Existence and stability of solutions for a system of quadratic integral equations in Banach algebras

Said Baghdad

doi:10.2478/aupcsm-2020-0015

.blurhash-client-img { display: none !important; }

Existence and stability of solutions for a system of quadratic integral equations in Banach algebras

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

Volume 19 (2020): Issue 1 (December 2020)

By: Said Baghdad

Open Access

|Dec 2020

Abstract

The aim of this paper is to prove the existence and stability of solutions of a system of quadratic integral equations in the Banach algebra of continuous and bounded functions on unbounded rectangle. The main tool used in our considerations is the multiple fixed point theorem which is a consequence of Darbo’s fixed point theorem and the technique associated with measures of noncompactness. We also present an illustrative example.

References

[1] Abbas, Saïd et al. “Existence and attractivity results for coupled systems of nonlinear Volterra-Stieltjes multidelay fractional partial integral equations.” Abstr. Appl. Anal. (2018): Art. ID 8735614. Cited on 204.
Search in Google Scholar Back to article
[2] Abbas, Saïd, and Mouffak Benchohra, and Juan J. Nieto Roig. “Global attractivity of solutions for nonlinear fractional order Riemann-Liouville Volterra-Stieltjes partial integral equations.” Electron. J. Qual. Theory Differ. Equ. (2012): Article no. 81. Cited on 204.
Search in Google Scholar Back to article
[3] Abbas, Saïd et al. “Existence and stability results for nonlinear fractional order Riemann-Liouville Volterra-Stieltjes quadratic integral equations.” Appl. Math. Comput. 247 (2014): 319-328. Cited on 203.
Search in Google Scholar Back to article
[4] Aghajani, Asadollah, and Ali Shole Haghighi. “Existence of solutions for a system of integral equations via measure of noncompactness.” Novi Sad J. Math. 44, no. 1 (2014): 59-73. Cited on 204, 206 and 209.
Search in Google Scholar Back to article
[5] Aghajani, Asadollah, Reza Allahyari, and Mohammad Mursaleen. “A generalization of Darbo’s theorem with application to the solvability of systems of integral equations.” J. Comput. Appl. Math. 260 (2014): 68-77. Cited on 204 and 206.
Search in Google Scholar Back to article
[6] Akhmerov, R.R. et al. Measures of Noncompactness and Condensing Operators Vol. 55 of Operator Theory: Advances and Applications. Basel: Birkhäuser Verlag, 1992. Cited on 204, 205 and 206.10.1007/978-3-0348-5727-7
Search in Google Scholar Back to article
[7] Baghdad, Said, and Mouffak Benchohra. “Global existence and stability results for Hadamard-Volterra-Stieltjes integral equations.” Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 68, no. 2 (2019): 1387-1400. Cited on 204.
Search in Google Scholar Back to article
[8] Banaś, Józef. “Existence results for Volterra-Stieltjes quadratic integral equations on an unbounded interval.” Math. Scand. 98, no. 1 (2006): 143-160. Cited on 204.
Search in Google Scholar Back to article
[9] Banaś, Józefet et al. “Monotonic solutions of a class of quadratic integral equations of Volterra type.” Comput. Math. Appl. 49, no. 5-6 (2005): 943-952. Cited on 203, 204 and 206.
Search in Google Scholar Back to article
[10] Banaś, Józef, and Szymon Dudek. “The technique of measures of noncompactness in Banach algebras and its applications to integral equations.” Abstr. Appl. Anal. (2013): Art. ID 537897. Cited on 204, 205 and 207.
Search in Google Scholar Back to article
[11] Banaś, Józef, and Kazimierz Goebel. Measures of noncompactness in Banach spaces vol. 60 of Lecture Notes in Pure and Applied Mathematics. New York: Marcel Dekker, Inc., 1980. Cited on 204, 205 and 206.
Search in Google Scholar Back to article
[12] Banaś, Józef, Millenia Lecko, and Wagdy Gomaa El-Sayed. “Existence theorems for some quadratic integral equations.” J. Math. Anal. Appl. 222, no. 1 (1998): 276-285. Cited on 203 and 204.
Search in Google Scholar Back to article
