Principles of delta fractional calculus on time scales and inequalities. (English) Zbl 1201.26001

Summary: Here we develop the Delta Fractional Calculus on Time Scales. Then we produce related integral inequalities of types: Poincaré, Sobolev, Opial, Ostrowski and Hilbert-Pachpatte. Finally, we give inequalities’ applications on the time scale \(\mathbb{R}\).


26A33 Fractional derivatives and integrals
26D15 Inequalities for sums, series and integrals
26E70 Real analysis on time scales or measure chains
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[1] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (2001), Birkhaüser: Birkhaüser Boston · Zbl 0978.39001
[2] Bohner, M.; Guseinov, G. S., Multiple Lebesgue integration on time scales, Advances in Difference Equations, 1-12 (2006), Article ID 26391
[3] Agarwal, R.; Bohner, M., Basic Calculus on time scales and some of its applications, Results in Mathematics, 35, 1-2, 3-22 (1999) · Zbl 0927.39003
[4] Agarwal, R.; Bohner, M.; Peterson, A., Inequalities on time scales: a survey, Mathematical Inequalities & Applications, 4, 4, 535-557 (2001) · Zbl 1021.34005
[6] Bohner, M.; Guseinov, G., Double integral calculus of variations on time scales, Computers & Mathematics with Applications, 54, 45-57 (2007) · Zbl 1131.49019
[7] Bohner, M.; Luo, H., Singular second-order multipoint dynamic boundary value problems with mixed derivatives, Advances in Difference Equations, 1-15 (2006), Article ID 54989 · Zbl 1139.39024
[8] Guseinov, G., Integration on time scales, Journal of Mathematical Analysis and Applications, 285, 107-127 (2003) · Zbl 1039.26007
[9] Higgins, R.; Peterson, A., Cauchy functions and Taylor’s formula for time scales \(T\), (Aulbach, B.; Elaydi, S.; Ladas, G., Proc. Sixth. Internat. Conf. on Difference equations. Proc. Sixth. Internat. Conf. on Difference equations, New Progress in Difference Equations (2001), Chapman & Hall/CRC: Chapman & Hall/CRC Augsburg, Germany), 299-308 · Zbl 1065.39032
[11] Liu, Wenjun; Anh Ngô, Quôc; Chen, Wenbing, Ostrowski type inequalities on time scales for double integrals, Acta Applicandae Mathematicae, 110, 477-497 (2010) · Zbl 1194.26030
[12] Whittaker, E. T.; Watson, G. N., A Course in Modern Analysis (1927), Cambridge University Press · Zbl 0108.26903
[13] Bohner, M.; Guseinov, G., The Convolution on time scales, Abstract and Applied Analysis, 2007 (2007), Article ID 58373, 24 pages · Zbl 1155.39010
[14] Atici, F.; Eloe, P., A transform method in discrete fractional calculus, International Journal of Difference Equations, 2, #2, 165-176 (2007)
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