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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 $\bbfR$.

26A33Fractional derivatives and integrals (real functions)
26D15Inequalities for sums, series and integrals of real functions
26E70Real analysis on time scales or measure chains
Full Text: DOI
[1] Bohner, M.; Peterson, A.: Dynamic equations on time scales: an introduction with applications, (2001) · Zbl 0978.39001
[2] Bohner, M.; Guseinov, G. S.: Multiple Lebesgue integration on time scales, Advances in difference equations, 1-12 (2006) · Zbl 1139.39023
[3] Agarwal, R.; Bohner, M.: Basic calculus on time scales and some of its applications, Results in mathematics 35, No. 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, No. 4, 535-557 (2001) · Zbl 1021.34005
[5] G. Anastassiou, Time scales inequalities, International Journal of Difference Equations, 2010 (in press). · Zbl 1205.26029
[6] Bohner, M.; Guseinov, G.: Double integral calculus of variations on time scales, Computers mathematics with applications 54, 45-57 (2007) · Zbl 1131.49019 · doi:10.1016/j.camwa.2006.10.032
[7] Bohner, M.; Luo, H.: Singular second-order multipoint dynamic boundary value problems with mixed derivatives, Advances in difference equations, 1-15 (2006) · Zbl 1139.39024 · doi:10.1155/ADE/2006/54989
[8] Guseinov, G.: Integration on time scales, Journal of mathematical analysis and applications 285, 107-127 (2003) · Zbl 1039.26007 · doi:10.1016/S0022-247X(03)00361-5
[9] Higgins, R.; Peterson, A.: Cauchy functions and Taylor’s formula for time scales T, New progress in difference equations, 299-308 (2001) · Zbl 1065.39032
[10] S. Hilger, Ein Maßketten kalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universität Würzburg, Germany, 1988. · Zbl 0695.34001
[11] Liu, Wenjun; Ngô, Quôc Anh; Chen, Wenbing: Ostrowski type inequalities on time scales for double integrals, Acta applicandae mathematicae 110, 477-497 (2010) · Zbl 1194.26030 · doi:10.1007/s10440-009-9456-y
[12] Whittaker, E. T.; Watson, G. N.: A course in modern analysis, (1927) · Zbl 53.0180.04
[13] Bohner, M.; Guseinov, G.: The convolution on time scales, Abstract and applied analysis 2007 (2007) · Zbl 1155.39010 · doi:10.1155/2007/58373
[14] Atici, F.; Eloe, P.: A transform method in discrete fractional calculus, International journal of difference equations 2, No. #2, 165-176 (2007)
[15] G. Anastassiou, Discrete fractional calculus and inequalities (2009) (submitted for publication).
[16] G. Anastassiou, Nabla discrete fractional calculus and Nabla inequalities, Mathematics and Computer Modelling (2009), in press (doi:10.1016/j.mcm.2009.11.006). · Zbl 1190.26001