Asymptotic Integration of Some Classes of Fractional Differential Equations

Milan Medveď

doi:10.2478/tmmp-2013-00010

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Asymptotic Integration of Some Classes of Fractional Differential Equations

Tatra Mountains Mathematical Publications

Volume 54 (2013): Issue 1 (April 2013)

By: Milan Medveď

Open Access

|Jul 2013

Abstract

In this paper we deal with the problem of asymptotic integration of nonlinear higher order fractional differential equations of the Caputo’s type. We give some conditions under which all global solutions of these equations behave like linear functions as t -> ∞.

References

[1] AGARWAL, R. P.-DJEBALI, S.-MOUSSAOUI, T.-MUSTAFA, O. G.: On the asymptoticintegration of nonlinear differential equations, J. Comput. Appl. Math. 202 (2007), 352-376.10.1016/j.cam.2005.11.038
Search in Google Scholar
[2] BǍLEANU, D.-MUSTAFA, O. G.-AGARWAL, R. P.: Asymptotically linear solutionsfor some linear fractional differential equations, Abstr. Appl. Anal., Vol. 2010, Article ID 865139, 8 p.10.1155/2010/865139
Search in Google Scholar
[3] BǍLEANU, D.-MUSTAFA, O. G.-AGARWAL, R. P.: Asymptotic integration of (1+α)-order fractional differential equatios, Comput. Math. Appl. 62 (2011), 1492-1500.10.1016/j.camwa.2011.03.021
Search in Google Scholar
[4] BIHARI, I.: Researches of the boundedness and stabiliy of solutions of non-linear differentialequations, Acta Math. Acad. Hungar. 8 (1957), 261-278.10.1007/BF02020315
Search in Google Scholar
[5] CALIGO, D.: Comportamento asintotico degli integrali dell’equazione y′′(x)+A(x)y(x) = = 0, nell’ipotesi limx→+∞ A(x) = 0, Boll. Un. Mat. Ital. 6 (1941), 286-295.
Search in Google Scholar
[6] CAPUTO, M.: Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. Royal Astronom. Soc. 13 (1967), 529-535.10.1111/j.1365-246X.1967.tb02303.x
Search in Google Scholar
[7] COHEN, D. S.: The asymptotic behavior of a class of nonlinear differntial equations, Proc. Amer. Math. Soc. 18 (1967), 607-609.10.1090/S0002-9939-1967-0212289-3
Search in Google Scholar
[8] CONSTANTIN, A.: On the asymptotic behavior of second order nonlinear differentialequations, Rend. Math. Appl. 13 (1993), 627-634.
Search in Google Scholar
[9] CONSTANTIN, A.: On the existence of positive solutions of second order differentialequations, Ann. Mat. Pura Appl. (4) 184 (2005), 131-138.10.1007/s10231-004-0100-1
Search in Google Scholar
[10] DANNAN, F. M.: Integral inequalities of Gronwall-Bellman-Bihari type and asymptoticbehavior of certain second order nonlinear differential equations, J. Math. Anal. Appl. 108 (1985), 151-164.10.1016/0022-247X(85)90014-9
Search in Google Scholar
[11] FURATI, M.-TATAR, N. E.: Power-type estimates for nonlinear fractional differentialequations, Nonlinear Anal. 62 (2001), 1025-1036.10.1016/j.na.2005.04.010
Search in Google Scholar
[12] KILBAS, A. A.-SRIVASTAVA, H. M.-TRUJILLO, J. J.: Theory and Applicationsof Fractional Differential Equations, in: North-Holland Math. Stud., Vol. 204, Elsevier, Amsterdam, 2006.
Search in Google Scholar
[13] KUSANO, T.-TRENCH, W. F.: Global existence of second order differential equationswith integrable coefficients, J. London Math. Soc. 31 (1985), 478-486.10.1112/jlms/s2-31.3.478
Search in Google Scholar
[14] KUSANO, T.-TRENCH,W. F.: Existence of global solutions with prescribed asymptoticbehavior for nonlinear ordinary differential equations, Ann. Mat. Pura Appl. (4) 142 (1985), 381-392.10.1007/BF01766602
