Research Article
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Kamenev-type oscillation criteria for second order matrix differential systems with damping

Year 2018, Volume: 47 Issue: 5, 1248 - 1267, 16.10.2018

Yasemin Başcı

Abstract

By using the positive linear functional, including the generalized averaging technique, some new Kamenev-type oscillation criteria are established for the second order matrix differential system \[ (r(t)P(t)\psi(X(t))K(X'(t)))'+p(t)R(t)\psi(X(t))K(X'(t))+Q(t)F(X'(t))G(X(t))=0. \]

The results improve and generalize those given in some previous papers.

Keywords

Matrix differential system Oscillation Averaging technique Generalized averaging technique Kamenev-type oscillation

References

  • Baker, J.W. Oscillation theorems for a second-order damped nonlinear differential equation, SIAM J. Appl. Math. 25, 37-40, 1973.
  • Butler, G.J. The oscillatory behavior of a second-order nonlinear differential equation with damping, J. Math. Anal. Appl. 57, 273-289, 1977.
  • Butler, G.J., Erbe, L.H. Oscillation results for self-adjoint differential systems, J. Math. Anal. Appl. 115, 470-481, 1986.
  • Erbe, L.E., Kong, Q., Ruan, S. Kamenev-type theorems for second-order matrix differential systems, Proc. Amer. Math. Soc. 117, 957-962, 1993.
  • Etgen, G.J., Pawlowski, J.F. Oscillation criteria for second-order self-adjoint matrix differential systems, Pacific J. Math. 66, 99-110, 1976.
  • Grace, S.R. Oscillation theorems for second-order nonlinear differential equations with damping, Math. Nachr. 141, 117-127,1989.
  • Grace, S.R. Oscillation criteria for second-order nonlinear differential equations with damp- ing, J. Austral. Math. Soc. Ser. A, 49, 43-54, 1990.
  • Grace, S.R., Lalli, B.S. Integral averaging technique for the oscillation of second-order non- linear differential equations, J. Math. Anal. Appl. 149, 277-311, 1990.
  • Grace, S.R., Lalli, B.S., Yeh, C.C. Oscillation theorems for nonlinear second-order differential equations with a nonlinear damping term, SIAM J. Math. Anal. 15, 1082-1093, 1984.
  • Hartman, P. On nonoscillatory linear differential equations of second-order, Amer. J. Math. 74, 389-400, 1952.
  • Kamenev, I.V. An integral criterion for oscillation of linear differential equations, Mat. Zametki, 23, 249-251, 1978.
  • Kong, Q. Interval criteria for oscillation of second-order linear ordinary differential equa- tions, J. Math. Anal. Appl. 229, 258-270, 1999.
  • Li, H.J. Oscillation criteria for second-order linear differential equations, J. Math. Anal. Appl. 194, 217-234, 1995.
  • Li, W.T. Oscillation of certain second-order nonlinear differential equations, J. Math. Anal. Appl. 217, 1-14, 1998.
  • Li, W.T., Agarwal,R.P. Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations, J. Math. Anal. Appl. 245, 171-188, 2000.
  • Li, W.T., Agarwal, R.P. Interval oscillation criteria for second-order nonlinear equations with damping, Computers Math. Appl. 40, 217-230, 2000.
  • Meng, F., Wang, J., Zheng, Z. A note on Kamenev-type theorem for second-order matrix differential systems, Proc. Amer. Math. Soc. 126, 391-395, 1998.
  • Parhi, N., Praharaj, P. Oscillation of linear second order matrix differential equations, J. Math. Anal. Appl. 221, 287-305, 1998.
  • Philos, Ch.G. Oscillation theorems for linear differential equations of second-order, Arch. Math.(Basel), 53, 482-492, 1989.
