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Basic calculus on time scales and some of its applications. (English) Zbl 0927.39003
The main result of this paper is a unification of the continuous and discrete Taylor’s formulae which is at the same time an extension to the case of so-called time scale. A time scale $T$ is a closed subset of the reals, and for a function $g:T\to \bbfR$ it is possible to introduce a derivative $g^\Delta$ and an integral $\int_a^b g(\tau) \Delta\tau$ in a certain manner. Basic tools of calculus on time scales such as versions of Taylor’s formula, l’Hôspital’s rule, and Kneser’s theorem are developed. The established Taylor’s formula is applied to obtain some results in interpolation theory. Applications of these results in the study of asymptotic and oscillatory behavior of solutions of higher-order equations on time scales are given $(y^\Delta= f(t,y,y^\Delta,\dots, y^{\Delta^{n-1}})$, $y^{\Delta^i} (\alpha)= \alpha_i$ for $0\leq i\leq n-1)$.

39A10Additive difference equations
39A11Stability of difference equations (MSC2000)
26A24Differentiation of functions of one real variable
28A25Integration with respect to measures and other set functions
Full Text: DOI
[1] R. P. Agarwal. Properties of solutions of higher order nonlinear difference equations I. An. st. Univ. Iasi, 31:165--172, 1985. · Zbl 0599.39001
[2] R. P. Agarwal. Properties of solutions of higher order nonlinear difference equations II. An. st. Univ. Iasi, 29:85--96, 1983.
[3] R. P. Agarwal. Difference calculus with applications to difference equations. In W.Walter, editor, International Series of Numerical Mathematics, volume 71, pages 95-110, Basel, 1984. Birkhäuser. General Inequalities 4, Oberwolfach. · Zbl 0592.39001
[4] R. P. Agarwal. Difference Equations and Inequalities. Marcel Dekker, Inc., New York, 1992. · Zbl 0925.39001
[5] R. P. Agarwal and M. Bohner. Quadratic functionals for second order matrix equations on time scales. Nonlinear Analysis, 1997. To appear. · Zbl 0938.49001
[6] R. P. Agarwal and B. S. Lalli. Discrete polynomial interpolation, Green’s functions, maximum principles, error bounds and boundary value problems. Computers Math. Applic., 25:3--39, 1993. · Zbl 0772.65092 · doi:10.1016/0898-1221(93)90169-V
[7] R. P. Agarwal, Q. Sheng, and P. J. Y. Wong. Abel-Gontscharoff boundary value problems. Mathl. Comput. Modelling, 17(7):37--55, 1993. · Zbl 0779.34016 · doi:10.1016/0895-7177(93)90067-9
[8] R. P. Agarwal and P. J. Y. Wong. Abel-Gontscharoff interpolation error bounds for derivatives. Proc. Roy. Soc. Edinburgh, 119A:367--372, 1991. · Zbl 0767.41010
[9] R. P. Agarwal and P. J. Y. Wong. Advanced Topics in Difference Equations. Kluwer Academic Publishers, Dordrecht, 1997. · Zbl 0878.39001
[10] B. Aulbach and S. Hilger. Linear dynamic processes with inhomogeneous time scale. In Nonlinear dynamics and Quantum Dynamical Systems. Akademie Verlag, Berlin, 1990. · Zbl 0719.34088
[11] L. Erbe and S. Hilger. Sturmian theory on measure chains. Differential equations and dynamical systems, 1(3):223--246, 1993. · Zbl 0868.39007
[12] S. Hilger. Analysis on measure chains -- a unified approach to continuous and discrete calculus. Result. Math., 18:19--56, 1990. · Zbl 0722.39001 · doi:10.1007/BF03323153
[13] B. Kaymakçalan, V. Lakshmikantham, and S. Sivasundaram. Dynamic Systems on Measure Chains. Kluwer Academic Publishers, Boston, 1996. · Zbl 0869.34039
[14] A. Kneser. Untersuchungen über die reellen Nullstellen der Integrale linearer Differentialgleichungen. Math. Ann., 42:409--435, 1893. · Zbl 25.0522.01 · doi:10.1007/BF01444165
[15] A. Levin. A bound for a function with monotonely distributed zeros of successive derivatives. Mat. Sb., 64:396--409, 1964.
[16] H. Onose. Oscillatory property of certain non-linear ordinary differential equations. Proc. Japan Acad., 44:232--237, 1968. · Zbl 0162.39702
[17] P. J. Y. Wong. Best error estimates for discrete Abel-Gontscharoff interpolation. 1997. Preprint. · Zbl 0921.41005