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Power-type estimates for a nonlinear fractional differential equation. (English) Zbl 1078.34028

Abstract: The authors are concerned with a family of nonlinear fractional differential equations. They prove that solutions of such equations with weighted initial data exist globally and decay as a power function.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
26A33 Fractional derivatives and integrals
34C11 Growth and boundedness of solutions to ordinary differential equations
45D05 Volterra integral equations
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[1] Askhabov, S.N., Integral equations of convolution type with power non-linearity, Colloq. math., 62, 49-65, (1991) · Zbl 0738.45003
[2] Bainov, D.; Simeonov, P., Integral inequalities and applications, vol. 57, (1992), Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0759.26012
[3] Barrett, J.H., Differential equations of non-integer order, Canad. J. math., 6, 4, 529-541, (1954) · Zbl 0058.10702
[4] Bushell, P.J., On a class of Volterra and Fredholm non-linear integral equations, Math. proc. Cambridge philos. soc., 79, 329-335, (1979) · Zbl 0316.45003
[5] Bushell, P.J.; Okrasinski, W., Nonlinear Volterra integral equations and the apery identities, Bull. London math. soc., 24, 478-484, (1992) · Zbl 0767.45002
[6] Butler, G.; Rogers, T., A generalization of a lemma of bihari and applications to pointwise estimates for integral equations, J. math. anal. appl., 33, 77-81, (1971) · Zbl 0209.42503
[7] Campos, L.M.B.C., On the solution of some simple fractional differential equations, Internat. J. math. sci., 13, 3, 481-496, (1990) · Zbl 0711.34019
[8] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. math. anal. appl., 204, 2, 609-625, (1996) · Zbl 0881.34005
[9] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[10] Fujiwara, M., On the integration and differentiation of an arbitrary order, Tohoku math. J., 37, 110-121, (1933) · JFM 59.0415.03
[11] Furati, K.M.; Tatar, N.-e., An existence result for a nonlocal nonlinear fractional differential problem, J. fract. calc., 26, 43-51, (2004) · Zbl 1101.34001
[12] Karapetyants, N.K.; Kilbas, A.A.; Saigo, M., On the solution of non-linear Volterra convolution equation with power nonlinearity, J. integral equations, 8, 429-445, (1996) · Zbl 0874.45002
[13] Karapetyants, N.K.; Kilbas, A.A.; Saigo, M.; Samko, S.G., Upper and lower bounds for solutions of nonlinear Volterra convolution integral equations with power nonlinearity, J. integral equations appl., 12, 4, 421-448, (2000) · Zbl 0982.45003
[14] 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
[15] Kilbas, A.A.; Bonilla, B.; Trujillo, J.J., Fractional integrals and derivatives and differential equations of fractional order in weighted spaces of continuous functions (Russian), Dokl. nats. akad. nauk belar., 44, 6, (2000) · Zbl 1177.26011
[16] Kilbas, A.A.; Saigo, M., On solutions of integral equations of abel – volterra type, Differential integral equations, 8, 993-1011, (1995) · Zbl 0823.45002
[17] A.A. Kilbas, M. Saigo, On solution in closed from of nonlinear integral and differential equations of fractional order, Univalent Functions and the Briot-Bouquet Differential Equations (Japanese) (Kyoto, 1996), Surikaisekiken-Kyusho Kokyuroku, vol. 963, 1996, pp. 39-50. · Zbl 0924.45010
[18] Kilbas, A.A.; Saigo, M., On solutions of a nonlinear abel – volterra integral equation, J. math. anal. appl., 229, 41-60, (1999) · Zbl 0917.45004
[19] Kilbas, A.A.; Trujillo, J.J., Differential equations of fractional ordermethods, results and problem, Appl. anal., 78, 1-2, 153-192, (2001) · Zbl 1031.34002
[20] Kirane, M.; Tatar, N.-e., Global existence and stability of some semi-linear problems, Arch. math. (Brno) tomus, 36, 33-44, (2000) · Zbl 1048.34102
[21] Kirane, M.; Tatar, N.-e., Convergence rates for a reaction – diffusion system, Z. anal. anwend. (J. anal. appl.), 20, 2, 347-357, (2001) · Zbl 0984.35080
[22] Mazouzi, S.; Tatar, N.-e., Global existence for some integrodifferential equations with delay subject to nonlocal conditions, Z. anal. anwend., 21, 1, 249-256, (2002) · Zbl 1008.45009
[23] Medved, M., A new approach to an analysis of henry type integral inequalities and their bihari type versions, J. math. anal. appl., 214, 349-366, (1997) · Zbl 0893.26006
[24] Medved, M., Singular integral inequalities and stability of semilinear parabolic equations, Arch. math. (Brno) tomus, 24, 183-190, (1998) · Zbl 0915.34057
[25] M.W. Michalski, Derivatives of non integer order and their applications, Dissertationes Mathematicae, Polska Akademia Nauk, Instytut Matematyczny Warszawa, 1993. · Zbl 0880.26007
[26] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[27] Nestell, M.K.; Chandehari, On a separable nonlinear Volterra integral equations with power nonlinearity, Libertas math., 17, 175-181, (1979)
[28] Okrasinski, W., On the existence and uniqueness of nonnegative solution of a certain nonlinear convolution equation, Ann. polon. math., 36, 61-72, (1979) · Zbl 0412.45006
[29] Okrasinski, W., On a nonlinear convolution equation occurring in the theory of water perlocation, Ann. polon. math., 37, 223-229, (1980) · Zbl 0451.45004
[30] Okrasinski, W., Nonlinear Volterra equations and physical applications, Extracta math., 4, 51-80, (1989)
[31] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York, London · Zbl 0428.26004
[32] Podlubny, I., Fractional differential equations, mathematics in sciences and engineering, vol. 198, (1999), Academic Press San Diego
[33] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives, theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[34] Schneider, W.R., The general solution of a non-linear integral equation of convolution type, Z. angew. math. phys., 33, 140-142, (1982) · Zbl 0493.45007
[35] Tatar, N.-e., Exponential decay for a semilinear problem with memory, Arab J. math. sci., 7, 1, 29-45, (2001) · Zbl 0992.45007
[36] A.Z.-A.M. Tazali, Local existence theorems for ordinary differential equations of fractional order, Ordinary and Partial Differential Equations (Dundee, 1982), Lecture Notes in Mathematics, vol. 964, 1982, pp. 652-665.
[37] Zhang, S., The existence of a positive solution for a nonlinear fractional differential equation, J. math. anal. appl., 252, 804-812, (2000) · Zbl 0972.34004
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