Fixed point approach to the stability of an integral equation in the sense of Ulam-Hyers-Rassias
Abstract
In this paper, by using the classical Banach contraction principle, we investigate and establish the stability in the sense of Ulam–Hyers and in the sense of Ulam–Hyers–Rassias for the integral equation which defines the mild solutions of an abstract Cauchy problem in Banach spaces.
Keywords
Ulam–Hyers stability; Ulam–Hyers–Rassias stability; Banach contraction principle; C_0-semi group; abstract Cauchy problem; mild solutions; Banach spaces
References
2. Aoki T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan. 2 (1950), 64–66.
3. Bourgin D.G., Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949), 385–397.
4. Cădariu L., Radu V., Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Art. ID 4.
5. Diaz J.B., Margolis B., A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309.
6. Gajda Z., On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434.
7. Hyers D.H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224.
8. Hyers D.H., The stability of homomorphisms and related topics, Global gnalysis–analysis on manifolds (T.M. Rassias ed.), Teubner-Texte Math., B. 57, Teubner, Leipzig, 1983, pp. 140–153.
9. Hyers D.H., Isac G., Rassias Th.M., Stability of Functional Equation in Several Variables, Birkhäuser, Basel, 1998.
10. Jung S.-M., Hyers–Ulam–Rassias stability of Jensen’s equation and its application, Proc. Amer. Math. Soc. 126 (1998), 3137–3143.
11. Jung S.-M., Hyers–Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004), no. 10, 1135–1140.
12. Jung S.-M., Hyers–Ulam stability of linear differential equations of first order. II, Appl. Math. Lett. 19 (2006), no. 9, 854–858.
13. Jung S.-M., Hyers–Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl. 320 (2006), no. 2, 549–561.
14. Jung S.-M., A fixed point approach to the stability of differential equations y' = F(x,y), Bull. Malays. Math. Sci. Soc. (2) 33 (2010), no. 1, 47–56.
15. Kuczma M., Functional equations in a single variable, Monographs math., vol. 46, PWN, Warszawa, 1968.
16. Miura T., Miyajima S., Takahasi S.-E., A characterization of Hyers–Ulam stability of first order linear differential operators, J. Math. Anal. Appl. 286 (2003), no. 1, 136–146.
17. Miura T., Miyajima S., Takahasi S.-E., Hyers–Ulam stability of linear differential operator with constant coefficients, Math. Nachr. 258 (2003), 90–96.
18. Obloza M., Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat. 13 (1993), 259–270.
19. Obloza M., Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat. 14 (1997), 141–146.
20. Park C., Hyers–Ulam–Rassias stability of homomorphisms in quasi-Banach algebras, Bull. Sci. Math. 132 (2008), 87–96.
21. Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
22. Radu V., The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91–96.
23. Rassias Th.M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
24. Rassias Th.M., The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352–378.
25. Rassias Th.M., On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284.
26. Rassias Th.M., On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23–130.
27. Rassias Th.M., Functional Equations, Inequalities and Applications, Kluwer Academic, Dordrecht, Boston and London, 2003.
28. Rus I.A., Metrical fixed point theorems, University of Cluj-Napoca, Department of Mathematics, 1979.
29. Rus I.A., Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math. 26 (2010), no. 1, 103–107.
30. Takahasi S.-E., Miura T., Miyajima S., On the Hyers–Ulam stability of the Banach space valued differential equation y' = λy, Bull. Korean Math. Soc. 39 (2002), no. 2, 309–315.
31. Ulam S.M., Problems in Modern Mathematics, Chapter VI, Science Editions Wiley, New York, 1960.
32. Ulam S.M., A Collection of the Mathematical Problems, Interscience Publ., New York, 1960.
33. Wang J., Some further generalizations of the Hyers–Ulam Rassias stability of functional equations, J. Math. Anal. Appl. 263 (2001), 406–423.
Department of Mathematics, Faculty of Sciences, University Cadi Ayyad, Morocco Morocco
Department of Mathematics, Faculty of Sciences, University Cadi Ayyad, Morocco Morocco
Department of Mathematics, Faculty of Sciences, University Cadi Ayyad, Morocco Morocco
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