Abstract
In this paper, we prove some new dynamic inequalities of Opial type involving higher-order derivatives of two functions, with two different weights on time scales. From these inequalities, we will derive some special cases and give an improvement of some versions of recent results.
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Agarwal, R.P., Bohner, M.: Basic calculus on time scales and some of its applications. Results Math. 35(1–2), 3–22 (1999)
Agarwal, R.P., Bohner, M., O’Regan, D., Osman, M.M., Saker, S.H.: A general dynamic inequality of Opial type. Appl. Math. Inf. Sci. 10(1), 1–5 (2016)
Agarwal, R.P., Pang, P.Y.H.: Opial Inequalities with Applications in Differential and Difference Equations. Kluwer Academic Publishers, Dordrechet (1995)
Agarwal, R.P., Pang, P.Y.H.: Opial-type inequalities involving higher order derivatives. J. Math. Anal. Appl. 189, 85–103 (1995)
Agarwal, R.P., Pang, P.Y.H.: Remarks on the generalizations of Opial’s inequality. J. Math. Anal. Appl. 190, 559–577 (1995)
Agarwal, R.P., Pang, P.Y.H.: Sharp Opial-type inequalities involving higher order derivatives of two functions. Math. Nachr. 174, 5–20 (1995)
Alzer, H.: An Opial-type inequality involving higher-order derivatives of two functions. Appl. Math. Lett. 10(4), 123–128 (1997)
Beesack, P.R.: Elementary proofs of some Opial-type integral inequalities. J. Anal. Math. 36, 1–14 (1979)
Beesack, P.R., Das, K.M.: Extensions of Opial’s inequality. Pacific J. Math. 26(2), 215–232 (1968)
Bohner, M., Kaymakçalan, B.: Opial inequalities on time scales. Ann. Polon. Math. 77(1), 11–20 (2001)
Bohner, M., Osman, M.M., Saker, S.H.: General higher order dynamic Opial inequalities with applications. Dynam. Systems Appl. 26, 65–80 (2017)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)
Das, K.M.: An inequality similar to Opial’s inequality. Proc. Amer. Math. Soc. 22, 258–261 (1969)
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1934)
Higgins, R.J., Peterson, A.: Cauchy functions and Taylor’s formula for time scales \({\mathbb{T}}\), Proceeding of six international conference on differential equations and dynamical systems, Roca Raton, FL, CRC, 299–308 (2004)
Nhan, N.D.V., Phung, T.D., Duc, D.T., Tuan, V.K.: Opial and Lyapunov inequalities on time scales and their applications to dynamic equations. Kodai Math. J. 40(2), 254–277 (2017)
Olech, Z.: A simple proof of a certain result of Z. Opial. Ann. Polon. Math. 8, 61–63 (1960)
Opial, Z.: Sur uné inegalité. Ann. Polon. Math. 8, 29–32 (1960)
Pachpatte, B.G.: On Opial-type integral inequalities. J. Math. Anal. Appl. 120, 547–556 (1986)
Pachpatte, B.G.: On some new generalizations of Opial inequality. Demonstrat. Math. 19, 281–291 (1986)
Saker, S.H., Osman, M.M., O’Regan, D., Agarwal, R.P.: Some new Opial dynamic inequalities with weighted functions on time scales. Math. Inequal. Appl. 18(3), 1171–1187 (2015)
Wong, F.H., Lian, W.C., Yu, S.L., Yeh, C.C.: Some generalizations of Opial’s inequalities on time scales. Taiwan. J. Math. 12(2), 463–471 (2008)
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Osman, M.M., Saker, S.H. & Anderson, D.R. Two Weighted Higher-Order Dynamic Inequalities of Opial Type with Two Functions. Qual. Theory Dyn. Syst. 21, 57 (2022). https://doi.org/10.1007/s12346-022-00592-z
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DOI: https://doi.org/10.1007/s12346-022-00592-z