×

Integral inequalities with ‘maxima’ and their applications to Hadamard type fractional differential equations. (English) Zbl 1372.26009

Summary: In this paper, some new integral inequalities with ‘maxima’ are established involving Hadamard integral. Applications to Hadamard fractional differential equations with ‘maxima’ are also presented.

MSC:

26A33 Fractional derivatives and integrals
26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pachpatte BG: On some generalizations of Bellman’s lemma.J. Math. Anal. Appl. 1995, 5:141-150. · Zbl 0305.26010
[2] Pachpatte BG: Inequalities for Differential and Integral Equations. Academic Press, New York; 1998. · Zbl 1032.26008
[3] Pachpatte BG: Explicit bounds on certain integral inequalities.J. Math. Anal. Appl. 2002, 267:48-61. · Zbl 0996.26008 · doi:10.1006/jmaa.2001.7743
[4] Pachpatte BG: Integral and Finite Difference Inequalities and Applications. Elsevier, Netherlands; 2006. · Zbl 1104.26015
[5] Lipovan O: A retarded Gronwall-like inequality and its applications.J. Math. Anal. Appl. 2000, 252:389-401. · Zbl 0974.26007 · doi:10.1006/jmaa.2000.7085
[6] Belarbi S, Dahmani Z: On some new fractional integral inequalities.J. Inequal. Pure Appl. Math. 2009.,10(3): Article ID 86 · Zbl 1184.26011
[7] Dahmani Z: On Minkowski and Hermit-Hadamard integral inequalities via fractional integration.Ann. Funct. Anal. 2010, 1:51-58. · Zbl 1205.26031 · doi:10.15352/afa/1399900993
[8] Dahmani Z: New inequalities in fractional integrals.Int. J. Nonlinear Sci. 2010, 9:493-497. · Zbl 1394.26002
[9] Dahmani Z: The Riemann-Liouville operator to generate some new inequalities.Int. J. Nonlinear Sci. 2011, 12:452-455. · Zbl 1394.26001
[10] Shao, J.; Meng, F., Gronwall-Bellman type inequalities and their applications to fractional differential equations, No. 2013 (2013) · Zbl 1297.34010
[11] Thiramanus, P.; Tariboon, J.; Ntouyas, SK, Henry-Gronwall integral inequalities with ‘maxima’ and their applications to fractional differential equations, No. 2014 (2014) · Zbl 1470.26039
[12] Hadamard J: Essai sur l’étude des fonctions données par leur développement de Taylor.J. Math. Pures Appl. 1892, 8:101-186.
[13] Kilbas, AA; Srivastava, HM; Trujillo, JJ, North-Holland Mathematics Studies 204 (2006), Amsterdam · Zbl 1092.45003
[14] Butzer PL, Kilbas AA, Trujillo JJ: Compositions of Hadamard-type fractional integration operators and the semigroup property.J. Math. Anal. Appl. 2002, 269:387-400. · Zbl 1027.26004 · doi:10.1016/S0022-247X(02)00049-5
[15] Butzer PL, Kilbas AA, Trujillo JJ: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals.J. Math. Anal. Appl. 2002, 269:1-27. · Zbl 0995.26007 · doi:10.1016/S0022-247X(02)00001-X
[16] Butzer PL, Kilbas AA, Trujillo JJ: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals.J. Math. Anal. Appl. 2002, 270:1-15. · Zbl 1022.26011 · doi:10.1016/S0022-247X(02)00066-5
[17] Kilbas AA: Hadamard-type fractional calculus.J. Korean Math. Soc. 2001, 38:1191-1204. · Zbl 1018.26003
[18] Kilbas AA, Trujillo JJ: Hadamard-type integrals asG-transforms.Integral Transforms Spec. Funct. 2003, 14:413-427. · Zbl 1043.26004 · doi:10.1080/1065246031000074443
[19] Chinchane VL, Pachpatte DB: A note on some integral inequalities via Hadamard integral.J. Fract. Calc. Appl. 2013, 4:1-5.
[20] Chinchane VL, Pachpatte DB: On some integral inequalities using Hadamard fractional integral.Malaya J. Mat. 2012, 1:62-66. · Zbl 1368.26016
[21] Sroysang B: A study on Hadamard fractional integral.Int. J. Math. Anal. 2013, 7:1903-1906. · Zbl 1285.26011
[22] Sudsutad, W.; Ntouyas, SK; Tariboon, J., Fractional integral inequalities via Hadamard’s fractional integral, No. 2014 (2014) · Zbl 1474.26143
[23] Kilbas, AA; Srivastava, HM; Trujillo, JJ, North-Holland Mathematics Studies 204 (2006), Amsterdam · Zbl 1092.45003
[24] Bainov, D.; Hristova, S., Pure and Applied Mathematics (2011), New York
[25] Hristova S, Stefanova K: Some integral inequalities with maximum of the unknown functions.Adv. Dyn. Syst. Appl. 2011, 6:57-69.
[26] Tariboon, J.; Thiramanus, P.; Ntouyas, SK, Dynamic integral inequalities on time scales with ‘maxima’, No. 2013 (2013) · Zbl 1295.34100
[27] Kuczma M: An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality. Birkhäuser, Basel; 2009. · Zbl 1221.39041 · doi:10.1007/978-3-7643-8749-5
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.