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On generalized Hyers-Ulam stability of admissible functions. (English) Zbl 1237.39033
Summary: We consider the Hyers-Ulam stability for the following fractional differential equations in sense of Srivastava-Owa fractional operators (derivative and integral) defined in the unit disk: $$D^\beta_z f(z) = G(f(z)$$, $$D^\alpha_z f(z)$$, $$zf'(z); z)$$, $$0 < \alpha < 1 < \beta \leq 2$$, in a complex Banach space. Furthermore, a generalization of the admissible functions in complex Banach spaces is imposed, and applications are illustrated.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 34A08 Fractional ordinary differential equations and fractional differential inclusions
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##### References:
 [1] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, NY, USA, 1961. · Zbl 0086.24101 [2] S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1964. · Zbl 0137.24201 [3] D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222-224, 1941. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222 [4] T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. · Zbl 0398.47040 · doi:10.2307/2042795 [5] D. H. Hyers, “The stability of homomorphisms and related topics,” in Global Analysis-Analysis on Manifolds, vol. 57 of Teubner-Texte zur Mathematik, pp. 140-153, Teubner, Leipzig, Germany, 1983. · Zbl 0517.22001 [6] D. H. Hyers and T. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125-153, 1992. · Zbl 0806.47056 · doi:10.1007/BF01830975 · eudml:137488 [7] D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications No. 34, Birkhäuser, Boston, Mass, USA, 1998. · Zbl 0907.39025 [8] Y. Li and L. Hua, “Hyers-Ulam stability of a polynomial equation,” Banach Journal of Mathematical Analysis, vol. 3, no. 2, pp. 86-90, 2009. · Zbl 1192.39022 · emis:journals/BJMA/tex_v3_n2_a10.pdf · eudml:225269 [9] M. Bidkham, H. A. S. Mezerji, and M. E. Gordji, “Hyers-Ulam stability of polynomial equations,” Abstract and Applied Analysis, vol. 2010, Article ID 754120, 7 pages, 2010. · Zbl 1201.39012 · doi:10.1155/2010/754120 · eudml:228037 [10] M. J. Rassias, “Generalised Hyers-Ulam product-sum stability of a Cauchy type additive functional equation,” European Journal of Pure and Applied Mathematics, vol. 4, no. 1, pp. 50-58, 2011. · Zbl 1218.39023 · www.ejpam.com [11] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011. · Zbl 1221.39038 [12] J. Wang, L. Lv, and Y. Zhou, “Ulam stability and data dependence for fractional differential equations with Caputo derivative,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 63, pp. 1-10, 2011. · Zbl 1340.34034 [13] R. W. Ibrahim, “Approximate solutions for fractional differential equation in the unit disk,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 64, pp. 1-11, 2011. · Zbl 1340.34018 [14] R. W. Ibrahim, “Ulam stability for fractional differential equation in complex domain,” Abstract and Applied Analysis, vol. 2012, Article ID 649517, 8 pages, 2012. · Zbl 1239.34106 · doi:10.1155/2012/649517 [15] R. W. Ibrahim, “Generalized Ulam-Hyers stability for fractional differential equations,” International Journal of Mathematics. In press. · Zbl 1256.34004 [16] K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229-248, 2002. · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194 [17] R. W. Ibrahim and S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 1-10, 2007. · Zbl 1123.34302 · doi:10.1016/j.jmaa.2006.12.036 [18] S. Momani and R. W. Ibrahim, “On a fractional integral equation of periodic functions involving Weyl-Riesz operator in Banach algebras,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1210-1219, 2008. · Zbl 1136.45010 · doi:10.1016/j.jmaa.2007.08.001 [19] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, London, UK, 1999. · Zbl 0924.34008 [20] R. W. Ibrahim, “Initial and boundary value problems for inclusions involving Caputo’s fractional derivatives,” Pure Mathematics and Applications, vol. 18, no. 1-2, pp. 1-14, 2007. · Zbl 1224.34023 [21] O. P. Agrawal, “Fractional optimal control of a distributed system using Eigenfunctions,” Journal of Computational and Nonlinear Dynamics, vol. 3, no. 2, Article ID 021204, 2008. · doi:10.1115/1.2833873 [22] G. M. Mophou, “Optimal control of fractional diffusion equation,” Computers & Mathematics with Applications, vol. 61, no. 1, pp. 68-78, 2011. · Zbl 1228.35263 · doi:10.1016/j.camwa.2011.03.025 [23] J. Cao and H. Chen, “Some results on impulsive boundary value problem for fractional differential inclusions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 11, pp. 1-24, 2011. · Zbl 1261.34003 [24] R. A. Khan, M. Rehman, and J. Henderson, “Existence and uniqueness for nonlinear differential equations,” Fractional Differential Equations, vol. 1, no. 1, pp. 29-43, 2011. [25] S. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 804-812, 2000. · Zbl 0972.34004 · doi:10.1006/jmaa.2000.7123 [26] R. W. Ibrahim and M. Darus, “Subordination and superordination for analytic functions involving fractional integral operator,” Complex Variables and Elliptic Equations, vol. 53, no. 11, pp. 1021-1031, 2008. · Zbl 1155.30006 · doi:10.1080/17476930802429131 [27] R. W. Ibrahim and M. Darus, “Subordination and superordination for univalent solutions for fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 871-879, 2008. · Zbl 1147.30009 · doi:10.1016/j.jmaa.2008.05.017 [28] R. W. Ibrahim, “Existence and uniqueness of holomorphic solutions for fractional Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 380, no. 1, pp. 232-240, 2011. · Zbl 1214.30027 · doi:10.1016/j.jmaa.2011.03.001 [29] H. M. Srivastava and S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood Ltd., Chichester, UK, 1989. · Zbl 0683.00012 [30] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherland, 2006. · Zbl 1092.45003 [31] J. Sabatier, O. P. Agrawal, and J. A. Machado, Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007. · Zbl 1116.00014 · doi:10.1007/978-1-4020-6042-7 [32] E. Hille and R. S. Phillips, Functional Analysis and Semi-Group, American Mathematical Society Colloquium Publications Vol. 31, American Mathematical Society, Providence, RI, USA, 1957. · Zbl 0078.10004 · www.ams.org [33] S. S. Miller and P. T. Mocanu, Differential Subordinantions: Theory and Applications., vol. 225 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000. · Zbl 0954.34003 [34] S. Owa, K. Nishimoto, S. K. Lee, and N. E. Cho, “A note on certain fractional operator,” Bulletin of the Calcutta Mathematical Society, vol. 83, no. 2, pp. 87-90, 1991. · Zbl 0754.30009
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