Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Małgorzata Klimek; Marek Błasik

Open Mathematics (2012)

  • Volume: 10, Issue: 6, page 1981-1994
  • ISSN: 2391-5455

Abstract

top
Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.

How to cite

top

Małgorzata Klimek, and Marek Błasik. "Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order." Open Mathematics 10.6 (2012): 1981-1994. <http://eudml.org/doc/269411>.

@article{MałgorzataKlimek2012,
abstract = {Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.},
author = {Małgorzata Klimek, Marek Błasik},
journal = {Open Mathematics},
keywords = {Fractional derivative; Fractional differential equation; Fixed point condition; Equivalent norms/metrics; Initial value problem; fractional derivative; fractional differential equation; fixed point condition; equivalent norms; equivalent metrics; initial value problem},
language = {eng},
number = {6},
pages = {1981-1994},
title = {Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order},
url = {http://eudml.org/doc/269411},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Małgorzata Klimek
AU - Marek Błasik
TI - Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 1981
EP - 1994
AB - Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.
LA - eng
KW - Fractional derivative; Fractional differential equation; Fixed point condition; Equivalent norms/metrics; Initial value problem; fractional derivative; fractional differential equation; fixed point condition; equivalent norms; equivalent metrics; initial value problem
UR - http://eudml.org/doc/269411
ER -

References

top
  1. [1] Baleanu D., Diethelm K., Scalas E., Trujillo J.J., Fractional Calculus, Ser. Complex. Nonlinearity Chaos, 3, World Scientific, Singapore, 2012 Zbl1248.26011
  2. [2] Bǎleanu D., Mustafa O.G., On the global existence of solutions to a class of fractional differential equations, Comput. Math. Appl., 2010, 59(5), 1835–1841 http://dx.doi.org/10.1016/j.camwa.2009.08.028[Crossref] 
  3. [3] Bielecki A., Une remarque sur la méthode de Banach-Cacciopoli-Tikhonov dans la théorie des équations différentielles ordinaires, Bull. Acad. Polon. Sci. Cl. III, 1956, 4, 261–264 Zbl0070.08103
  4. [4] Błasik M., Klimek M., On application of contraction principle to solve two-term fractional differential equations, Acta Mechanica et Automatica, 2011, 5(2), 5–10 
  5. [5] Deng J., Ma L., Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 2010, 23(6), 676–680 http://dx.doi.org/10.1016/j.aml.2010.02.007[Crossref] 
  6. [6] Diethelm K., The Analysis of Fractional Differential Equations, Lecture Notes in Math., 2004, Springer, Berlin, 2010 
  7. [7] El-Raheem Z.F.A., Modification of the application of a contraction mapping method on a class of fractional differential equation, Appl. Math. Comput., 2003, 137(2–3), 371–374 http://dx.doi.org/10.1016/S0096-3003(02)00136-4[Crossref] Zbl1034.34070
  8. [8] Kilbas A.A., Srivastawa H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., 204, Elsevier, Amsterdam, 2006 http://dx.doi.org/10.1016/S0304-0208(06)80001-0[Crossref] 
  9. [9] Kilbas A.A., Trujillo J.J., Differential equations of fractional order: methods, results and problems. I, Appl. Anal., 2001, 78(1–2), 153–192 http://dx.doi.org/10.1080/00036810108840931[Crossref] Zbl1031.34002
  10. [10] Kilbas A.A., Trujillo J.J., Differential equations of fractional order: methods, results and problems. II, Appl. Anal., 2002, 81(2), 435–493 http://dx.doi.org/10.1080/0003681021000022032[Crossref] 
  11. [11] Klimek M., On Solutions of Linear Fractional Differential Equations of a Variational Type, Czestochowa University of Technology, Czestochowa, 2009 
  12. [12] Klimek M., On contraction principle applied to nonlinear fractional differential equations with derivatives of order α ∈ (0,1), Banach Center Publ., 2011, 95, 325–338 http://dx.doi.org/10.4064/bc95-0-19 Zbl1273.34011
  13. [13] Klimek M., Sequential fractional differential equations with Hadamard derivative, Commun. Nonlinear Sci. Numer. Simul., 2011, 16(12), 4689–4697 http://dx.doi.org/10.1016/j.cnsns.2011.01.018[Crossref] Zbl1242.34009
  14. [14] Klimek M., Błasik M., Existence-uniqueness result for nonlinear two-term sequential FDE, In: 7th European Nonlinear Dynamics Conference (ENOC 2011), Rome, July 24–29, 2011, available at http://w3.uniroma1.it/dsg/enoc2011/proceedings/pdf/Klimek_Blasik.pdf 
  15. [15] Kosmatov N., Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear Anal., 2009, 70(7), 2521–2529 http://dx.doi.org/10.1016/j.na.2008.03.037[Crossref] 
  16. [16] Lakshmikantham V., Leela S., Vasundhara Devi J., Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, 2009 Zbl1188.37002
  17. [17] Miller K.S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993 Zbl0789.26002
  18. [18] Podlubny I., Fractional Differential Equations, Math. Sci. Engrg., 198, Academic Press, San Diego, 1999 
  19. [19] ur Rehman M., Khan R.A., Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, Appl. Math. Lett., 2010, 23(9), 1038–1044 http://dx.doi.org/10.1016/j.aml.2010.04.033[Crossref] 
  20. [20] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives, Gordon and Breach, Amsterdam, 1993 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.