  • Rogovchenko, Y.V. Oscillation criteria for certain nonlinear differential equations, J. Math. Anal. Appl. 229, 399-416, 1999.
  • Rogovchenko, Y.V., Tuncay, F. Interval oscillation of second-order nonlinear differential equation with a damping term, Discrete and Continuous Dynamical Systems Supplement, 883-891, 2007.
  • Rogovchenko, Y.V., Tuncay, F. Oscillation criteria for second-order nonlinear differential equations with damping, Nonlinear Analysis, 69, 208-221, 2008.
  • Sun, Y.G. New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping, J. Math. Anal. Appl. 291, 341-351, 2004.
  • Sun, Y.G., Meng, F.W. Oscillation results for matrix differential systems with damping, Appl. Mathl. Comput. 175, 212-220, 2006.
  • Swanson, C.A. Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York, 1968.
  • Tiryaki, A., Ayanlar, B. Oscillation theorems for nonlinear second-order differential equa- tions with damping, Acta Math. Hungar. 89 (1-2), 1-13, 2000.
  • Tiryaki, A., Çakmak, D. Integral averages and oscillation criteria of second-order nonlinear differential equations, Comput. Math. Appl. 47, 1495-1506, 2004.
  • Tiryaki, A., Zafer, A. Oscillation criteria for second-order nonlinear differential equations with damping, Turk. J. Math. 24, 185-196, 2000.
  • Wang, Q.R. Oscillation criteria for second-order matrix differential systems, Arch. Math.(Basel), 76, 385-390, 2001.
  • Wang, Q.R. Interval criteria for oscillation of certain matrix differential systems, J. Math. Anal. Appl. 276, 373-395, 2002.
  • Wang, Q.R. Oscillation of self-adjoint matrix differential systems, Appl. Math. Letters, 17, 1299-1305, 2004.
  • Wang, Q.R.,Wu, X.M., Zhu, S.M. Kamenev-type oscillation criteria for second-order matrix differential systems, Appl. Math. Letters, 16, 821-826, 2003.
  • Wintner, A. A criterion of oscillatory stability, Quart. Appl. Math. 7, 115-117, 1949.
  • Wong, J.S.W. Oscillation criteria for a forced second-order linear differential equations, J Math. Anal. Appl. 231, 235-240, 1999.
  • Wong, J.S.W. On Kamenev-type oscillation theorems for second-order differential equations with damping, J. Math. Anal. Appl. 258, 244-257, 2001.
  • Yan, J. Oscillation theorems for second-order linear differential equations with damping, Proc. Amer. Math. Soc. 9, 276-282, 1986.
  • Yancong, X., Fanwei, M. New oscillation critera for second-order nonlinear matrix differential equations, Appl. Math. J. Chinese Univ. Set. B, 21 (3), 313-319, 2006.
  • Yang, Q., Cheng, S.S. Kamenev-type oscillation criteria for second-order matrix differential systems with damping, Ann. Polon. Math. 85 (2), 145-152, 2005.
  • Yang, Q., Tang, Y. Oscillation theorems for certain second-order self-adjoint matrix differential systems, J. Math. Anal. Appl. 288, 565-585, 2003.
  • Yang, X.J. Oscillation criteria for certain second-order matrix differential equations, J. Math. Anal. Appl. 265, 285-295, 2002.
  • Yeh, C.C. Oscillation theorems for nonlinear second-order differential equations with damped term, Proc. Amer. Math. Soc. 84, 397-402, 1982.
  • Zheng, Z.W., Meng, F.W., Yu, Y.H. On the oscillation of second order matrix differential systems, Acta. Math. Sinica. 41 (6), 1231-1238, 1998.
  • Zhuang, R.K. Interval criteria for oscillatory of second order matrix differential systems, Acta. Math. Sinica. 44 (6), 1037-1044, 2001.

