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Fractional integral inequalities and applications. (English) Zbl 1189.26044
Summary: Fractional integral inequality results when $0 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.

##### MSC:
 26D15 Inequalities for sums, series and integrals of real functions 26A33 Fractional derivatives and integrals (real functions) 34A08 Fractional differential equations
##### References:
 [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 · doi:10.1006/jmaa.2001.7194 [3] Diethelm, K.; Ford, N. J.: Multi-order fractional differential equations and their numerical solution, Amc 154, 621-640 (2004) · Zbl 1060.65070 · doi:10.1016/S0096-3003(03)00739-2 [4] Diethelm, K.; Freed, A. D.: On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, Scientific computing in chemical engineering II: Computational fluid dynamics, reaction engineering, and molecular properties, 217-224 (1999) [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) [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) · Zbl 0818.26003 [9] Oldham, B.; Spanier, J.: The fractional calculus, (1974) [10] Podlubny, I.: Fractional differential equations, (1999) [11] Lakshmikantham, V.; Leela, S.; Vasundhara, D. J.: Theory of fractional dynamic systems, (2009) [12] Lakshmikantham, V.; Vatsala, A. S.: Theory of fractional differential inequalities and applications, Commun. appl. Anal. 11, No. 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) · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025 [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 · doi:10.1016/j.aml.2007.09.006 [15] Lakshmikantham, V.; Vatsala, A. S.: Generalized quasilinearization for nonlinear problems, (1998) [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 · doi:10.1016/j.jmaa.2006.05.061