Existence of solutions of infinite systems of nonlinear Volterra integral equations on a Fréchet space

Authors
Department of Mathematics, Mashhad Branch, Islamic Azad Uni- versity, Mashhad, Iran.
Abstract
In this paper, we introduce a Fréchet space and define a measure of noncompactness on it. For credit and the application to our theorems, in the application section of this paper, we present a theorem which shows the existence of solution of infinite system of nonlinear Volterra integral equations, and by using the technique of measures of noncompactness together Darbo fixed point theorem we guarantee the existence of solution in this Fréchet space . Finally, we present some examples which show the efficiency and applicability of our theorem../files/site1/files/72/5Abstract.pdf
Keywords

1. Agarwal R. P., Meehan M., O'Regan D., "Fixed Point Theory and Applications", Cambridge university press, Vol. 141 (2001).## 2. Banas J., Goebel K., "Measures of noncompactness in Banach spaces", Lecture Notes in Pure and Applied Math. Vol. 60, Dekker, NewYork (1980). ## 3. Darbo G., "Punti uniti in trasformazioni a codominio non compatto", Rend. Sem. Mat. Uni. Padova, 24 (1955) 84-92. ## 4. Deimling K., "Ordinary differential equations in Banach spaces", Lecture Notes in Math., Vol. 596, Springer Verlag, Berlin (1977). ## 5. Kuratowski K., "Sur les espaces complets", Fund. Math. 15 (1930) 301-309. ## 6. Latefa B., Dejebali S., "Solvability of functional integral equations in the Frechet space ", Mediter. J. Math. (2016). ## 7. Mursaleen M., Mohiuddine S. A., "Applications of measures of noncompactness to the infinite system of differential equations in spaces", Nonlinear Anal. 75 (2012) 2111-2115. ## 8. Mursaleen M., Alotaibi A., "Infinite system of differential equations in some BK spaces", Abstr. Applied Anal, (2012), Article ID 863483, 20. ## 9. Olszowy L., "Solvability of infinite systems of singular integral equations in Frechet space of continuous functions", Computers and Mathematics with Applications, 59 (2010) 2794-2801. ## 10. Olszowy L.,"On some measure of noncompactness in Frechet space of continuous functions", Nonlinear Anal. 71 (2009) 5157-5163. ## 11. M.N. Oguzt¨oreli, On the neural equations of cowan and stein, Util. Math. 2 (1972) 305-315. ## 12. Persidskii K. P., "Countable systems of differential equations and stability of their solutions III: Fundamental theorems on stability of solutions of countable many differential equations", Izv. Akad. Nauk Kazach. SSR 9 (1961) 11-34. ## 13. Rzepka R., Sadarangani K., "On solutions of an infinite system of singular integral equations", Math. Comput. Modelling 45 (2007) 1265-1271. ## 14. Zautykov O. A., "Countable systems of differential equations and their applications", Differ. Uravn. 1 (1965) 162-170. ## 15. Zautykov O. A., Valeev K. G., "Infinite systems of differential equations, Izdat", ”Nauka ” Kazach. SSR, Alma-Ata (1974). ## 16. مؤسسۀ نشرعلوم نوین،" آنالیزتابعی"، والتررودین، ترجمۀ علی اکبرعالم زاده، (1383). ##