Nonlinear Fredholm integral equations and majorant functions

• J. A. Ezquerro
• M. A. Hernández-Verón
Original Paper

Abstract

From the majorant principle of Kantorovich and the theoretical significance of Newton’s method, we obtain domains of existence and uniqueness of solution for nonlinear Fredholm integral equations, so that these domains are defined from the positive real zeros of a scalar function that we call majorant function. We illustrate this study with three Fredholm integral equations, where separable and nonseparable kernels are involved, by obtaining domains of existence and uniqueness of solution, approximating solutions numerically and giving a priori and a posteriori error estimates of the approximations.

Keywords

Fredholm integral equation Newton’s method Majorant function Domain of existence of solution Domain of uniqueness of solution Error estimates

Mathematics Subject Classification (2010)

45G10 47H99 65J15

References

1. 1.
Ahues, M.: Newton methods with Hoolder̈ derivative. Numer. Func. Anal. and Optimiz. 25(5–6), 1–17 (2004)
2. 2.
Altürk, A.: Numerical solution of linear and nonlinear Fredholm integral equations by using weighted mean-value theorem. SpringerPlus 5, 1962 (2016)
3. 3.
Awawdeh, F., Adawi, A., Al-Shara, S.: A numerical method for solving nonlinear integral equations. Int. Math. Forum 4, 805–817 (2009)
4. 4.
Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambridge (2004)
5. 5.
Carutasu, V.: Numerical solution of two-dimensional nonlinear Fredholm integral equations of the second kind by spline functions. Gen. Math. 9(1–2), 31–48 (2001)
6. 6.
Ezquerro, J.A., Hernández, M.A.: The Newton method for Hammerstein equations. J. Comput. Anal. Appl. 7(4), 437–446 (2005)
7. 7.
Ezquerro, J.A., González, D., Hernández, M.A.: A variant of the Newton-Kantorovich theorem for nonlinear integral equations of mixed Hammerstein type. Appl. Math. Comput. 218, 9536–9546 (2012)
8. 8.
Ezquerro, J.A., Hernández-Verón, M.A.: Newton’s Method: an Updated Approach of Kantorovich’s Theory. Birkhäuser, Cham (2017)
9. 9.
Gragg, W.B., Tapia, R.A.: Optimal error bounds for the Newton-Kantorovich theorem. SIAM J. Numer. Anal. 11, 10–13 (1974)
10. 10.
Gutiérrez, J.M., Hernández, M.A., Salanova, M.A.: On the approximate solution of some Fredholm integral equations by Newton’s method. Southwest J. Pure Appl. Math. 1, 1–9 (2004)
11. 11.
Jafari Emamzadeh, M., Tavassoli Kajani, M.: Nonlinear Fredholm integral equation of the second kind with quadrature methods. Journal of Mathematical Extension 4(2), 51–58 (2010)
12. 12.
Ibrahim, I.A.: On the existence of solutions of functional integral equation of Urysohn type. Comput. Math. Appl. 57(10), 1609–1614 (2009)
13. 13.
Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, New York (1982)
14. 14.
Lepik, Ü., Tamme, E.: Solution of nonlinear Fredholm integral equations via the Haar wavelet method. Proc. Estonian Acad. Sci. Phys. Math. 56(1), 17–27 (2007)
15. 15.
Moore, C.: Picard iterations for solution of nonlinear equations in certain Banach spaces. J. Math. Anal. Appl. 245(2), 317–325 (2000)
16. 16.
Nadir, M., Khirani, A.: Adapted Newton-Kantorovich method for nonlinear integral equations. J. Math. Stat. 12(3), 176–181 (2016)
17. 17.
Ostrowski, A.M.: Solution of Equations and Systems of Equations. Academic Press, New York (1966)
18. 18.
Potra, F.A., Ptàk, V.: Nondiscrete Induction and Iterative Processes. Number 103 in Research Notes in Mathematics. Wiley, Boston-London-Melbourne (1984)
19. 19.
Potra, F.A.: The Kantorovich theorem and interior point methods. Math. Program., Ser. A 102, 47–70 (2005)
20. 20.
Potra, F.A.: A superquadratic variant of Newton’s method. SIAM J. Numer. Anal. 55(6), 2863–2884 (2017)
21. 21.
Rall, L.B.: Computational Solution of Nonlinear Operator Equations. Robert E Krieger Publishing Company, Michigan (1979)
22. 22.
Rashidinia, J., Zarebnia, M.: New approach for numerical solution of Hammerstein integral equations. Appl. Math. Comput. 185, 147–154 (2007)
23. 23.
Rashidinia, J., Parsa, A.: Analytical-numerical solution for nonlinear integral equations of Hammerstein type. International Journal of Mathematical Modelling and Computations 2(1), 61–69 (2012)Google Scholar
24. 24.
Ray, S.S., Sahu, P.K.: Numerical methods for solving Fredholm integral equations of second kind. Abstr. Appl. Anal. Art. ID 426916, 17 (2013)Google Scholar
25. 25.
Saberi-Nadja, J., Heidari, M.: Solving nonlinear integral equations in the Urysohn form by Newton-Kantorovich-quadrature method. Comput. Math. Applic. 60, 2018–2065 (2010)
26. 26.
Wazwaz, A.M.: A First Course in Integral Equations. World Scientific, Singapore (1997)