Oscillation results for a fractional order dynamic equation on time scales with conformable fractional derivative

Springer Science and Business Media LLC - Tập 2018 Số 1 - 2018
Qinghua Feng1, Fanwei Meng2
1School of Mathematics and Statistics, Shandong University of Technology, Zibo, China
2School of Mathematical Sciences, Qufu Normal University, Qufu, China

Tóm tắt

Từ khóa


Tài liệu tham khảo

Zheng, Z., Meng, F.: On oscillation properties for linear Hamiltonian systems. Rocky Mt. J. Math. 39(1), 343–358 (2009)

Huang, Y., Meng, F.: Oscillation criteria for forced second-order nonlinear differential equations with damping. J. Comput. Appl. Math. 224, 339–345 (2009)

Liu, H., Meng, F., Liu, P.: Oscillation and asymptotic analysis on a new generalized Emden–Fowler equation. Appl. Math. Comput. 219(5), 2739–2748 (2012)

Li, L., Meng, F., Zheng, Z.: Some new oscillation results for linear Hamiltonian systems. Appl. Math. Comput. 208, 219–224 (2009)

Zheng, Z., Wang, X., Han, H.: Oscillation criteria for forced second order differential equations with mixed nonlinearities. Appl. Math. Lett. 22, 1096–1101 (2009)

Liu, L., Bai, Y.: New oscillation criteria for second-order nonlinear neutral delay differential equations. J. Comput. Appl. Math. 231, 657–663 (2009)

Meng, F., Huang, Y.: Interval oscillation criteria for a forced second-order nonlinear differential equations with damping. Appl. Math. Comput. 218, 1857–1861 (2011)

Shao, J., Zheng, Z., Meng, F.: Oscillation criteria for fractional differential equations with mixed nonlinearities. Adv. Differ. Equ. 2013, 323 (2013)

Liu, H., Meng, F.: Oscillation criteria for second order linear matrix differential systems with damping. J. Comput. Appl. Math. 229(1), 222–229 (2009)

Liu, H., Meng, F.: Interval oscillation criteria for second-order nonlinear forced differential equations involving variable exponent. Adv. Differ. Equ. 2016, 291 (2016)

Zheng, Z.: Oscillation criteria for matrix Hamiltonian systems via summability method. Rocky Mt. J. Math. 39(5), 1751–1766 (2009)

Feng, Q., Meng, F.: Oscillation of solutions to nonlinear forced fractional differential equations. Electron. J. Differ. Equ. 2013, 169 (2013)

Hilger, S.: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)

Wang, J., Meng, F., Gu, J.: Estimates on some power nonlinear Volterra–Fredholm type dynamic integral inequalities on time scales. Adv. Differ. Equ. 2017, 257 (2017)

Liu, H.: A class of retarded Volterra–Fredholm type integral inequalities on time scales and their applications. J. Inequal. Appl. 2017, 293 (2017)

Li, W.N.: Some delay integral inequalities on time scales. Comput. Math. Appl. 59, 1929–1936 (2010)

Agarwal, R.P., Bohner, M., O’Regan, D., Peterson, A.: Dynamic equations on time scales: a survey. J. Comput. Appl. Math. 141(1–2), 1–26 (2006)

Gu, J., Meng, F.: Some new nonlinear Volterra–Fredholm type dynamic integral inequalities on time scales. Appl. Math. Comput. 245, 235–242 (2014)

Meng, F., Shao, J.: Some new Volterra–Fredholm type dynamic integral inequalities on time scales. Appl. Math. Comput. 223, 444–451 (2013)

Wang, T., Xu, R.: Some integral inequalities in two independent variables on time scales. J. Math. Inequal. 6(1), 107–118 (2012)

Feng, Q., Meng, F., Zheng, B.: Gronwall–Bellman type nonlinear delay integral inequalities on times scales. J. Math. Anal. Appl. 382, 772–784 (2011)

Wang, T., Xu, R.: Bounds for some new integral inequalities with delay on time scales. J. Math. Inequal. 6(1), 1–12 (2012)

Du, L., Xu, R.: Some new Pachpatte type inequalities on time scales and their applications. J. Math. Inequal. 6(2), 229–240 (2012)

Feng, Q., Meng, F.: Generalized Ostrowski type inequalities for multiple points on time scales involving functions of two independent variables. J. Inequal. Appl. 2012, 74 (2012)

