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Existence and uniqueness results for fractional differential equations with boundary value conditions

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EN
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EN
In this paper, we study the existence and uniqueness of fractional differential equations with boundary value conditions. A new generalized singular type Gronwall inequality is given to obtain important a priori bounds. Existence and uniqueness results of solutions are established by virtue of fractional calculus and fixed point method under some weak conditions. An example is given to illustrate the results.
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629--643
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Bibliogr. 22 poz.
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Bibliografia
  • [1] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. 109 (2010), 973–1033.
  • [2] K. Balachandran, J.Y. Park, Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear Anal. 71 (2009), 4471–4475.
  • [3] K. Balachandran, S. Kiruthika, J.J. Trujillo, Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011), 1970–1977.
  • [4] M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1–12.
  • [5] G.M. N’Guérékata, A Cauchy problem for some fractional differential abstract differential equation with nonlocal conditions, Nonlinear Anal. 70 (2009), 1873–1876.
  • [6] G.M. N’Guérékata, Corrigendum: A Cauchy problem for some fractional differential equations, Commun. Math. Anal. 7 (2009), 11–11.
  • [7] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, [in:] North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.
  • [8] K.S. Miller, B. Ross, An introduction to the fractional calculus and differential equations, John Wiley, New York, 1993.
  • [9] G.M. Mophou, G.M. N’Guérékata, Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Appl. Math. Comput. 216 (2010), 61–69.
  • [10] V. Lakshmikantham, S. Leela, J.V. Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers, 2009.
  • [11] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
  • [12] JinRong Wang, W. Wei, Y. Yang, Fractional nonlocal integrodifferential equations of mixed type with time-varying generating operators and optimal control, Opuscula Math. 30 (2010), 217–234.
  • [13] JinRong Wang, Y. Yang, W. Wei, Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces, Opuscula Math. 30 (2010), 361–381.
  • [14] JinRong Wang, W. Wei, Y. Yang, On some impulsive fractional differential equations in Banach spaces, Opuscula Math. 30 (2010), 507–525.
  • [15] JinRong Wang, W. Wei, Y. Yang, Fractional nonlocal integrodifferential equations and its optimal control in Banach spaces, J. KSIAM 14 (2010), 79–91.
  • [16] JinRong Wang, Yong Zhou, Time optimal control problem of a class of fractional distributed systems, Int. J. Dyn. Diff. Eq. 3 (2011), 363–382.
  • [17] JinRong Wang, Yong Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. 12 (2011), 262–272.
  • [18] JinRong Wang et al., A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions, Opuscula Math. 31 (2011), 119–135.
  • [19] JinRong Wang, W. Wei, Yong Zhou, Fractional finite time delay evolution systems and optimal controls in infinite dimensional spaces, J. Dyn. Contr. Syst. 17 (2011), in press.
  • [20] Yong Zhou, Existence and uniqueness of fractional functional differential equations with unbounded delay, Int. J. Dyn. Diff. Eq. 1 (2008), 239–244.
  • [21] Yong Zhou, Feng Jiao, Existence of mild solutions for fractional neutral evolution equations, Comp. Math. Appl. 59 (2010), 1063–1077.
  • [22] Yong Zhou, Feng Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. 11 (2010), 4465–4475.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGHS-0003-0009
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