[13] Banaś, Józef, and Leszek Olszowy. “On a class of measures of noncompactness in Banach algebras and their application to nonlinear integral equations.” Z. Anal. Anwend. 28, no. 4 (2009): 475-498. Cited on 204, 205 and 207.
Search in Google Scholar Back to article
[14] Banaś, Józef, and Donal O’Regan. “On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order.” J. Math. Anal. Appl. 345, no. 1 (2008): 573-582. Cited on 203, 204 and 210.
Search in Google Scholar Back to article
[15] Banaś, Józef, Méndez, and Kishin B. Sadarangani. “On a class of Urysohn-Stieltjes quadratic integral equations and their applications.” J. Comput. Appl. Math. 113, no. 1-2 (2000): 35-50. Cited on 203 and 204.
Search in Google Scholar Back to article
[16] Banaś, Józef, and Beata Rzepka. “Monotonic solutions of a quadratic integral equation of fractional order.” J. Math. Anal. Appl. 332, no. 2 (2007): 1371-1379. Cited on 203 and 204.
Search in Google Scholar Back to article
[17] Banaś, Józef, and Kishin B. Sadarangani. “Solvability of Volterra-Stieltjes operator-integral equations and their applications.” Comput. Math. Appl. 41, no. 12 (2001): 1535-1544. Cited on 204.
Search in Google Scholar Back to article
[18] Chandrasekhar, Subrahmanyan. Radiative transfer. New York: Dover Publications, 1960. Cited on 203.
Search in Google Scholar Back to article
[19] Corduneanu, Constantin. Integral equations and stability of feedback systems. Vol. 104 of Mathematics in Science and Engineering. New York, London: Academic Press, 1973. Cited on 203.
Search in Google Scholar Back to article
[20] Dhage, Bapurao Chandrabhan. “A coupled hybrid fixed point theorem involving the sum of two coupled operators in a partially ordered Banach space with applications.” Differ. Equ. Appl. 9, no. 4 (2017): 453-477. Cited on 203, 204 and 206.
Search in Google Scholar Back to article
[21] Dhage, Bapurao Chandrabhan and Sidheshwar Sangram Bellale. “Local asymptotic stability for nonlinear quadratic functional integral equations.” Electron. J. Qual. Theory Differ. Equ. (2008): Art. no. 10. Cited on 203, 204 and 210.
Search in Google Scholar Back to article
[22] Kilbas, Anatoly Aleksandrovich, Hari Mohan Srivastava, and Juan J. Trujillo. Theory and applications of fractional differential equations Vol. 204 of North-Holland Mathematics Studies. Amsterdam: Elsevier Science B.V., 2006. Cited on 207.
Search in Google Scholar Back to article
[23] Lebesgue, Henri Leon. Leçons sur l’intégration et la recherche des fonctions primitives professées au Collège de France. Cambridge Library Collection. Cambridge: Cambridge University Press, 2009. Reprint of the 1904 original. Cited on 209.10.1017/CBO9780511701825
Search in Google Scholar Back to article
[24] Isidor Pavlovič Natanson. Theory of functions of a real variable Vol. 85 of North-Holland Mathematics Studies. Berlin: Akademie-Verlag, 1981. Cited on 208 and 209.
Search in Google Scholar Back to article
[25] Schwabik, Štefan, Milan Tvrdý and Otto Vejvoda. Differential and integral equations. Dordrecht, Boston, London: D. Reidel Publishing Co., 1979. Cited on 203, 208 and 209.
Search in Google Scholar Back to article
[26] Sikorski, Roman. Funkcje rzeczywiste. Warszawa: Państwowe Wydawnictwo Naukowe, 1958. Cited on 209.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/aupcsm-2020-0015 | Journal eISSN: 2300-133X | Journal ISSN: 2081-545X

Journal RSS Feed

Language: English

Page range: 203 - 218

Submitted on: Aug 18, 2020

Accepted on: Nov 13, 2020

Published on: Dec 31, 2020

Published by: Pedagogical University of Cracow

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

quadratic integral equations of fractional order,

fixed-point theorems,

Banach algebra,

measure of noncompactness,

stability

Related subjects:

Mathematics,

General mathematics

© 2020 Said Baghdad, published by Pedagogical University of Cracow
This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 License.

Volume 19 (2020): Issue 1 (December 2020)