Feng, Q., Meng, F.: Some generalized Ostrowski–Grüss type integral inequalities. Comput. Math. Appl. 63, 652–659 (2012)

Bohner, M., Matthews, T.: The Grüss inequality on time scales. Commun. Math. Anal. 3(1), 1–8 (2007)

Feng, Q., Meng, F., Zhang, Y., Zheng, B., Zhou, J.: Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations. J. Inequal. Appl. 2011, 29 (2011)

Xu, R., Meng, F., Song, C.: On some integral inequalities on time scales and their applications. J. Inequal. Appl. 2010, 464976 (2010)

Grace, S.R., Agarwal, R.P., Bohner, M., O’Regan, D.: Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations. Commun. Nonlinear Sci. Numer. Simul. 14, 3463–3471 (2009)

Saker, S.H.: Oscillation of third-order functional dynamic equations on time scales. Sci. China Math. 12, 2597–2614 (2011)

Agarwal, R.P., Bohner, M., Saker, S.H.: Oscillation of second order delay dynamic equations. Can. Appl. Math. Q. 13, 1–18 (2005)

Shi, Y., Han, Z., Sun, Y.: Oscillation criteria for a generalized Emden–Fowler dynamic equation on time scales. Adv. Differ. Equ. 2016, 3 (2016)

Hassan, T.S., Kong, Q.: Oscillation criteria for higher-order nonlinear dynamic equations with Laplacians and a deviating argument on time scales. Math. Methods Appl. Sci. 40(11), 4028–4039 (2017)

Erbe, L., Hassan, T.S., Peterson, A.: Oscillation of third-order functional dynamic equations with mixed arguments on time scales. J. Appl. Math. Comput. 34, 353–371 (2010)

Li, T., Saker, S.H.: A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales. Commun. Nonlinear Sci. Numer. Simul. 19, 4185–4188 (2014)

Han, Z., Li, T., Sun, S., Cao, F.: Oscillation criteria for third order nonlinear delay dynamic equations on time scales. Ann. Pol. Math. 99, 143–156 (2010)

Hassan, T.S.: Oscillation criteria for higher order quasilinear dynamic equations with Laplacians and a deviating argument. J. Egypt. Math. Soc. 25(2), 178–185 (2016)

Erbe, L., Hassan, T.S.: Oscillation of third order nonlinear functional dynamic equations on time scales. Differ. Equ. Dyn. Syst. 18(1), 199–227 (2010)

Sun, Y.B., Han, Z., Sun, Y., Pan, Y.: Oscillation theorems for certain third order nonlinear delay dynamic equations on time scales. Electron. J. Qual. Theory Differ. Equ. 2011, 75 (2011)

Agarwal, R.P., Bohner, M., Li, T.: Oscillatory behavior of second-order half-linear damped dynamic equations. Appl. Math. Comput. 254, 408–418 (2015)

Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: A Philos-type theorem for third-order nonlinear retarded dynamic equations. Appl. Math. Comput. 249, 527–531 (2014)

Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Oscillation criteria for second-order dynamic equations on time scales. Appl. Math. Lett. 31, 34–40 (2014)

Bohner, M., Hassan, T.S., Li, T.: Fite–Hille–Wintner-type oscillation criteria for second-order halflinear dynamic equations with deviating arguments. Indag. Math. 29(2), 548–560 (2018)

Bohner, M., Li, T.: Kamenev-type criteria for nonlinear damped dynamic equations. Sci. China Math. 58(7), 1445–1452 (2015)

Chatzarakis, G.E., Li, T.: Oscillation criteria for delay and advanced differential equations with nonmonotone arguments. Complexity 2018, 8237634 (2018)

Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: Oscillation of fourth-order delay dynamic equations. Sci. China Math. 58(1), 143–160 (2015)

Agarwal, R.P., Bohner, M., Peterson, A.: Inequalities on time scales: a survey. Math. Inequal. Appl. 4, 535–557 (2001)

Benkhettou, N., Hassani, S., Torres, D.F.M.: A conformable fractional calculus on arbitrary time scales. J. King Saud Univ., Sci. 28, 93–98 (2016)

Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)

Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1988)

Hassan, T.S.: Oscillation of third order nonlinear delay dynamic equations on time scales. Math. Comput. Model. 49, 1573–1586 (2009)

Bohner, M.: Some oscillation criteria for first order delay dynamic equations. Far East J. Appl. Math. 18(3), 289–304 (2005)