A uniqueness criterion for fractional differential equations with Caputo derivative. (English) Zbl 1268.34008

Summary: We investigate the uniqueness of solutions to an initial value problem associated with a nonlinear fractional differential equation of order \(\alpha \in (0,1)\). The differential operator is of Caputo type whereas the nonlinearity cannot be expressed as a Lipschitz function. Instead, the Riemann-Liouville derivative of this nonlinearity verifies a special inequality.


34A08 Fractional ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
Full Text: DOI


[1] Agarwal, R.P., Lakshmikantham, V.: Uniqueness and nonuniqueness criteria for ordinary differential equations. World Scientific, Singapore (1993) · Zbl 0785.34003
[2] Băleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. World Scientific, Boston (2012) · Zbl 1248.26011
[3] Băleanu, D., Avkar, T.: Lagrangians with linear velocities within Riemann–Liouville fractional derivatives. Nuovo Cimento B 119, 73–79 (2004)
[4] Băleanu, D., Mustafa, O.G.: On the asymptotic integration of a class of sublinear fractional differential equations. J. Math. Phys. 50, 123520 (2009) · Zbl 1373.34004
[5] Băleanu, D., Mustafa, O.G.: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59, 1835–1841 (2010) · Zbl 1189.34006
[6] Băleanu, D., Mustafa, O.G., O’Regan, D.: A Nagumo-like uniqueness theorem for fractional differential equations. J. Phys. A 44, 392003 (2011) · Zbl 1229.26013
[7] Băleanu, D., Agarwal, R.P., Mustafa, O.G., Coşulschi, M.: Asymptotic integration of some nonlinear differential equations with fractional time derivative. J. Phys. A 44, 055203 (2011) · Zbl 1238.26008
[8] Bhalekar, S., Daftardar-Gejji, V., Băleanu, D., Magin, R.L.: Transient chaos in fractional Bloch equations. Comput. Math. Appl. (2012). doi: 10.1016/j.camwa.2012.01.069 · Zbl 1268.34009
[9] Delavari, H., Băleanu, D., Sadati, J.: Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 67(4), 2433–2439 (2012) · Zbl 1243.93081
[10] Agrawal, O.P., Defterli, O., Băleanu, D.: Fractional optimal control problems with several state and control variables. J. Vib. Control 16(13), 1967–1976 (2010) · Zbl 1269.49002
[11] Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964) · Zbl 0125.32102
[12] Herzallah, M.A.E., Băleanu, D.: Fractional Euler–Lagrange equations revisited. Nonlinear Dyn. (2012). doi: 10.1007/s11071-011-0319-5 · Zbl 1256.26004
[13] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland, New York (2006) · Zbl 1092.45003
[14] Lovelady, D.L., Martin, R.H. Jr.: A global existence theorem for a nonautonomous differential equation in a Banach space. Proc. Am. Math. Soc. 35, 445–449 (1972) · Zbl 0222.34059
[15] Lovelady, D.L.: A necessary and sufficient condition for exponentially bounded existence and uniqueness. Bull. Aust. Math. Soc. 8, 133–135 (1973) · Zbl 0246.34060
[16] McShane, E.J.: Linear functionals on certain Banach spaces. Proc. Am. Math. Soc. 1, 402–408 (1950) · Zbl 0039.11802
[17] Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) · Zbl 0789.26002
[18] Mustafa, O.G., O’Regan, D.: On the Nagumo uniqueness theorem. Nonlinear Anal. TMA 74, 6383–6386 (2011) · Zbl 1252.34011
[19] Mustafa, O.G.: On the uniqueness of flow in a recent tsunami model. Appl. Anal. (2011). doi: 10.1080/00036811.2011.569499 (on-line)
[20] Mustafa, O.G.: A Nagumo-like uniqueness result for a second order ODE. Monatshefte Math. (2011). doi: 10.1007/s00605-011-0324-2 (on-line) · Zbl 1256.34005
[21] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[22] Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) · Zbl 0925.00005
[23] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach, New York (1993) · Zbl 0818.26003
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.