Year 2018, Volume: 47 Issue: 5, 1248 - 1267, 16.10.2018

Yasemin Başcı

Abstract

References

  • Baker, J.W. Oscillation theorems for a second-order damped nonlinear differential equation, SIAM J. Appl. Math. 25, 37-40, 1973.
  • Butler, G.J. The oscillatory behavior of a second-order nonlinear differential equation with damping, J. Math. Anal. Appl. 57, 273-289, 1977.
  • Butler, G.J., Erbe, L.H. Oscillation results for self-adjoint differential systems, J. Math. Anal. Appl. 115, 470-481, 1986.
  • Erbe, L.E., Kong, Q., Ruan, S. Kamenev-type theorems for second-order matrix differential systems, Proc. Amer. Math. Soc. 117, 957-962, 1993.
  • Etgen, G.J., Pawlowski, J.F. Oscillation criteria for second-order self-adjoint matrix differential systems, Pacific J. Math. 66, 99-110, 1976.
  • Grace, S.R. Oscillation theorems for second-order nonlinear differential equations with damping, Math. Nachr. 141, 117-127,1989.
  • Grace, S.R. Oscillation criteria for second-order nonlinear differential equations with damp- ing, J. Austral. Math. Soc. Ser. A, 49, 43-54, 1990.
  • Grace, S.R., Lalli, B.S. Integral averaging technique for the oscillation of second-order non- linear differential equations, J. Math. Anal. Appl. 149, 277-311, 1990.
  • Grace, S.R., Lalli, B.S., Yeh, C.C. Oscillation theorems for nonlinear second-order differential equations with a nonlinear damping term, SIAM J. Math. Anal. 15, 1082-1093, 1984.
  • Hartman, P. On nonoscillatory linear differential equations of second-order, Amer. J. Math. 74, 389-400, 1952.
  • Kamenev, I.V. An integral criterion for oscillation of linear differential equations, Mat. Zametki, 23, 249-251, 1978.
  • Kong, Q. Interval criteria for oscillation of second-order linear ordinary differential equa- tions, J. Math. Anal. Appl. 229, 258-270, 1999.
  • Li, H.J. Oscillation criteria for second-order linear differential equations, J. Math. Anal. Appl. 194, 217-234, 1995.
  • Li, W.T. Oscillation of certain second-order nonlinear differential equations, J. Math. Anal. Appl. 217, 1-14, 1998.
  • Li, W.T., Agarwal,R.P. Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations, J. Math. Anal. Appl. 245, 171-188, 2000.
  • Li, W.T., Agarwal, R.P. Interval oscillation criteria for second-order nonlinear equations with damping, Computers Math. Appl. 40, 217-230, 2000.
  • Meng, F., Wang, J., Zheng, Z. A note on Kamenev-type theorem for second-order matrix differential systems, Proc. Amer. Math. Soc. 126, 391-395, 1998.
  • Parhi, N., Praharaj, P. Oscillation of linear second order matrix differential equations, J. Math. Anal. Appl. 221, 287-305, 1998.
  • Philos, Ch.G. Oscillation theorems for linear differential equations of second-order, Arch. Math.(Basel), 53, 482-492, 1989.
  • Rogovchenko, Y.V. Oscillation criteria for certain nonlinear differential equations, J. Math. Anal. Appl. 229, 399-416, 1999.
  • Rogovchenko, Y.V., Tuncay, F. Interval oscillation of second-order nonlinear differential equation with a damping term, Discrete and Continuous Dynamical Systems Supplement, 883-891, 2007.
  • Rogovchenko, Y.V., Tuncay, F. Oscillation criteria for second-order nonlinear differential equations with damping, Nonlinear Analysis, 69, 208-221, 2008.
  • Sun, Y.G. New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping, J. Math. Anal. Appl. 291, 341-351, 2004.
  • Sun, Y.G., Meng, F.W. Oscillation results for matrix differential systems with damping, Appl. Mathl. Comput. 175, 212-220, 2006.
  • Swanson, C.A. Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York, 1968.
  • Tiryaki, A., Ayanlar, B. Oscillation theorems for nonlinear second-order differential equa- tions with damping, Acta Math. Hungar. 89 (1-2), 1-13, 2000.
  • Tiryaki, A., Çakmak, D. Integral averages and oscillation criteria of second-order nonlinear differential equations, Comput. Math. Appl. 47, 1495-1506, 2004.
  • Tiryaki, A., Zafer, A. Oscillation criteria for second-order nonlinear differential equations with damping, Turk. J. Math. 24, 185-196, 2000.
  • Wang, Q.R. Oscillation criteria for second-order matrix differential systems, Arch. Math.(Basel), 76, 385-390, 2001.
  • Wang, Q.R. Interval criteria for oscillation of certain matrix differential systems, J. Math. Anal. Appl. 276, 373-395, 2002.
  • Wang, Q.R. Oscillation of self-adjoint matrix differential systems, Appl. Math. Letters, 17, 1299-1305, 2004.
  • Wang, Q.R.,Wu, X.M., Zhu, S.M. Kamenev-type oscillation criteria for second-order matrix differential systems, Appl. Math. Letters, 16, 821-826, 2003.
  • Wintner, A. A criterion of oscillatory stability, Quart. Appl. Math. 7, 115-117, 1949.
  • Wong, J.S.W. Oscillation criteria for a forced second-order linear differential equations, J Math. Anal. Appl. 231, 235-240, 1999.
  • Wong, J.S.W. On Kamenev-type oscillation theorems for second-order differential equations with damping, J. Math. Anal. Appl. 258, 244-257, 2001.
  • Yan, J. Oscillation theorems for second-order linear differential equations with damping, Proc. Amer. Math. Soc. 9, 276-282, 1986.
  • Yancong, X., Fanwei, M. New oscillation critera for second-order nonlinear matrix differential equations, Appl. Math. J. Chinese Univ. Set. B, 21 (3), 313-319, 2006.
  • Yang, Q., Cheng, S.S. Kamenev-type oscillation criteria for second-order matrix differential systems with damping, Ann. Polon. Math. 85 (2), 145-152, 2005.
  • Yang, Q., Tang, Y. Oscillation theorems for certain second-order self-adjoint matrix differential systems, J. Math. Anal. Appl. 288, 565-585, 2003.
  • Yang, X.J. Oscillation criteria for certain second-order matrix differential equations, J. Math. Anal. Appl. 265, 285-295, 2002.
  • Yeh, C.C. Oscillation theorems for nonlinear second-order differential equations with damped term, Proc. Amer. Math. Soc. 84, 397-402, 1982.
  • Zheng, Z.W., Meng, F.W., Yu, Y.H. On the oscillation of second order matrix differential systems, Acta. Math. Sinica. 41 (6), 1231-1238, 1998.
  • Zhuang, R.K. Interval criteria for oscillatory of second order matrix differential systems, Acta. Math. Sinica. 44 (6), 1037-1044, 2001.
There are 43 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Yasemin Başcı

