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

Idczak, Dariusz

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 171-195

Abstract (VO)

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. equations containing the Dirac measures.

Détail

DOI : 10.21136/CMJ.2017.0455-15

Classification : 26A45, 34A37
Keywords: finite fractional variation; weak $\sigma $-additive fractional; derivative; fractional impulsive equation; Dirac measure; Cauchy formula

Bibliographie

[1] Abada, N., Agarwal, R. P., Benchohra, M., Hammouche, H.: Impulsive semilinear neutral functional differential inclusions with multivalued jumps. Appl. Math., Praha 56 (2011), 227-250. | DOI | MR | JFM

[2] Bainov, D. D., Simeonov, P. S.: 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 | JFM

[3] Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions. Contemporary Mathematics and Its Applications 2, Hindawi Publishing Corporation, New York (2006). | DOI | MR | JFM

[4] Benchohra, M., Slimani, B. A.: Existence and uniqueness of solutions to impulsive fractional differential equations. Electron. J. Differ. Equ. (electronic only) 2009 (2009), Paper No. 10, 11 pages. | MR | JFM

[5] Bonanno, G., Rodríguez-López, R., Tersian, S.: Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17 (2014), 717-744. | DOI | MR | JFM

[6] Bourdin, L.: Existence of a weak solution for fractional Euler-Lagrange equations. J. Math. Anal. Appl. 399 (2013), 239-251. | DOI | MR | JFM

[7] Bourdin, L., Idczak, D.: 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 | JFM

[8] Brezis, H.: Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maî trise, Masson, Paris French (1983). | MR | JFM

[9] Gayathri, B., Murugesu, R., Rajasingh, J.: Existence of solutions of some impulsive fractional integrodifferential equations. Int. J. Math. Anal., Ruse 6 (2012), 825-836. | MR | JFM

[10] Halanay, A., Wexler, D.: Qualitative Theory of Impulse Systems. Russian Mir, Moskva (1971). | JFM

[11] Haloi, R., Kumar, P., Pandey, D. N.: 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

[12] Hildebrandt, T. H.: On systems of linear differentio-Stieltjes-integral equations. Ill. J. Math. 3 (1959), 352-373. | DOI | MR | JFM

[13] Idczak, D.: 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 | MR | JFM

[14] Idczak, D., Kamocki, R.: 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 | MR | JFM

[15] Idczak, D., Walczak, S.: Fractional Sobolev spaces via Riemann-Liouville derivatives. J. Funct. Spaces Appl. 2013 (2013), Article ID 128043, 15 pages. | DOI | MR | JFM

[16] Kurzweil, J.: Generalized ordinary differential equations. Czech. Math. J. 8 (1958), 360-388. | MR | JFM

[17] Kurzweil, J.: 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 | MR | JFM

[18] Kurzweil, J.: Linear differential equations with distributions as coefficients. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 7 (1959), 557-560. | MR | JFM

[19] Lakshmikantham, V., Bainov, D. D., Simeonov, P. S.: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics 6, World Scientific, Singapore (1989). | DOI | MR | JFM

[20] ojasiewicz, S. Ł: An Introduction to the Theory of Real Functions. A Wiley-Interscience Publication, John Wiley & Sons, Chichester (1988). | MR | JFM

[21] Rodríguez-López, R., Tersian, S.: Multiple solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17 (2014), 1016-1038. | DOI | MR | JFM

[22] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993). | MR | JFM

[23] Samoilenko, A. M., Perestyuk, N. A.: Impulsive Differential Equations. World Scientific Series on Nonlinear Science, Series A. 14, World Scientific, Singapore (1995). | MR | JFM

[24] Schwartz, L.: Méthodes mathématiques pour les sciences physiques. Enseignement des Sciences, Hermann, Paris French (1961). | MR | JFM

[25] Stallard, F. W.: Functions of bounded variation as solutions of differential systems. Proc. Am. Math. Soc. 13 (1962), 366-373. | DOI | MR | JFM

[26] Wang, J., ckan, M. Fe\u, Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8 (2011), 345-361. | DOI | MR | JFM

[27] Wyderka, Z.: Linear differential equations with measures as coefficients and the control theory. Čas. PěstováníMat. 114 (1989), 13-27. | MR | JFM

[28] Wyderka, Z.: Linear Differential Equations with Measures as Coefficients and Control Theory. Prace Naukowe Uniwersytetu Śl\c askiego w Katowicach 1413, Wydawnictwo Uniwersytetu Śl\c askiego, Katowice (1994). | MR | JFM