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Boundary Value Problems for Impulsive Fractional Differential Equations with Nonlocal Conditions

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Advances in Applied Mathematics and Approximation Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 41))

Abstract

In this study, we discuss some existence results for the solutions to impulsive fractional differential equations with nonlocal conditions by using contraction mapping principle and Krasnoselskii’s fixed point theorem.

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References

  1. R.P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann–Liouville fractional derivative, Adv. Differential Equations, (2009) 47. Art. ID 981728.

    Google Scholar 

  2. R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math., 109 (3) (2010), 973–1033.

    Google Scholar 

  3. B. Ahmad and J.J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray–Schauder degree theory, Topol. Methods Nonlinear Anal., 35 (2010) 295–304.

    Google Scholar 

  4. B. Ahmad and S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal. Hybrid Syst., 4 (2010) 134–141.

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Ahmad and S. Sivasundaram, On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order,  Appl. Math. Comput., 217(2) (2010), 480–487.

    Google Scholar 

  6. Z. Bai and H. Lü, Positive solutions for the boundary value problem of nonlinear fractional differential equations, J. Math. Anal. Appl., 311 (2005) 495–505.

    Google Scholar 

  7. M. Benchohra and B.A. Slimani, Existence and uniqueness of solutions to impulsive fractional differential equations, Electron. J. Differential Equations 2009(2009), no. 10, pp. 1–11.

    Google Scholar 

  8. M. Benchohra, S. Hamani, and S. K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal., 71, no. 7–8, (2009) 2391–2396.

    Google Scholar 

  9. L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonloncal Cauchy problem, J. Math. Anal. Appl., 162 (1991) 494–505.

    Google Scholar 

  10. L. Byszewski and V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. 40 (1991) 11–19.

    Google Scholar 

  11. Y.K. Chang, J. J. Nieto and Zhi-Han Zhao, Existence results for a nondensely-defined impulsive neutral differential equation with state-dependent delay, Nonlinear Anal. Hybrid Syst. 4 (3) (2010) 593–599.

    Google Scholar 

  12. K. Diethelm, A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in: F. Keil, W. Mackens, H. Voss, J. Werther (Eds.), Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, 1999, pp. 217–224.

    Google Scholar 

  13. W.G. Glockle and T.F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (1995) 46–53.

    Article  Google Scholar 

  14. N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann Liouville fractional derivatives, Rheol. Acta., 45 (5) (2006) 765–772.

    Google Scholar 

  15. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

    Book  MATH  Google Scholar 

  16. E.R. Kaufmann and E. Mboumi, Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. Theory Differ. Equ. 3 (2008) 1–11.

    Google Scholar 

  17. A.A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.

    Google Scholar 

  18. M.A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, New York, 1964.

    Google Scholar 

  19. V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

    Book  MATH  Google Scholar 

  20. F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 1997, pp. 291–348.

    Google Scholar 

  21. G.M. Mophou, Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Anal. 72 (2010) 1604–1615.

    Google Scholar 

  22. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

    MATH  Google Scholar 

  23. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002) 367–386.

    MathSciNet  MATH  Google Scholar 

  24. Y.V. Rogovchenko, Impulsive evolution systems: main results and new trends, Dyn. Contin. Discrete Impuls. Syst. 3 (1997) 57–88.

    Google Scholar 

  25. J. Sabatier, O.P. Agrawal and J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007.

    MATH  Google Scholar 

  26. H.A.H. Salem, On the fractional m-point boundary value problem in reflexive Banach space and the weak toplogies, J. Comput. Appl. Math. 224 (2009) 565–572.

    Google Scholar 

  27. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.

    MATH  Google Scholar 

  28. A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.

    MATH  Google Scholar 

  29. Smart, D.R., Fixed Point Theorems, Cambridge University press, Cambridge (1980)

    MATH  Google Scholar 

  30. L. Yang and H.Chen, Nonlocal Boundary Value Problem for Impulsive Differential Equations of Fractional Order, Advances in Difference Equations (2011) 16 pages Article ID 404917, doi:10.1155/2011/404917.

    Google Scholar 

  31. S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differential Equations 36 (2006) 1–12.

    Google Scholar 

  32. W. Zhong and W. Lin, Nonlocal and multiple-point boundary value problem for fractional differential equations, Comput. Math. Appl., 59 (3) (2010) 1345–1351.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Yuzuncu Yil University, Van, Turkey

    Hilmi Ergören & M. Giyas Sakar

Authors
  1. Hilmi Ergören
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  2. M. Giyas Sakar
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Correspondence to Hilmi Ergören .

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Editors and Affiliations

  1. , Department of Mathematical Sciences, The University of Memphis, Memphis, 38152, Tennessee, USA

    George A. Anastassiou

  2. , Department of Mathematics, TOBB Economics and Technology University, Sögütözü, Ankara, 06530, Turkey

    Oktay Duman

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Ergören, H., Sakar, M.G. (2013). Boundary Value Problems for Impulsive Fractional Differential Equations with Nonlocal Conditions. In: Anastassiou, G., Duman, O. (eds) Advances in Applied Mathematics and Approximation Theory. Springer Proceedings in Mathematics & Statistics, vol 41. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6393-1_18

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