Lerch’s theorem on nabla time scales
-
Matej Dolník
und
Tomáš Kisela
Abstract
The paper discusses uniqueness of Laplace transform considered on nabla time scales. As the main result, a nabla time scales analogue of Lerch’s theorem ensuring uniqueness of Laplace image is proved for so-called simply periodic time scales. Moreover, several presented counterexamples demonstrate that the uniqueness of Laplace image does not occur on general time scales when the nabla approach is employed. Other special properties of Laplace transform on nabla time scales, such as potential disconnectedness of domain of convergence, are addressed as well.
The research has been supported by the grant 17-03224S of the Czech Science Foundation.
(Communicated by Ján Borsík)
References
[1] Agarwal, R. P.: Certain fractional q-integrals and q-derivatives, Math. Proc. Camb. Philos. Soc. 66 (1969), 365–370.10.1017/S0305004100045060Suche in Google Scholar
[2] Ahrendt, Ch. R.: Properties of the Generalized Laplace Transform and Transport Partial Dynamic Equation on Time Scales, Dissertation, University of Nebraska at Lincoln, 2010.Suche in Google Scholar
[3] Akin-Bohner, E.—Bohner, M.: Exponential functions and Laplace transforms for alpha derivatives. In: Proceedings of the Sixth International Conference on Difference Equations, Augsburg, Germany, 2001, pp. 231–238.Suche in Google Scholar
[4] Bastos, N. R. O.—Mozyrska, D.—Torres, D. F. M.: Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform, Int. J. Math. Comput. 11 (2011), 1–9.Suche in Google Scholar
[5] Bohner, M.—Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001.10.1007/978-1-4612-0201-1Suche in Google Scholar
[6] Čermák, J.—Nechvátal, L.: On (q, h)-analogue of fractional calculus, J. Nonlinear Math. Phys. 17 (2010), 51–68.10.1142/S1402925110000593Suche in Google Scholar
[7] Čermák, J.—Kisela, T.: Introduction to stability theory of linear fractional difference equations. In: Fractional Calculus: Theory, Mathematics Research Developments, 2014, pp. 117–162.Suche in Google Scholar
[8] Davis, J. M.—Gravagne, I. A.—Jackson, B. J.—Marks II, R. J.—Ramos, A. A.: The Laplace transform on time scales revisited, J. Math. Anal. Appl. 332(2) (2007), 1291–1307.10.1016/j.jmaa.2006.10.089Suche in Google Scholar
[9] Diaz, R.—Osler, T. J.: Differences of fractional order, Math. Comput. 28 (1974), 185–202.10.1090/S0025-5718-1974-0346352-5Suche in Google Scholar
[10] Karpuz, B.: On uniqueness of the Laplace transform on time scales, PanAmer. Math. J. 21 (2011), 101–110.Suche in Google Scholar
[11] Kisela, T.: Power functions and essentials of fractional calculus on isolated time scales, Adv. Differ. Equ. 2013:259 (2013), 18 p.10.1186/1687-1847-2013-259Suche in Google Scholar
[12] Ortigueira, M. D.—Torres, D. F. M.—Trujillo, J. J.: Exponentials and Laplace transform on nonuniform time scales, Commun. Nonlinear Sci. Numer. Simul. 39 (2016), 252–270.10.1016/j.cnsns.2016.03.010Suche in Google Scholar
[13] Podlubný, I: Fractional Differential Equations, Academic Press, New York, 1999.Suche in Google Scholar
[14] Rahmat, R. S. R.: Integral transform methods for solving fractional dynamic equations on time scales, Abstr. Appl. Anal. 2014 (2014), 10 p.10.1155/2014/261348Suche in Google Scholar
[15] Watson, E. J.: Laplace Transforms and Applications, Van Nostrand Reinhold Company, New York, 1981.Suche in Google Scholar
[16] Zhu, J.—Wu, L.: Fractional Cauchy problem with Caputo nabla derivative on time scales, Abstr. Appl. Anal. 2015 (2015), 23 p.10.1155/2015/486054Suche in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- Professor Anatolij Dvurečenskij
- A generating theorem of punctured surface triangulations with inner degree at least 4
- Natural extension of EF-spaces and EZ-spaces to the pointfree context
- On weak isometries in directed groups
- On self-injectivity of the f-ring Frm(𝓟(ℝ), L)
- On the extensions of discrete valuations in number fields
- On certain classes of generalized derivations
- Nilpotent elements in medial semigroups
- Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
- Sharp coefficient bounds for starlike functions associated with the Bell numbers
- Sufficient conditions for Carathéodory functions and applications to univalent functions
- Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks
- Linear directional differential equations in the unit ball: solutions of bounded L-index
- On oscilatory fourth order nonlinear neutral differential equations – IV
- Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms
- Lerch’s theorem on nabla time scales
-
- Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula
- Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
- On the amenability of a class of Banach algebras with application to measure algebra
- A bivariate Kumaraswamy-exponential distribution with application
- Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data
- Corrigendum to ”on the congruent number problem over integers of cyclic number fields”, Math. Slovaca 66(3) (2016), 561–564
Artikel in diesem Heft
- Regular papers
- Professor Anatolij Dvurečenskij
- A generating theorem of punctured surface triangulations with inner degree at least 4
- Natural extension of EF-spaces and EZ-spaces to the pointfree context
- On weak isometries in directed groups
- On self-injectivity of the f-ring Frm(𝓟(ℝ), L)
- On the extensions of discrete valuations in number fields
- On certain classes of generalized derivations
- Nilpotent elements in medial semigroups
- Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
- Sharp coefficient bounds for starlike functions associated with the Bell numbers
- Sufficient conditions for Carathéodory functions and applications to univalent functions
- Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks
- Linear directional differential equations in the unit ball: solutions of bounded L-index
- On oscilatory fourth order nonlinear neutral differential equations – IV
- Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms
- Lerch’s theorem on nabla time scales
-
- Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula
- Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
- On the amenability of a class of Banach algebras with application to measure algebra
- A bivariate Kumaraswamy-exponential distribution with application
- Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data
- Corrigendum to ”on the congruent number problem over integers of cyclic number fields”, Math. Slovaca 66(3) (2016), 561–564