Publication Date October 16, 2018
Published in Issue Year 2018 Volume: 47 Issue: 5

Cite

APA Başcı, Y. (2018). Kamenev-type oscillation criteria for second order matrix differential systems with damping. Hacettepe Journal of Mathematics and Statistics, 47(5), 1248-1267.
AMA Başcı Y. Kamenev-type oscillation criteria for second order matrix differential systems with damping. Hacettepe Journal of Mathematics and Statistics. October 2018;47(5):1248-1267.
Chicago Başcı, Yasemin. “Kamenev-Type Oscillation Criteria for Second Order Matrix Differential Systems With Damping”. Hacettepe Journal of Mathematics and Statistics 47, no. 5 (October 2018): 1248-67.
EndNote Başcı Y (October 1, 2018) Kamenev-type oscillation criteria for second order matrix differential systems with damping. Hacettepe Journal of Mathematics and Statistics 47 5 1248–1267.
IEEE Y. Başcı, “Kamenev-type oscillation criteria for second order matrix differential systems with damping”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 5, pp. 1248–1267, 2018.
ISNAD Başcı, Yasemin. “Kamenev-Type Oscillation Criteria for Second Order Matrix Differential Systems With Damping”. Hacettepe Journal of Mathematics and Statistics 47/5 (October2018), 1248-1267.
JAMA Başcı Y. Kamenev-type oscillation criteria for second order matrix differential systems with damping. