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Mathematica Slovaca

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Extremal solutions of Cauchy problems for abstract fractional differential equations

1School of Mathematics and Computer Science, Guizhou Normal College Guiyang, 550018, Guizhou, China

2Department of Mathematics, Xiangtan University Xiangtan, 411105, Hunan, China

3Department of Mathematical and Numerical Mathematics, Comenius University Bratislava, SK-842 15, Bratislava, Slovakia

© 2013 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Citation Information: Mathematica Slovaca. Volume 63, Issue 4, Pages 769–792, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: 10.2478/s12175-013-0134-1, August 2013

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Published Online:


In this paper, we study the extremal solutions of Cauchy problems for abstract fractional differential equations. Some definitions such as L 1-Lipschitz-like, L 1-Carathéodory-like and L 1-Chandrabhan-like are introduced. By virtue of the singular integral inequalities with several nonlinearities due to Medved’, the properties of solutions are given. By using a hybrid fixed point theorem due to Dhage, existence results for extremal solutions are established. Finally, we present an example to illustrate our main results.

MSC: Primary 26A33, 06D35, 34A40.

Keywords: fractional differential equations; extremal solutions; existence; fixed point method

  • [1] AIZICOVICI, S.— PAPAGEORGIOU, N. S.: Extremal solutions to a class of multivalued integral equations in Banach space, J. Appl. Math. Stoc. Anal. 5 (1992), 205–220. http://dx.doi.org/10.1155/S1048953392000170 [CrossRef]

  • [2] BALACHANDRAN, K.— PARK, J. Y.: Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear Anal. 71 (2009), 4471–4475. http://dx.doi.org/10.1016/j.na.2009.03.005 [CrossRef]

  • [3] BALACHANDRAN, K.— KIRUTHIKA, S.— TRUJILLO, J. J.: Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011), 1970–1977. http://dx.doi.org/10.1016/j.cnsns.2010.08.005 [Web of Science] [CrossRef]

  • [4] BENCHOHRA, M.— HENDERSON, J.— NTOUYAS, S. K.— OUAHAB, A.: Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2008), 1340–1350. http://dx.doi.org/10.1016/j.jmaa.2007.06.021 [CrossRef]

  • [5] BENCHOHRA, M.— HENDERSON, J.— NTOUYAS, S. K.— OUAHAB, A.: Existence results for fractional functional differential inclusions with infinite delay and application to control theory, Fract. Calc. Appl. Anal. 11 (2008), 35–56.

  • [6] DHAGE, B. C.: On existence of extremal solutions of nonlinear functional integral equations in Banach algebras, J. Appl. Math. Stoc. Anal. 2004 (2004), 271–282. http://dx.doi.org/10.1155/S1048953304308038 [CrossRef]

  • [7] DHAGE, B. C.: Existence of extremal solutions for discontinuous functional integral equations, Applied Math. Letters 19 (2006), 881–886. http://dx.doi.org/10.1016/j.aml.2005.08.023 [CrossRef]

  • [8] DIETHELM, K.: The Analysis of Fractional Differential Equations, Lecture Notes in Math., Springer, New York, 2010. http://dx.doi.org/10.1007/978-3-642-14574-2 [CrossRef]

  • [9] HEIKKILÄ, S.— LAKSHMIKANTHAM, V.: Monotone Iterative for Discontinuous Nonlinear Differential Equations, Monogr. Textbooks Pure Appl. Math. 181, Marcel Dekker, Inc., New York, 1994.

  • [10] HENRY, D.: Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-Heidelberg-New York, 1981.

  • [11] JANKOWSKI, T.: Fractional differential equations with deviating arguments, Dyn. Syst. Appl. 17 (2008), 677–684.

  • [12] KILBAS, A. A.— SRIVASTAVA, H. M.— TRUJILLO, J. J.: Theory and Applications of Fractional Differential Equations. North-HollandMath. Stud. 204, Elsevier Science B.V., Amsterdam, 2006. http://dx.doi.org/10.1016/S0304-0208(06)80001-0 [CrossRef]

  • [13] LAKSHMIKANTHAM, V.— LEELA, S.— DEVI, J. V.: Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambrigge, 2009.

  • [14] MEDVEĎ, M.: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions, J. Math. Anal. Appl. 214 (1997), 349–366. http://dx.doi.org/10.1006/jmaa.1997.5532 [CrossRef]

  • [15] MEDVEĎ, M.: Integral inequalities and global solutions of semilinear evolution equations, J. Math. Anal. Appl. 267 (2002), 643–650. http://dx.doi.org/10.1006/jmaa.2001.7798 [CrossRef]

  • [16] MEDVEĎ, M.: On the existence of global solutions of evolution equations, Demonstratio Math. XXXVII (2004), 871–882.

  • [17] MEDVEĎ, M.: Singular integral inequalities with several nonlinearities and integral equations with singular kernels, Nonlinear Oscil. 11 (2007), 70–79. http://dx.doi.org/10.1007/s11072-008-0015-7 [Web of Science] [CrossRef]

  • [18] MILLER, K. S.— ROSS, B.: An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.

  • [19] MOPHOU, G. M.— N’GUÉRÉKATA, G. M.: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Appl. Math. Comput. 216 (2010), 61–69. http://dx.doi.org/10.1016/j.amc.2009.12.062 [CrossRef]

  • [20] NIETO, J. J.— RODRIGUEZ-LÓPEZ, R.: Existence of extremal solutions for quadratic fuzzy equations, Fixed Point Theory Appl. 2005 (2005), 321–342. http://dx.doi.org/10.1155/FPTA.2005.321 [CrossRef]

  • [21] N’GUÉRÉKATA, G. M.: A Cauchy problem for some fractional differential abstract differential equation with nonlocal conditions, Nonlinear Anal. 70 (2009), 1873–1876. http://dx.doi.org/10.1016/j.na.2008.02.087 [CrossRef]

  • [22] N’GUÉRÉKATA, G. M.: Corrigendum: A Cauchy problem for some fractional differential equations, Commun. Math. Anal. 7 (2009), 11–11.

  • [23] PINTO, M.: Integral inequalties of Bihari-type and applications, Funkc. Ekvacioj 33 (1990), 387–403.

  • [24] PODLUBNY, I.: Fractional Differential Equations, Academic Press, San Diego, 1999.

  • [25] TARASOV, V. E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, HEP, 2010.

  • [26] WANG, J.— ZHOU, Y.: A class of fractional evolution equations and optimal controls, Nonlinear Anal. 12 (2011), 262–272. http://dx.doi.org/10.1016/j.nonrwa.2010.06.013 [CrossRef]

  • [27] WANG, J.— ZHOU, Y.— WEI, W.: A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011), 4049–4059. http://dx.doi.org/10.1016/j.cnsns.2011.02.003 [CrossRef]

  • [28] ZHOU, Y.— JIAO, F.: Existence of extremal solutions for discontinuous fractional functional differential equations, Int. J. Dyn. Diff. Eq. 2 (2008), 237–252.

  • [29] ZHOU, Y.— JIAO, F.: Existence of mild solutions for fractional neutral evolution equations, Comp. Math. Appl. 59 (2010), 1063–1077. http://dx.doi.org/10.1016/j.camwa.2009.06.026 [CrossRef]

  • [30] ZHOU, Y.— JIAO, F.: Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. 11 (2010), 4465–4475. http://dx.doi.org/10.1016/j.nonrwa.2010.05.029 [CrossRef]

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