A few complex equations constituted by an operator consisting of fractional calculus and their consequences. (English) Zbl 1300.30026
Summary: A few complex (differential) equations constituted by certain operators consisting of fractional calculus are first presented and some of their comprehensive consequences relating to (analytic and) geometric function theory are then pointed out.
MSC:
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 26A33 | Fractional derivatives and integrals |
References:
| [1] | A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003 |
| [2] | A. A. Kilbas, O. I. Marichev, and S. G. Samko, Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. · Zbl 0818.26003 |
| [3] | A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methodsresults and problems-I,” Applicable Analysis, vol. 78, pp. 153-192, 2001. · Zbl 1031.34002 · doi:10.1080/00036810108840931 |
| [4] | V. Kiryakova, Generalized Fractional Calculus and Applications, Longman Sci. Tech., Harlow, UK, John Wiley & Sons, New York, NY, USA, 1994. · Zbl 0882.26003 |
| [5] | V. Laksmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, U.K., 2009. · Zbl 1188.37002 |
| [6] | R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,” Indiana University Mathematics Journal, vol. 28, pp. 673-688, 1979. · Zbl 0421.47033 · doi:10.1512/iumj.1979.28.28046 |
| [7] | K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equation, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002 |
| [8] | K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002 |
| [9] | K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011 |
| [10] | H. M. Srivastava and S. Owa, Eds., Univalent Functions, Fractional Calculus and Their Applications, Halsted Press, John Wiley & Sons, New York, NY, USA, 1989. · Zbl 0683.00012 |
| [11] | H. Pollard, “The complete monotonic character of the Mittag-Leffler function Erf(x),” Bulletin of the American Mathematical Society, vol. 54, pp. 1115-1116, 1948. · Zbl 0033.35902 · doi:10.1090/S0002-9904-1948-09132-7 |
| [12] | B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers and Mathematics with Applications, vol. 58, no. 9, pp. 1838-1843, 2009. · Zbl 1205.34003 · doi:10.1016/j.camwa.2009.07.091 |
| [13] | R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973-1033, 2010. · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6 |
| [14] | A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an unbounded domain,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 580-586, 2009. · Zbl 1179.26015 · doi:10.1016/j.na.2009.06.106 |
| [15] | D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3692-3705, 2008. · Zbl 1166.34033 · doi:10.1016/j.na.2007.10.004 |
| [16] | R. I. Avery and J. Henderson, “Two positive fixed points of nonlinear operators on ordered Banach spaces,” Communications on Applied Nonlinear Analysis, vol. 8, pp. 27-36, 2001. · Zbl 1014.47025 |
| [17] | M. Belmekki, J. J. Nieto, and R. Rodríguez-López, “Existence of periodic solution for a nonlinear fractional differential equation,” Boundary Value Problems, vol. 2009, Article ID 324561, 2009. · Zbl 1181.34006 · doi:10.1155/2009/324561 |
| [18] | B. Bonilla, M. Rivero, L. Rodríguez-Germá, and J. J. Trujillo, “Fractional differential equations as alternative models to nonlinear differential equations,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 79-88, 2007. · Zbl 1120.34323 · doi:10.1016/j.amc.2006.08.105 |
| [19] | Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions with boundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605-609, 2009. · Zbl 1165.34313 · doi:10.1016/j.mcm.2008.03.014 |
| [20] | R. Dehghant and K. Ghanbari, “Triple positive solutions for boundary value problem of a nonlinear fractional differential equation,” Bulletin of the Iranian Mathematical Society, vol. 33, no. 2, pp. 1-14, 2007. · Zbl 1148.34008 |
| [21] | 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 |
| [22] | E. R. Kaufmann and E. Mboumi, “Positive solutions of a boundary value problem for a nonlinear fractional differential equation,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1-11, 2008. · Zbl 1183.34007 |
| [23] | V. Lakshmikantham and S. Leela, “Nagumo-type uniqueness result for fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2886-2889, 2009. · Zbl 1177.34003 · doi:10.1016/j.na.2009.01.169 |
| [24] | V. Laksmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, pp. 2677-2682, 2008. · Zbl 1161.34001 |
| [25] | W. R. Schneider, “Completely monotone generalized Mittag-Leffler functions,” Expositiones Mathematicae, vol. 14, pp. 3-16, 1996. · Zbl 0843.60024 |
| [26] | S. Zhang, “Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 2087-2093, 2009. · Zbl 1172.26307 · doi:10.1016/j.na.2009.01.043 |
| [27] | S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, “On the solution of the fractional nonlinear Schrödinger equation,” Physics Letters A, vol. 372, no. 5, pp. 553-558, 2008. · Zbl 1217.81068 · doi:10.1016/j.physleta.2007.06.071 |
| [28] | 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 |
| [29] | F. Zang, “Existence results of positive solutions to boundary value problem for fractional differential equation,” Positivity, vol. 13, no. 3, pp. 583-599, 2008. · Zbl 1202.26018 · doi:10.1007/s11117-008-2260-5 |
| [30] | O. Altinta\cs, H. Irmak, and H. M. Srivastava, “Fractional calculus and certain starlike functions with negative coefficients,” Computers and Mathematics with Applications, vol. 30, no. 2, pp. 9-15, 1995. · Zbl 0838.30011 · doi:10.1016/0898-1221(95)00073-8 |
| [31] | M.-P. Chen, H. Irmak, and H. M. Srivastava, “Some families of multivalently analytic functions with negative coefficients,” Journal of Mathematical Analysis and Applications, vol. 214, no. 2, pp. 674-690, 1997. · Zbl 0883.30011 |
| [32] | M.-P. Chen, H. Irmak, and H. M. Srivastava, “A certain subclass of analytic functions involving operators of fractional calculus,” Computers and Mathematics with Applications, vol. 35, no. 5, pp. 83-91, 1998. · Zbl 0921.30012 · doi:10.1016/S0898-1221(98)00007-8 |
| [33] | H. Irmak and N. Tuneski, “Fractional calculus operator and certain applications in geometric function theory,” Sarajevo Journal of Mathematics, vol. 6, no. 18, pp. 51-57, 2010. · Zbl 1200.30014 |
| [34] | P. L. Duren, Grundlehren der Mathematischen Wissenchaffen, Springer, New York, NY, USA, 1983. |
| [35] | A. W. Goodman, Univalent Functions, vol. 1-2, Polygonal Publishing House, Washington, DC, USA, 1983. · Zbl 1041.30501 |
| [36] | M. Nunokawa, “On properties of non-caratheodory functions,” Proceedings of the Japan Academy A, vol. 68, pp. 152-153, 1992. · Zbl 0773.30020 · doi:10.3792/pjaa.68.152 |
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.
