Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 171-195, 2017

Functions of finite fractional variation and their applications to fractional impulsive equations

Dariusz Idczak

Received August 22, 2015.   First published February 24, 2017.

Dariusz Idczak, Faculty of Mathematics and Computer Science, University of Lódź, Stefana Banacha 22, 90-238 Lódź, Poland, e-mail: idczak@math.uni.lodz.pl

Abstract: We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak $\sigma$-additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a $\sigma$-additive term - we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e.\^^Mequations containing the Dirac measures.

Keywords: finite fractional variation; weak $\sigma$-additive fractional; derivative; fractional impulsive equation; Dirac measure; Cauchy formula

Classification (MSC 2010): 26A45, 34A37

DOI: 10.21136/CMJ.2017.0455-15

Full text available as PDF.


References:
  [1] N. Abada, R. P. Agarwal, M. Benchohra, H. Hammouche: Impulsive semilinear neutral functional differential inclusions with multivalued jumps. Appl. Math., Praha 56 (2011), 227-250. DOI 10.1007/s10492-011-0004-5 | MR 2810245 | Zbl 1224.34207
  [2] D. D. Bainov, P. S. Simeonov: Systems with Impulse Effect. Stability, Theory and Applications. Ellis Horwood Series in Mathematics and Its Applications, Ellis Horwood Limited, Chichester; Halsted Press, New York (1989). MR 1010418 | Zbl 0683.34032
  [3] M. Benchohra, J. Henderson, S. Ntouyas: Impulsive Differential Equations and Inclusions. Contemporary Mathematics and Its Applications 2, Hindawi Publishing Corporation, New York (2006). DOI 10.1155/9789775945501 | MR 2322133 | Zbl 1130.34003
  [4] M. Benchohra, B. A. Slimani: Existence and uniqueness of solutions to impulsive fractional differential equations. Electron. J. Differ. Equ. (electronic only) 2009 (2009), Paper No. 10, 11 pages. MR 2471119 | Zbl 1178.34004
  [5] G. Bonanno, R. Rodríguez-López, S. Tersian: Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17 (2014), 717-744. DOI 10.2478/s13540-014-0196-y | MR 3260304 | Zbl 1308.34010
  [6] L. Bourdin: Existence of a weak solution for fractional Euler-Lagrange equations. J. Math. Anal. Appl. 399 (2013), 239-251. DOI 10.1016/j.jmaa.2012.10.008 | MR 2993851 | Zbl 06125381
  [7] L. Bourdin, D. Idczak: A fractional fundamental lemma and a fractional integration by parts formula - Applications to critical points of Bolza functionals and to linear boundary value problems. Adv. Differ. Equ. 20 (2015), 213-232. MR 3311433 | Zbl 1309.26007
  [8] H. Brezis: Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maitrise, Masson, Paris French (1983). MR 0697382 | Zbl 0511.46001
  [9] B. Gayathri, R. Murugesu, J. Rajasingh: Existence of solutions of some impulsive fractional integrodifferential equations. Int. J. Math. Anal., Ruse 6 (2012), 825-836. MR 2905181 | Zbl 1252.45004
  [10] A. Halanay, D. Wexler: Qualitative Theory of Impulse Systems. Russian Mir, Moskva (1971). Zbl 0226.34001
  [11] R. Haloi, P. Kumar, D. N. Pandey: Sufficient conditions for the existence and uniqueness of solutions to impulsive fractional integro-differential equations with deviating arguments. J. Fract. Calc. Appl. 5 (2014), 73-84. MR 3234097
  [12] T. H. Hildebrandt: On systems of linear differentio-Stieltjes-integral equations. Ill. J. Math. 3 (1959), 352-373. MR 0105600 | Zbl 0088.31101
  [13] D. Idczak: Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation. Czech. Math. J. 48 (1998), 145-171. DOI 10.1023/A:1022427914423 | MR 1614025 | Zbl 0930.26006
  [14] D. Idczak, R. Kamocki: On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in $\Bbb R^n$. Fract. Calc. Appl. Anal. 14 (2011), 538-553. DOI 10.2478/s13540-011-0033-5 | MR 2846375 | Zbl 1273.34010
  [15] D. Idczak, S. Walczak: Fractional Sobolev spaces via Riemann-Liouville derivatives. J. Funct. Spaces Appl. 2013 (2013), Article ID 128043, 15 pages. DOI 10.1155/2013/128043 | MR 3144452 | Zbl 1298.46033
  [16] J. Kurzweil: Generalized ordinary differential equations. Czech. Math. J. 8 (1958), 360-388. MR 0111878 | Zbl 0094.05804
  [17] J. Kurzweil: On generalized ordinary differential equations possessing discontinuous solutions. PMM, J. Appl. Math. Mech. 22 37-60 (1958), translation from Prikl. Mat. Mekh. 22 27-45 (1958). DOI 10.1016/0021-8928(58)90082-0 | MR 0111876 | Zbl 0102.07003
  [18] J. Kurzweil: Linear differential equations with distributions as coefficients. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 7 (1959), 557-560. MR 0111887 | Zbl 0117.34401
  [19] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics 6, World Scientific, Singapore (1989). DOI 10.1142/0906 | MR 1082551 | Zbl 0719.34002
  [20] S. Łojasiewicz: An Introduction to the Theory of Real Functions. A Wiley-Interscience Publication, John Wiley & Sons, Chichester (1988). MR 0952856 | Zbl 0653.26001
  [21] R. Rodríguez-López, S. Tersian: Multiple solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17 (2014), 1016-1038. DOI 10.2478/s13540-014-0212-2 | MR 3254678 | Zbl 1312.34024
  [22] S. G. Samko, A. A. Kilbas, O. I. Marichev: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993). MR 1347689 | Zbl 0818.26003
  [23] A. M. Samoilenko, N. A. Perestyuk: Impulsive Differential Equations. World Scientific Series on Nonlinear Science, Series A. 14, World Scientific, Singapore (1995). MR 1355787 | Zbl 0837.34003
  [24] L. Schwartz: Méthodes mathématiques pour les sciences physiques. Enseignement des Sciences, Hermann, Paris French (1961). MR 0143360 | Zbl 0101.41301
  [25] F. W. Stallard: Functions of bounded variation as solutions of differential systems. Proc. Am. Math. Soc. 13 (1962), 366-373. DOI 10.2307/2034939 | MR 0138835 | Zbl 0108.08203
  [26] J. Wang, M. Feckan, Y. Zhou: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8 (2011), 345-361. DOI 10.4310/DPDE.2011.v8.n4.a3 | MR 2901608 | Zbl 1264.34014
  [27] Z. Wyderka: Linear differential equations with measures as coefficients and the control theory. Čas. PěstováníMat. 114 (1989), 13-27. MR 0990112 | Zbl 0664.34013
  [28] Z. Wyderka: Linear Differential Equations with Measures as Coefficients and Control Theory. Prace Naukowe Uniwersytetu Ślaskiego w Katowicach 1413, Wydawnictwo Uniwersytetu Ślaskiego, Katowice (1994). MR 1292252 | Zbl 0813.34058


Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer need to access the articles on their site, which is http://link.springer.com/journal/10587.

[Previous Article] [Next Article] [Contents of This Number] [Contents of Czechoslovak Mathematical Journal]