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On the existence of solutions of fractional integro-differential equations

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Abstract

Under some suitable conditions, we prove the solvability of a large class of nonlinear fractional integro-differential equations by establishing some fractional integral inequalities and using the nonlinear alternative Leray-Schauder type. The uniqueness of solutions is also proved in some situations.

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References

  1. R.P. Agarwal and B. Ahmad, Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62 (2011), 1200–1214.

    Article  MathSciNet  MATH  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 (2010), 973–1033.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Aghajani, J. Bánas, and Y. Jalilian, Existence of solutions for a class of nonlinear Volterra singular integral equations. Comput. Math. Appl. 62 (2011), 1215–1227.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Anguraj, P. Karthikeyan, and J. J. Trujillo, Existence of solutions to fractional mixed integrodifferential equations with nonlocal initial condition. Adv. Differ. Equ. 2011, Art. ID 690653, 12 pp.

  5. K. Balachandran, and S. Kiruthika, Existence results for fractional integrodifferential equations with nonlocal condition via resolvent operators. Comput. Math. Appl., 62 (2011), 1350–1358.

    Article  MathSciNet  MATH  Google Scholar 

  6. K. Balachandran, S. Kiruthika, and J.J. Trujillo, Existence results for fractional impulsive integrodifferential equations in Banach spaces. Commun. Nonlin. Sci. Numer. Sim. 16 (2011), 1970–1977.

    Article  MathSciNet  MATH  Google Scholar 

  7. K. Balachandran and J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Analysis: Th. Meth. Appl. 72 (2010), 4587–4593.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods. World Scientific Publishing Company, 2012.

  9. D. Baleanu and O. G. Mustafa, On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59 (2010), 1835–1841.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Caballero, J. Harjani, and K. Sadarangani, On existence and uniqueness of positive solutions to a class of fractional boundary value problems. Bound. Val. Probl. 2011: 25; http://www.boundaryvalueproblems.com/content/2011/1/25

  11. Ph. Clément, G. Gripenberg, and S-O. Londen, Schauder estimates for equations with fractional derivatives. Trans. Amer. Math. Soc. 352 (2000), 2239–2260.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Das, Functional Fractional Calculus for System Identification and Controls. Springer-Verlag, Berlin, Heidelberg, 2008.

    MATH  Google Scholar 

  13. S. Das, Functional Fractional Calculus. Springer-Verlag, Berlin, Heidelberg, 2011.

    Book  MATH  Google Scholar 

  14. Z. Hu and W. Liu, Solvability for fractional order boundary value problems at resonance. Bound. Val. Probl. 2011: 20; http://www.boundaryvalueproblems.com/content/2011/1/20

  15. K. Diethelm and N. J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2002), 229–248.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Idczak and R. Kamocki, On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in Rn. Fract. Calc. Appl. Anal. 14, No 4 (2011), 538–553; DOI: 10.2478/s13540-011-0033-5, http://www.springerlink.com/content/1311-0454/14/4/

    Article  Google Scholar 

  17. A.A. Kilbas and S.A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously, differentiable functions. Diff. Eq. 45 (2005), 84–89.

    Article  MathSciNet  Google Scholar 

  18. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.

    MATH  Google Scholar 

  19. J. Klafter, S.C. Lim, and R. Metzler, Fractional Dynamics in Physics: Recent Advances. World Scientific, Singapore, 2011.

    Google Scholar 

  20. A.N. Kochubei, Fractional differential equations: α-entire solutions, regular and irregular singularities. Fract. Calc. Appl. Anal. 12, No 2 (2009), 135–158, http://www.math.bas.bg/~fcaa

    MathSciNet  MATH  Google Scholar 

  21. N. Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Analysis: Th. Meth. Appl. 70 (2009), 2521–2529.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Kostić, Abstract time-fractional equations: existence and growth of solutions, Fract. Cal. Appl. Anal. 14, No 2 (2011), 301–316; DOI: 10.2478/s13540-011-0018-4, http://www.springerlink.com/content/1311-0454/14/2/

    Google Scholar 

  23. C. Li, Z. Zhao, and Y. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl. 62 (2011), 855–875.

    Article  MathSciNet  MATH  Google Scholar 

  24. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London, 2010.