Search in Google Scholar
[15] LIPOVAN, O.: On the asymptotic behaviour of the solutions to a class of second ordernonlinear differential equations, Glasgow Math. J. 45 (2003), 179-187.10.1017/S0017089502001143
Search in Google Scholar
[16] MA, Q.-H.-PEČARIČ, J.-ZHANG, J.-M.: Integral inequalities of systems and the estimatefor solutions of certain nonlinear two-dimensional fractional differential systems, Comput. Math. Appl. 61 (2011), 3258-3267.10.1016/j.camwa.2011.04.008
Search in Google Scholar
[17] MEDVEĎ, M.: A new approach to an analysis of Henry type integral inequalities andtheir Bihari type versions, J. Math. Anal. Appl. 214 (1997), 349-366.10.1006/jmaa.1997.5532
Search in Google Scholar
[18] MEDVEĎ, M.: Integral inequalities and global solutions of semilinear evolution equations, J. Math. Anal. Appl. 37 (2002), 871-882.
Search in Google Scholar
[19] MEDVEĎ, M.-MOUSSAOUI, T.: Asymptotic integration of nonlinear Φ-Laplacian differentialequations, Nonlinear Anal. 72 (2010), 1-8.10.1016/j.na.2009.09.042
Search in Google Scholar
[20] MEDVEĎ, M.: On the asymptotic behavior of solutions of nonlinear differential equationsof integer and also of non-integer order, in: Proc. 9th Colloquium on the Qual. Theory of Differ. Equ., Szeged, Hungary, 2011, Electron. J. Qual. Theory Differ. Equ. 10 (2012), pp. 1-9.
Search in Google Scholar
[21] MEDVEĎ, M.-PEKÁRKOVÁ, E.: Long time behavior of second order differential equationswith p-Laplacian, Electron. J. Qual. Theory Differ. Equ. 108 (2008), 1-12.
Search in Google Scholar
[22] MILLER, K. S.-ROSS, B.: An Introduction to the Fractional Calculus and DifferentialEquations. John Wiley, New York, 1993.
Search in Google Scholar
[23] MUSTAFA, O. G.-ROGOVCHENKO, Y. V.: Global existence of solutions with prescribedasymptotic behavior for second-order nonlinear differential equations, Nonlinear Anal. 51 (2002), 339-368.10.1016/S0362-546X(01)00834-3
Search in Google Scholar
[24] PODLUBNY, I.: Fractional Differential Equations. Academic Press, San Diego, 1999.
Search in Google Scholar
[25] PHILOS, CH. G.-PURNARAS I. K.-TSAMATOS, P. CH.: Large time asymptotic topolynomials solutions for nonlinear differential equations, Nonlinear Anal. 59 (2004), 1157-1179. 10.1016/j.na.2004.08.011
Search in Google Scholar
[26] PRUDNIKOV, A. P.-BRYCHKOV, ZU. A.-MARICHEV, O. I.: Integral and Series, in: Elementary Functions, Vol. 1, Nauka, Moscow, 1981. (In Russian)
Search in Google Scholar
[27] ROGOVCHENKO, Y. V.: On asymptotics behavior of solutions for a class of secondorder nonlinear differential equations, Collect. Math. 49 (1998), 113-120.
Search in Google Scholar
[28] ROGOVCHENKOS. P.-ROGOVCHENKO, YU. V.: Asymptotics of solutions for a classof second order nonlinear differential equations, Portugal. Math. 57 (2000), 17-32.
Search in Google Scholar
[29] SAMKO, S. G.-KILBAS, A. A.-MARICHEV, O. I.: Fractional Integarals and Derivatives:Theory and Applications. Gordon and Breach Sci. Publ., New York, 1993.
Search in Google Scholar
[30] TONG, J.: The asymptotic behavior of a class of nonlinear differential equations of secondorder, Proc. Amer. Math. Soc. 84 (1982), 235-236.10.1090/S0002-9939-1982-0637175-4
Search in Google Scholar
[31] TRENCH, W. F.: On the asymptotic behavior of solutions of second order linear differentialequations, Proc. Amer. Math. Soc. 54 (1963), 12-14.10.1090/S0002-9939-1963-0142844-7
Search in Google Scholar