc.2006.10.015 |
| [10] | M. Grau-Sánchez, M. Noguera and S. Amat. On the approximation of derivatives using divided difference operators preserving the local convergence order of itera-tive methods. Journal of Computational and Applied Mathematics, 237(1):363-372, 2013. https://doi.org/10.1016/j.cam.2012.06.005. · Zbl 1308.65074 · doi:10.1016/j.cam.2012.06.005 |
| [11] | 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. · Zbl 1189.65092 · doi:10.1016/j.aml.2009.12.006 |
| [12] | M.A. Hernández and M.J. Rubio. A uniparametric family of iterative pro-cesses for solving nondifferentiable equations. Journal of Mathematical Anal-ysis and Applications, 275(2):821-834, 2002. https://doi.org/10.1016/S0022-247X(02)00432-8. · Zbl 1019.65036 · doi:10.1016/S0022-247X(02)00432-8 |
| [13] | M.A. Hernández and M.A. Salanova. A Newton-like iterative pro-cess 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. · Zbl 1079.65135 · doi:10.1216/jiea/1181075309 |
| [14] | M.A. Hernández-Verón, N. Yadav, Á. Magreñán, E. Martínez and S. Singh. An improvement of the Kurchatov method by means of a parametric modifica-tion. Mathematical Methods in the Applied Sciences, 45(11):6844-6860, 2022. https://doi.org/10.1002/mma.8209. · Zbl 1527.65035 · doi:10.1002/mma.8209 |
| [15] | R. Hongmin and W. Qingbiao. The convergence ball of the se-cant 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. · Zbl 1101.65057 · doi:10.1016/j.cam.2005.07.008 |
| [16] | 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. · Zbl 1301.65133 · doi:10.1007/s00009-014-0391-4 |
| [17] | F. Mirzaee and E. Hadadiyan. Using operational matrix for solving nonlinear class of mixed Volterra-Fredholm integral equations. Math-ematical Methods in the Applied Sciences, 40(10):3433-3444, 2017. https://doi.org/10.1002/mma.4237. · Zbl 1376.65157 · doi:10.1002/mma.4237 |
| [18] | 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. · Zbl 1461.65270 · doi:10.1002/mma.4907 |
| [19] | Y. Ordokhani and M. Razzaghi. Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via a collocation method and rational-ized Haar functions. Applied Mathematics Letters, 21(1):4-9, 2008. https://doi.org/10.1016/j.aml.2007.02.007. · Zbl 1133.65117 · doi:10.1016/j.aml.2007.02.007 |
| [20] | 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. · Zbl 0714.45001 · doi:10.1017/CBO9781139172028 |
| [21] | J. Rashidinia and M. Zarebnia. New approach for numerical solution of Ham-merstein integral equations. Applied Mathematics and Computation, 185(1):147-154, 2007. https://doi.org/10.1016/j.amc.2006.07.017. · Zbl 1110.65126 · doi:10.1016/j.amc.2006.07.017 |
| [22] | J. Saberi-Nadjafi and M. Heidari. Solving nonlinear integral equa-tions 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. · Zbl 1205.65340 · doi:10.1016/j.camwa.2010.07.046 |
| [23] | S.M. Shakhno. Nonlinear majorants for investigation of methods of linear inter-polation for the solution of nonlinear equations. In ECCOMAS 2004-European Congress on Computational Methods in Applied Sciences and Engineering, 2004. |
| [24] | A.-M. Wazwaz. A First Course in Integral Equations. World Scientific Publishing Company, 2015. https://doi.org/10.1142/9570. · Zbl 1327.45001 · doi:10.1142/9570 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
