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Volume 63, Issue 4 (Aug 2013)


Extremal solutions of Cauchy problems for abstract fractional differential equations

JinRong Wang
  • School of Mathematics and Computer Science, Guizhou Normal College Guiyang, 550018, Guizhou, China
  • Email:
/ Yong Zhou
  • Department of Mathematics, Xiangtan University Xiangtan, 411105, Hunan, China
  • Email:
/ Milan Medveď
  • Department of Mathematical and Numerical Mathematics, Comenius University Bratislava, SK-842 15, Bratislava, Slovakia
  • Email:
Published Online: 2013-08-23 | DOI: https://doi.org/10.2478/s12175-013-0134-1


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/S1048953392000170CrossrefGoogle Scholar

  • [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.2009.03.005CrossrefGoogle Scholar

  • [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.2010.08.005Web of ScienceCrossrefGoogle Scholar

  • [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.2007.06.021CrossrefGoogle Scholar

  • [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. Google Scholar

  • [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/S1048953304308038CrossrefGoogle Scholar

  • [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.2005.08.023CrossrefGoogle Scholar

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

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

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

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

  • [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)80001-0CrossrefGoogle Scholar

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

  • [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.1997.5532CrossrefGoogle Scholar

  • [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.2001.7798CrossrefGoogle Scholar

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

  • [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/s11072-008-0015-7Web of ScienceCrossrefGoogle Scholar

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

  • [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.2009.12.062CrossrefGoogle Scholar

  • [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.2005.321CrossrefGoogle Scholar

  • [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.2008.02.087CrossrefGoogle Scholar

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

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

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

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

  • [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.2010.06.013CrossrefGoogle Scholar

  • [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.2011.02.003CrossrefGoogle Scholar

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

  • [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.2009.06.026CrossrefGoogle Scholar

  • [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.2010.05.029CrossrefGoogle Scholar

About the article

Published Online: 2013-08-23

Published in Print: 2013-08-01

Citation Information: Mathematica Slovaca, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.2478/s12175-013-0134-1.

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

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