Subordination and superordination for univalent solutions for fractional differential equations. (English) Zbl 1147.30009

In this paper the authors establish the existence and uniqueness of univalent solutions for fractional differential equations. The existence is obtained by applying the Schauder fixed point theorem while the uniqueness is obtained by the Banach fixed point theorem. Also some properties of this solution involving fractional differential subordination are given.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
26A33 Fractional derivatives and integrals
Full Text: DOI


[1] Goodman, A. W., Univalent Function (1983), Mariner Publishing Company, INC · Zbl 1041.30501
[2] Srivastava, H. M.; Owa, S., Univalent Functions, Fractional Calculus, and Their Applications (1989), Halsted Press, John Wiley and Sons: Halsted Press, John Wiley and Sons New York-Chichester-Brisbane-Toronto · Zbl 0683.00012
[3] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press London · Zbl 0918.34010
[4] Miller, K. S.; Ross, B., An Introduction to The Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons, Inc. · Zbl 0789.26002
[5] Oldham, K. B.; Spanier, J., The Fractional Calculus, Math. Sci. Eng. (1974), Academic Press: Academic Press New York/London · Zbl 0428.26004
[6] Balachandar, K.; Dauer, J. P., Elements of Control Theory (1999), Narosa Publishing House · Zbl 0965.93002
[7] Curtain, R. F.; Pritchard, A. J., Functional Analysis in Modern Applied Mathematics (1977), Academic Press · Zbl 0448.46002
[8] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives (Theory and Applications) (1993), Gordon and Breach: Gordon and Breach New York · Zbl 0818.26003
[9] Miller, S. S.; Mocanu, P. T., Subordinants of differential superordinations, Complex Variables, 48, 10, 815-826 (2003) · Zbl 1039.30011
[10] Miller, S. S.; Mocanu, P. T., Differential Subordinations: Theory and Applications, Pure Appl. Math., vol. 225 (2000), Dekker: Dekker New York · Zbl 0954.34003
[11] Shanmugam, T. N.; Ravichangran, V.; Sivasubramanian, S., Differential sandwich theorems for some subclasses of analytic functions, Aust. Math, Anal. Appl., 3, 1, 1-11 (2006) · Zbl 1091.30019
[12] Bulboaca, T., Classes of first-order differential superordinations, Demonstratio Math., 35, 2, 287-292 (2002) · Zbl 1010.30020
[13] Kiryakova, V., Generalized Fractional Calculus and Applications, Pitman Res. Notes Math. Ser., vol. 301 (1994), Longman/Wiley: Longman/Wiley New York · Zbl 0882.26003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.