    Book  MATH  Google Scholar 

  25. R. Magin, Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl., 59 (2010), 1586–1593.

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Magin, M. Ortigueira, I. Podlubny, and J. J. Trujillo, On the fractional signals and systems. Signal Process., 91 (2011), 350–371.

    Article  MATH  Google Scholar 

  27. F.C. Merala, T. J. Roystona, and R. Magin, Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlin. Sci. Numer. Sim. 15 (2010), 939–945.

    Article  Google Scholar 

  28. B.G. Pachpatte, Inequalities for Differential and Integral Equations. Academic Press, 1998.

  29. I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Beijing and Springer-Verlag, Berlin — Heidelberg, 2011.

  30. S. Pilipović and M. Stojanović, Fractional differential equations through Laguerre expansions in abstract spaces: Error estimates. Integr. Transf. Spec. Funct. 17 (2006), 877–887.

    Article  MATH  Google Scholar 

  31. I. Podlubny, Fractional Differential Equations. Academic Press New York, 1999.

    MATH  Google Scholar 

  32. W. Rui, Existence of solutions of nonlinear fractional differential equations at resonance. Elec. J. Qualit. Th. Diff. Eq. 66 (2011), 1–112.

    Google Scholar 

  33. J. Sabatier, O.P. Agrawal, and J.A. Tenreiro-Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, 2007.

  34. J. Sabatier, H.C. Nguyen, C. Farges, J.-Y. Deletage, X. Moreau, F. Guillemard, and B. Bavoux, Fractional models for thermal modeling and temperature estimation of a transistor junction. Adv. Differ. Eq. 2011, Article # 687363.

  35. 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 

  36. V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer-Verlag, Berlin — Heidelberg, 2010.

    MATH  Google Scholar 

  37. J. A. Tenreiro-Machado, V. Kiryakova, and F. Mainardi, Recent history of fractional calculus. Commun. Nonlin. Sci. Numer. Sim. 16 (2011), 1140–1153; doi:10.1016/j.cnsns.2010.05.027, http://www.sciencedirect.com/science/article/pii/S1007570410003205

    Article  MathSciNet  Google Scholar 

  38. J. A. Tenreiro-Machado, Time-delay and fractional derivatives, Adv. Differ. Eq. (2011), Article ID 934094.

  39. Y. Tian and Z. Bai, Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comput. Math. Appl. 59 (2010), 2601–2609.

    Article  MathSciNet  MATH  Google Scholar 

  40. Z. Wei, Q. Li, and J. Che, Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. and Appl. 367 (2010), 260–272.

    Article  MathSciNet  MATH  Google Scholar 

  41. S.B. Yuste and L. Acedo, An explicit finite difference method and a new von neumann-type stability analysis for fractional diffusion equations, SIAM J. on Numerical Analysis 42 (2005), 1862–1874.

    Article  MathSciNet  MATH  Google Scholar 

  42. E. Zeidler, Nonlinear Functional Analysis and Applications, I: Fixed Point Theorems. Springer-Verlag, New York, 1986.

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

    Asadollah Aghajani

  2. Department of Mathematics, Razi University, Kermanshah, 67149, Iran

    Yaghoub Jalilian

  3. Departamento de Análisis Matemático, Universidad de La Laguna, 38271, La Laguna, Tenerife, Spain

    Juan J. Trujillo

Authors
  1. Asadollah Aghajani
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  2. Yaghoub Jalilian
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  3. Juan J. Trujillo
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Correspondence to Asadollah Aghajani.

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Aghajani, A., Jalilian, Y. & Trujillo, J.J. On the existence of solutions of fractional integro-differential equations. fcaa 15, 44–69 (2012). https://doi.org/10.2478/s13540-012-0005-4

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