On the Existence of the Solutions of A Nonlinear Fredholm Integral Equation in Hölder Spaces

Year 2022, Volume: 10 Issue: 1, 16 - 26, 01.03.2022

Merve Temizer Ersoy

https://doi.org/10.36753/mathenot.756916

Cited By: 1

Abstract

In this article, we prove the theorem concerning the existence of the solutions for some nonlinear integral equations. As an application, we investigate the problem of existence of solutions of Fredholm integral equations using the technique of relative compactness in conjunction with fixed point theorem. Our solutions are placed in the space of functions satisfying the Hölder condition. Our work is more general than the previous works in [1-3]. In the last section, we show the efficiency of this approach on one numerical example.

Keywords

Hölder condition, Fredholm integral equation, Schauder fixed point theorem

References

• [1] Banas ́, J., Nalepa, R.:On the space of functions with growths tempered by a modulus of continuity and its applications. J. Funct. Space Appl. 13 pages, (2013). doi:http://dx.doi.org/10.1155/2013/820437
• [2] Caballero, J., Abdalla, M., Sadarangani, K.: Solvability of a quadratic integral equation of Fredholm type in Hölder spaces. Electron. J. Differ. Eq. 31, 1–10 (2014).
• [3] Peng, S., Wang, J., Chen, F.: A Quadratic integral equation in the space of funtions with tempered moduli of continuity. J. Appl. Math. and Informatics. 33, No. 3–4, 351–363 (2015).
• [4] Banas ́, J., Chlebowicz, A.: On an elementary inequality and its application in theory of integral equations. Journal of Mathematical Inequalities. 11 (2), 595–605 (2017).
• [5] Caballero, M. J., Nalepa, R., Sadarangani, K.: Solvability of a quadratic integral equation of Fredholm type with Supremum in Hölder Spaces. J. Funct. Space Appl. 7 pages, (2014).
• [6] Kulenovic, M. R. S.: Oscillation of the Euler differential equation with delay. Czech.Math. J. 45, 1–16 (1995).
• [7] Mures ̧an, V.: On a class of Volterra integral equations with deviating argument. Studia Univ.Babes-Bolyai Math. 44, 47–54 (1999).
• [8] Mures ̧an, V.: Volterra integral equations with iterations of linear modification of the argument. Novi Sad J. Math. 33 (2), 1–10 (2003).
• [9] Schauder, J.: Der Fixpunktsatz in Funktionalriiumen. Studia Math. 2, 171–180 (1930).
• [10] López, B., Harjani, J., Sadaragani, K.: Existence of positive solutions in the space of Lipschitz functions to a class of fractional differential equations of arbitrary order. Racsam, 112, 1281–1294 (2018).
• [11] Bacotiu, C.: Volterra-Fredholm nonlinear systems with modified argument via weakly Picard operators theory. Carpathian J. Math. 24, 1–19 (2008).
• [12] Benchohra, M., Darwish, M. A.: On unique solvability of quadratic integral equations with linear modification of theargument. Miskolc Math. Notes. 10, 3–10 (2009).
• [13] Dobritoiu, M.: Analysis of a nonlinear integral equation with modified argument from physics. Int. J. Math. Models 891–937 (1971).
• [14] Kato, T., Mcleod, J. B.: The functional-differential equation y′(x) = ay(λx) + by(x). Bull. Amer. Math. Soc. 77 (6),
• [15] Lauran, M.: Existence results for some differential equations with deviating argument. Filomat. 25, 21–31 (2011).
• [16] Mures ̧an,V.:Afunctional-integralequationwithlinearmodificationoftheargumentviaweaklyPicardoperators.Fixed Point Theory. 9, 189–197 (2008).
• [17] Mures ̧an, V.: A Fredholm-Volterra integro-differential equation with linear modification of the argument. J. Appl. Math. 3 (2), 147–158 (2010).
• [18] Agarwal, R. P., O’Regan, D.: Infinite interval problems for differential, difference and integral equations. Dordrecht, Kluwer Academic Publishers, Springer Netherlands, ISBN 978-94-015-9171-3 (2001).
• [19] Agarwal, R. P., O’Regan, D., Wong, P. J. Y.: Positive solutions of differential, difference and integral equations. Dordrecht, Kluwer Academic Publishers, Springer Netherlands (1999).
• [20] Case, K. M., Zweifel, P. F.: Linear Transport Theory. Addison Wesley (1967).
• [21] Chandrasekhar, S.: Radiative transfer. Dover Publications, New York (1960).
• [22] Hu, S., Khavani, M., Zhuang, W.: Integral equations arising in the kinetic theory of gases. J. Appl. Anal. 34, 261–266 (1989).
• [23] Kelly, C. T.: Approximation of solutions of some quadratic integral equations in transport theory. J. Int. Eq. 4, 221–237 (1982).
• [24] Banas ́, J., Lecko, M., El-Sayed, W. G.: Existence theorems of some quadratic integral equation. J. Math. Anal. Appl. 222, 276–285 (1998).
• [25] Banas ́, J., Caballero,J., Rocha J., Sadarangani, K.: Monotonic solutions of a class of quadratic integral equations of Volterra type. Comput. Math. Appl. , 49, 943–952 (2005).
• [26] Caballero, J., Rocha, J., Sadarangani, K.: On monotonic solutions of an integral equation of Volterra type. J. Comput. Appl. Math. 174, 119–133 (2005).
• [27] Darwish, M. A.: On solvability of some quadratic functional-integral equation in Banach algebras. Commun. Appl. Anal. 11, 441–450 (2007).
• [28] Darwish, M. A., Ntouyas, S. K.: On a quadratic fractional Hammerstein-Volterra integral equations with linear modification of the argument. Nonlinear Anal. Theor. 74, 3510–3517 (2011).
• [29] Darwish, M. A.: On quadratic integral equation of fractional orders. J. Math. Anal. Appl. 311, 112–119 (2005).
• [30] Agarwal, R. P., Banas ́, J., Banas ́, K., O’Regan, D.: Solvability of a quadratic Hammerstein integral equation in the class of functions having limits at infinity. J. Int. Eq. Appl. 23, 157–181 (2011).
• [31] Caballero, J., Darwish, M. A., Sadarangani, K.: Positive Solutions in the Space of Lipschitz Functions for Fractional Boundary Value Problems with Integral Boundary Conditions. Mediterr. J. Math. 14 (201) (2017).
• [32] Cabrera, I., Harjani, J., Sadarangani, K.: Existence and Uniqueness of Solutions for a Boundary Value Problem of Fractional Type with Nonlocal Integral Boundary Conditions in Hölder Spaces. Mediterr. J. Math. 15 (3) (2018).