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Solvability of Initial Value Problems With Fractional Order Differential Equations in Banach Spaces By α-Dense Curves

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Abstract

In this paper we study the existence of solutions for an initial value problem, posed in a given Banach space, with a fractional differential equation via densifiability techniques. For our goal, we will prove a new fixed point result (not based on measures of noncompactness) which is, in forms, a generalization of the well-known Darbo’s fixed point theorem but essentially different. Some illustrative examples are given.

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References

  1. A. Aghajani, J. Banaś and N. Sabzali, Some generalizations of Darbo fixed point theorem and applications. Bull. Belg. Math. Soc. Simon Stevin 20, No 2 (2013), 345–358.

    Article  MathSciNet  Google Scholar 

  2. A. Aghajani, E. Pourhadi and J.J. Trujillo, Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal. 16, No 4 (2013), 962–977; DOI: 10.2478/s13540-013-0059-y; https://www.degruyter.com/view/j/fca.2013.16.issue-4/issue-files/fca.2013.16.issue-4.xml.

    Article  MathSciNet  Google Scholar 

  3. R.R. Akhmerov, M.I. Kamenskii, A.S. Potapova, A.E. Rodkina and B.N. Sadovskii, Measure of Noncompactness and Condensing Operators. Birkhäuser Verlag, Basel (1992).

    Book  Google Scholar 

  4. R.P. Argawal, M. Benchohra and D. Seba, On the application of measure of noncompactness to the existence of solutions for fractional differential equations. Results Math. 55, No 3–4 (2009), 221–230; DOI: 10.1007/s00025-009-0434-5.

    MathSciNet  MATH  Google Scholar 

  5. R.P. Agarwal, M. Meehan and D. O’Regan, Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2004).

    MATH  Google Scholar 

  6. K. Aissani and M. Benchohra, Semilinear fractional order integro-differential equations with infinite delay in Banach spaces. Arch. Math. (Brno) 49, No 49 (2013), 105–117.

    Article  MathSciNet  Google Scholar 

  7. K. Aissani and M. Benchohra, Fractional integro-differential equations with state-dependent delay. Adv. Dyn. Syst. Appl. 9, No 1 (2014), 17–30.

    MathSciNet  MATH  Google Scholar 

  8. J.M. Ayerbe Toledano, T. Domínguez Benavides and G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory. Birkhäuser, Basel (1997).

    Book  Google Scholar 

  9. J. Banaś, On measures of noncompactness in Banach spaces. Comment. Math. Univ. Carolin. 21, No 21 (1980), 131–143.

    MathSciNet  MATH  Google Scholar 

  10. M. Belmekki, J.J. Nieto and R. Rodríguez-López, Existence of periodic solution for a nonlinear fractional differential equation. Bound. Value Probl. 2009 (2009), Article ID 324561, 18 pp.; DOI: 10.1155/2009/324561.

    Article  MathSciNet  Google Scholar 

  11. M. Benchohra, J.R. Graef and F.Z. Mostefai, Weak solutions for nonlinear fractional differential equations on reflexive Banach spaces. Elec. J. of Qualitative Theory of Diff. Eq. 2010, No 54 (2010), 1–10; DOI: 10.7494/OpMath.2012.32.1.31.

    MathSciNet  MATH  Google Scholar 

  12. Y. Cherruault and G. Mora, Optimisation Globale. Théorie des Courbes α-denses. Económica, Paris (2005).

    Google Scholar 

  13. E. Cuesta and J. Finat Codes, Image processing by means of a linear integro-differential equation. In: M.H. Hamza (Ed.), Visualization, Imaging, and Image Processing 2003, Paper # 91, ACTA Press, Calgary.

  14. W. Deng, Short memory principle and a predictor-corrector approach for fractional differential equations. J. Comput. Appl. Math. 206, No 1 (2007), 174–188; DOI: 10.1016/j.cam.2006.06.008.

    Article  MathSciNet  Google Scholar 

  15. X.W. Dong, J.R. Wang and Y. Zhou, On nonlocal problems for fractional differential equations in Banach spaces. Opuscula Math. 31, No 3 (2011), 341–357; DOI: 10.7494/OpMath.2011.31.3.341.

    Article  MathSciNet  Google Scholar 

  16. A.D. Freed and K. Diethelm, Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional-derivative kernels with application to the human calcaneal fat pad. Biomech. Model. Mechanobiol. 5, No 4 (2006), 203–215; DOI: 10.1007/s10237-005-0011-0.

    Article  Google Scholar 

  17. G. García and G. Mora, The degree of convex nondensifiability in Banach spaces. J. Convex Anal. 22, No 3 (2015), 871–888.

    MathSciNet  MATH  Google Scholar 

  18. J.L. Kelley, General Topology. Springer, New York (1955).

    MATH  Google Scholar 

  19. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier Science, Amsterdam (2006).

  20. K. Li, J. Peng and J. Gao, Nonlocal fractional semilinear differential equations in separable Banach spaces. Electron. J. Differential Equations 2013, No 7 (2013), 1–7.

    MathSciNet  MATH  Google Scholar 

  21. R. Metzler, W. Schick, H.-G. Kilian, T.F. Nonnenmacher, Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 103, No 16 (1998), 7180–7186; DOI: 10.1063/1.470346.

    Article  Google Scholar 

  22. G. Mora and Y. Cherruault, Characterization and generation of α-dense curves. Computers Math. Applic. 33, No 9 (1997), 83–91; DOI: 10.1016/S0898-1221(97)00067-9.

    Article  MathSciNet  Google Scholar 

  23. G. Mora and J.A. Mira, Alpha-dense curves in infinite dimensional spaces. Inter. Journal or Pure and Appl. Math. 5, No 4 (2003), 437–449.

    MathSciNet  MATH  Google Scholar 

  24. G. Mora and D.A. Redtwitz, Densifiable metric spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 105, No 1 (2011), 71–83; DOI: 10.1007/s13398-011-0005-y.

    Article  MathSciNet  Google Scholar 

  25. I. Podlubny, Fractional-order systems and fractional-order controllers. Technical Report UEF-03-94, Institute for Experimental Physics, Slovak Acad. Sci. (1994).

    Google Scholar 

  26. I. Podlubny, L. Dorcak and J. Misanek, Application of fractional-order derivatives to calculation of heat load intensity change in blast furnace walls. Trans. Tech. Univ. Kosice 5, No 2 (1995), 137–144.

    Google Scholar 

  27. H. Sagan, Space Filling Curves. Springer-Verlag, New York (1994).

    Book  Google Scholar 

  28. J.R. Wang, Y. Zhou and M. Feckan, Abstract Cauchy problem for fractional differential equations. Nonlinear Dynam. 71, No 4 (2013), 685–700; DOI: 10.1007/s11071-012-0452-9.

    Article  MathSciNet  Google Scholar 

  29. S. Willard, General Topology. Dover Pubs. Inc., New York (2004).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Universidad Nacional de Educación a Distancia (UNED) C/ Candalix S/N, 03202 Elche, Alicante, Spain

    Gonzalo García

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  1. Gonzalo García
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García, G. Solvability of Initial Value Problems With Fractional Order Differential Equations in Banach Spaces By α-Dense Curves. FCAA 20, 646–661 (2017). https://doi.org/10.1515/fca-2017-0034

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  • DOI: https://doi.org/10.1515/fca-2017-0034

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