## Integration on time scales.(English)Zbl 1039.26007

In this paper, the author defines the Riemann and Lebesgue integrals (more precisely, the Riemann and Lebesgue $$\Delta$$-integrals and $$\nabla$$-integrals) on time scales and studies their properties and relationship. In particular, the author presents results concerning the lower and upper Darboux sums, the Riemann sums, the Riemann and Lebesgue integrals, and certain mean value results on time scales, which are needed in the proofs. This paper will be useful for anyone interested in the theory of dynamic equations on time scales (measure chains).
Reviewer notes that Sections 1-5 are contained in the book [M. Bohner and A. Peterson, “Advances in dynamic equations on time scales” (2003; Zbl 1025.34001)] in Chapter 5 by M. Bohner and the author, and Appendix A is contained in the same book in Chapter 1 by M. Bohner, the author, and A. Peterson.

### MSC:

 26A42 Integrals of Riemann, Stieltjes and Lebesgue type 39A12 Discrete version of topics in analysis

Zbl 1025.34001
Full Text:

### References:

 [1] Agarwal, R.P.; Bohner, M., Basic calculus on time scales and some of its applications, Results math., 35, 3-22, (1999) · Zbl 0927.39003 [2] Ahlbrandt, C.D.; Bohner, M.; Ridenhour, J., Hamiltonian systems on time scales, J. math. anal. appl., 250, 561-578, (2000) · Zbl 0966.39010 [3] Aslim, G.; Guseinov, G.Sh., Weak semirings, ω-semirings, and measures, Bull. allahabad math. soc., 14, 1-20, (1999) · Zbl 1102.28301 [4] Atici, F.M.; Guseinov, G.Sh., On Green’s functions and positive solutions for boundary value problems on time scales, J. comput. appl. math., 141, 75-99, (2002) · Zbl 1007.34025 [5] Atici, F.M.; Guseinov, G.Sh.; Kaymakcalan, B., On Lyapunov inequality in stability theory for Hill’s equation on time scales, J. inequal. appl., 5, 603-620, (2000) · Zbl 0971.39005 [6] Aulbach, B.; Hilger, S., Linear dynamic processes with inhomogeneous time scale, (), 9-20 [7] Aulbach, B.; Neidhart, L., Integration on measure chains, (), to appear · Zbl 1083.26005 [8] Bohner, M.; Castillo, J., Mimetic methods on measure chains, Comput. math. appl., 42, 705-710, (2001) · Zbl 1107.34304 [9] Bohner, M.; Peterson, A., Dynamic equations on time scales: an introduction with applications, (2001), Birkhäuser Boston · Zbl 0978.39001 [10] Cohn, D.L., Measure theory, (1980), Birkhäuser Boston · Zbl 0436.28001 [11] Fischer, E., Intermediate real analysis, (1983), Springer-Verlag New York · Zbl 0506.26002 [12] Guseinov, G.Sh.; Kaymakcalan, B., On a disconjugacy criterion for second order dynamic equations on time scales, J. comput. appl. math., 141, 187-196, (2002) · Zbl 1014.34023 [13] Guseinov, G.Sh.; Kaymakcalan, B., On the Riemann integration on time scales, (), to appear · Zbl 1023.39009 [14] Guseinov, G.Sh.; Kaymakcalan, B., Basics of Riemann delta and nabla integration on time scales, J. difference equations appl., 8, 1001-1017, (2002) · Zbl 1023.39009 [15] Halmos, P.R., Measure theory, (1950), Van Nostrand Princeton [16] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001 [17] Hilger, S., Special functions, Laplace and Fourier transform on measure chains, Dynam. systems appl., 8, 471-488, (1999) · Zbl 0943.39006 [18] Kolmogorov, A.N.; Fomin, S.V., Introductory real analysis, (1975), Dover New York · Zbl 0213.07305 [19] Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B., Dynamic systems on measure chains, (1996), Kluwer Academic Dordrecht · Zbl 0869.34039 [20] Ross, K.A., Elementary analysis: the theory of calculus, (1990), Springer-Verlag New York [21] S. Sailer, Riemann-Stieltjes Integrale auf Zeitmengen (Schriftliche Hausarbeit, vorgelegt bei Prof. Dr. B. Aulbach), Universität Augsburg, 1992
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.