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Kou, Chunhai; Zhou, Huacheng; Yan, Ye

Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis. (English) Zbl 1235.34022

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 5975-5986 (2011).
The global existence of solutions on the half-axis for a classical initial value problem of fractional differential equations involving Riemann-Liouville fractional derivative is studied. The authors proof the main results using fixed-point theorems on Banach spaces.
Reviewer: Juan J. Trujillo (La Laguna)

Cited in 61 Documents

MSC:

34A08 Fractional ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations

Keywords:

fractional differential equations; initial value problem; global existence of solutions; Banach spaces; fixed-point theorems
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\textit{C. Kou} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 5975--5986 (2011; Zbl 1235.34022)
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References:

[1] Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific · Zbl 0998.26002
[2] Gaul, L.; Klein, P.; Kempfle, S., Damping description involving fractional operators, Mech. Syst. Signal Process., 5, 81-88 (1991)
[3] Diethelm, K.; Freed, A. D., On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, (Keil, F.; Mackens, W.; Voss, H.; Werther, J., Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties (1999), Springer-Verlag: Springer-Verlag Heidelberg), pp. 217-307
[4] Koeller, R. C., Application of fractional calculus to the theory of viscoelasticity, J. Appl. Mech, 51, 299-307 (1984) · Zbl 0544.73052
[5] Kumar, P.; Agrawal, O. P., An approximate method for numerical solution of fractional differential equations, Signal Process., 86, 2602-2610 (2006) · Zbl 1172.94436
[6] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5, 367-386 (2002) · Zbl 1042.26003
[7] Metzler, F.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F., Relaxation in filled polymers: a fractional calculus approach, J. Chem. Phys., 103, 7180-7186 (1995)
[8] Kilbas, A. A.; Srivastava, Hari M.; Trujillo, Juan J., (Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier Science B.V: Elsevier Science B.V Amsterdam) · Zbl 1092.45003
[9] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Differential Equations (1993), John Wiley: John Wiley New York · Zbl 0789.26002
[10] Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications (1999), Academic Press · Zbl 0924.34008
[11] Lakshmikantham, V.; Leela, S.; Vasundhara, J., Theory of Fractional Dynamic Systems (2009), Cambridge Academic Publishers: Cambridge Academic Publishers Cambridge · Zbl 1188.37002
[12] Lakshmikantham, V.; Vatsala, A. S., Basic theory of fractional differential equations, Nonlinear Anal., 69, 2677-2682 (2008) · Zbl 1161.34001
[13] Kilbas, A. A.; Trujillo, J. J., Differential equations of fractional order: Methods, results and problems I, Appl. Anal., 78, 153-192 (2001) · Zbl 1031.34002
[14] Kilbas, A. A.; Trujillo, J. J., Differential equations of fractional order: Methods, results and problems II, Appl. Anal., 81, 435-493 (2002) · Zbl 1033.34007
[15] Srivastava, H. M.; Saxena, R. K., Operators of fractional integration and their applications, Appl. Math. Comput., 118, 1-52 (2001) · Zbl 1022.26012
[16] Agarwal, R. P.; Benchohra, M.; Slimani, B. A., Existence results for differential equations with fractional order and impulses, Mem. Differential Equations, Math. Phys., 44, 1-21 (2008) · Zbl 1178.26006
[17] Ouahab, A., Some results for fractional boundary value problem of differential inclusions, Nonlinear Anal., 69, 3877-3896 (2008) · Zbl 1169.34006
[18] El-Sayed, A. M.A., On the fractional differential equation, Appl. Math. Comput., 49, 205-213 (1992) · Zbl 0757.34005
[19] El-Sayed, A. M.A., Fractional order diffusion-wave equations, Internat. J. Theoret. Phys., 35, 311-322 (1996) · Zbl 0846.35001
[20] El-Sayed, A. M.A., Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal., 33, 181-186 (1998) · Zbl 0934.34055
[21] Nieto, J. J., Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Appl. Math. Lett., 23, 1248-1251 (2010) · Zbl 1202.34019
[22] Li, C.; Deng, W., Remarks on fractional derivatives, Appl. Math. Comput., 187, 777-784 (2007) · Zbl 1125.26009
[23] Kosmatov, N., Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear Anal., 70, 2521-2529 (2009) · Zbl 1169.34302
[24] Muslim, M.; Conca, C.; Nandakumaran, A. K., Approximate of solutions to fractional integral equation, Comput. Math. Appl., 59, 1236-1244 (2010) · Zbl 1189.65310
[25] Stojanović, M., Existence-uniqueness result for a nonlinear \(n\)-term fractional equation, J. Math. Anal. Appl., 353, 244-245 (2009) · Zbl 1195.34014
[26] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204, 609-625 (1996) · Zbl 0881.34005
[27] Babakhani, A.; Gejji, V. D., Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl., 278, 434-442 (2003) · Zbl 1027.34003
[28] Yu, C.; Gao, G., Existence of fractional differential equations, J. Math. Anal. Appl., 310, 26-29 (2005) · Zbl 1088.34501
[29] Kou, C. H.; Liu, J.; Ye, Y., Existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equations, Disc. Dyn. Nat. Soc. (2010), Article ID 142175, 15 pages
[30] Kilbas, A. A.; Bonilla, B.; Trujillo, J. J., Existence and uniqueness theorems for nonlinear fractional differential equations, Demonstratio Math., 33, 3, 538-602 (2000) · Zbl 0964.34004
[31] Eidelman, S. D.; Kochubei, A. N., Cauchy problem for fractional diffusion equations, J. Differential Equations, 199, 211-255 (2004) · Zbl 1129.35427
[32] Zhang, S. Q., Monotone iterative method for initial value problem involving Riemann-Louville derivatives, Nonlinear Anal., 71, 2087-2093 (2009) · Zbl 1172.26307
[33] Wei, Z. L.; Li, Q. D.; Che, J. L., Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative, J. Math. Anal. Appl., 367, 260-272 (2010) · Zbl 1191.34008
[34] Zhao, X.; Ge, W., Unbounded solutions for fractional bounded value problems on the infinite interval, Acta Appl. Math., 109, 495-505 (2010) · Zbl 1193.34008
[35] Arara, A.; Benchohra, M.; Hamidi, N.; Nieto, J. J., Fractional order differential equations on an unbounded domain, Nonlinear Anal., 72, 580-586 (2010) · Zbl 1179.26015
[36] Agarwal, R. P.; Regan, D. O., Infinite Interval Problems for Differential, Difference and Integral Equations (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht
[37] Agarwal, R. P.; Regan, D. O., Boundary value problems of nonsingular type on the semi-infinite interval, Tohoku. Math. J., 51, 391C397 (1999)
[38] Granas, A.; Dugundji, J., Fixed Point Theory (2003), Springer Verlag · Zbl 1025.47002
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