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2012 | 10 | 6 | 1981-1994
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
6
Strony
1981-1994
Opis fizyczny
Daty
wydano
2012-12-01
online
2012-10-12
Twórcy
  • Institute of Mathematics, Częstochowa University of Technology, Dąbrowskiego 73, 42-200, Częstochowa, Poland, mklimek@im.pcz.pl
  • Institute of Mathematics, Częstochowa University of Technology, Dąbrowskiego 73, 42-200, Częstochowa, Poland, marek.blasik@gmail.com
Bibliografia
  • [1] Baleanu D., Diethelm K., Scalas E., Trujillo J.J., Fractional Calculus, Ser. Complex. Nonlinearity Chaos, 3, World Scientific, Singapore, 2012
  • [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] 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
  • [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] 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] Diethelm K., The Analysis of Fractional Differential Equations, Lecture Notes in Math., 2004, Springer, Berlin, 2010
  • [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]
  • [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] 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]
  • [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] Klimek M., On Solutions of Linear Fractional Differential Equations of a Variational Type, Czestochowa University of Technology, Czestochowa, 2009
  • [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
  • [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]
  • [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] 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] Lakshmikantham V., Leela S., Vasundhara Devi J., Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, 2009
  • [17] Miller K.S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993
  • [18] Podlubny I., Fractional Differential Equations, Math. Sci. Engrg., 198, Academic Press, San Diego, 1999
  • [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] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives, Gordon and Breach, Amsterdam, 1993
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
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