## Dynamic Systems and Applications xx (2005) pp-pp MULTIPLE INTEGRATION ON TIME SCALES

### BibTeX

@MISC{Bohner_dynamicsystems,

author = {Martin Bohner and Gusein and Sh. Guseinov},

title = {Dynamic Systems and Applications xx (2005) pp-pp MULTIPLE INTEGRATION ON TIME SCALES},

year = {}

}

### OpenURL

### Abstract

ABSTRACT. In this paper an introduction to integration theory for multivariable functions on time scales is given. Such an integral calculus can be used to develop a theory of partial dynamic equations on time scales in order to unify and extend the usual partial differential equations and partial difference equations. AMS (MOS) Subject Classification. 26B15, 28A25, 35R10. 1.

### Citations

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- 2001
(Show Context)
Citation Context ..., 28A25, 35R10. 1. INTRODUCTION A time scale is an arbitrary nonempty closed subset of the real numbers. For a general introduction to the calculus of time scales we refer the reader to the textbooks =-=[6, 7]-=-. In [5] a differential calculus for multivariable functions on time scales was presented by the authors in order to provide an instrument for introducing and investigating partial dynamic equations o... |

164 |
Mathematical Analysis
- Apostol
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Citation Context ...oof. Clearly, the above given Definition 2.1 and Definition 2.13 of the ∆-integral coincide in case T1 = T2 = R with the usual Darboux and Riemann definitions of the integral, respectively (see e.g., =-=[2, 11]-=-). Notice that the classical definitions of Darboux’s and Riemann’s integral do not depend on whether the subrectangles of the partition are taken closed, half-closed, or open. Moreover, if T1 = T2 = ... |

92 |
Analysis on measure chains - a unified approach to continuous and discrete calculus
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Citation Context ...ucing and investigating partial dynamic equations on time scales. The present paper continues [5] and discusses multiple integration on time scales. In the original papers of B. Aulbach and S. Hilger =-=[3, 10]-=- on single variable time scales calculus the concept of integral was defined by means of an antiderivative (or pre-antiderivative) of a function and called the Cauchy integral. Next by S. Sailer [12] ... |

58 |
Linear dynamic processes with inhomogeneous time scale
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(Show Context)
Citation Context ...ucing and investigating partial dynamic equations on time scales. The present paper continues [5] and discusses multiple integration on time scales. In the original papers of B. Aulbach and S. Hilger =-=[3, 10]-=- on single variable time scales calculus the concept of integral was defined by means of an antiderivative (or pre-antiderivative) of a function and called the Cauchy integral. Next by S. Sailer [12] ... |

23 |
Integration on time scales
- Guseinov
- 2003
(Show Context)
Citation Context ...Next by S. Sailer [12] the Darboux definition of the integral was used for integral calculus on time scales. Further Riemann and Lebesgue definitions of the integral on time scales were introduced in =-=[4, 7, 8, 9]-=- and a complete, to a considered extent, theory of integration for single variable time scales was developed. In [1], C. Ahlbrandt and C. Morian introduced double integrals over rectangles on time sca... |

10 |
Partial differential equations on time scales
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(Show Context)
Citation Context ...and Lebesgue definitions of the integral on time scales were introduced in [4, 7, 8, 9] and a complete, to a considered extent, theory of integration for single variable time scales was developed. In =-=[1]-=-, C. Ahlbrandt and C. Morian introduced double integrals over rectangles on time scales as iterated integrals defined by using antiderivatives of single variable functions, under the assumption that t... |

9 |
A Course of Mathematical Analysis
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(Show Context)
Citation Context ...oof. Clearly, the above given Definition 2.1 and Definition 2.13 of the ∆-integral coincide in case T1 = T2 = R with the usual Darboux and Riemann definitions of the integral, respectively (see e.g., =-=[2, 11]-=-). Notice that the classical definitions of Darboux’s and Riemann’s integral do not depend on whether the subrectangles of the partition are taken closed, half-closed, or open. Moreover, if T1 = T2 = ... |

4 |
Partial differentiation on time scales
- Guseinov
- 2004
(Show Context)
Citation Context ...R10. 1. INTRODUCTION A time scale is an arbitrary nonempty closed subset of the real numbers. For a general introduction to the calculus of time scales we refer the reader to the textbooks [6, 7]. In =-=[5]-=- a differential calculus for multivariable functions on time scales was presented by the authors in order to provide an instrument for introducing and investigating partial dynamic equations on time s... |

4 |
Basics of Riemann delta and nabla integration on time scales
- Guseinov, Kaymakçalan
(Show Context)
Citation Context ...Next by S. Sailer [12] the Darboux definition of the integral was used for integral calculus on time scales. Further Riemann and Lebesgue definitions of the integral on time scales were introduced in =-=[4, 7, 8, 9]-=- and a complete, to a considered extent, theory of integration for single variable time scales was developed. In [1], C. Ahlbrandt and C. Morian introduced double integrals over rectangles on time sca... |

2 |
Riemann–Stieltjes Integrale auf Zeitmengen. Universität Augsburg
- Sailer
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(Show Context)
Citation Context ..., 10] on single variable time scales calculus the concept of integral was defined by means of an antiderivative (or pre-antiderivative) of a function and called the Cauchy integral. Next by S. Sailer =-=[12]-=- the Darboux definition of the integral was used for integral calculus on time scales. Further Riemann and Lebesgue definitions of the integral on time scales were introduced in [4, 7, 8, 9] and a com... |

1 | s)∆2s + � b a ∆1t F (t, s)∆1t∆2s � d ψ(t) � ψ(t) ϕ(t) F (t, s)∆2s f(t, s)∆2s - unknown authors |