Fractional integral inequalities and applications. (English) Zbl 1189.26044

Summary: Fractional integral inequality results when \(0<q<1\) are developed when the nonlinear term is increasing in \(u\) and satisfies a one sided Lipschitz condition. Using the integral inequality result and the computation of the solution of the linear fractional equation of variable coefficients, Gronwall inequality results are established. This yields the results of \(q=1\) as a special case. As an application of this, the uniqueness and continuous dependence of the solution on the initial parameters of the nonlinear fractional differential equations are established.


26D15 Inequalities for sums, series and integrals
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
Full Text: DOI


[1] Caputo, M., Linear models of dissipation whose \(Q\) is almost independent, II, Geophys. J. R. astron., 13, 529-539, (1967)
[2] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, Jmaa, 265, 229-248, (2002) · Zbl 1014.34003
[3] Diethelm, K.; Ford, N.J., Multi-order fractional differential equations and their numerical solution, Amc, 154, 621-640, (2004) · Zbl 1060.65070
[4] Diethelm, K.; Freed, A.D., On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, (), 217-224
[5] Glöckle, W.G.; Nonnenmacher, T.F., A fractional calculus approach to self similar protein dynamics, Biophys. J., 68, 46-53, (1995)
[6] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier, North Holland · Zbl 1092.45003
[7] Metzler, R.; Schick, W.; Kilian, H.G.; Nonnenmacher, T.F., Relaxation in filled polymers: A fractional calculus approach, J. chem. phys., 103, 7180-7186, (1995)
[8] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives, theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[9] Oldham, B.; Spanier, J., The fractional calculus, (1974), Academic Press New York, London · Zbl 0292.26011
[10] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[11] Lakshmikantham, V.; Leela, S.; Vasundhara, D.J., Theory of fractional dynamic systems, (2009), Cambridge Scientific Publishers · Zbl 1188.37002
[12] Lakshmikantham, V.; Vatsala, A.S., Theory of fractional differential inequalities and applications, Commun. appl. anal., 11, July-October, 395-402, (2007) · Zbl 1159.34006
[13] Lakshmikantham, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear anal. TMA, 69, (2008), 3337-3343 · Zbl 1162.34344
[14] Lakshmikantham, V.; Vatsala, A.S., General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. lett., 21, 828-834, (2008) · Zbl 1161.34031
[15] Lakshmikantham, V.; Vatsala, A.S., Generalized quasilinearization for nonlinear problems, (1998), Kluwer Academic Publishers Boston · Zbl 0997.34501
[16] Ye, Y.; Gao, J.; Ding, Y., A generalized Gronwall inequality and its application to a fractional differential equation, J. math. anal. appl., 328, 1075-1081, (2007) · Zbl 1120.26003
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.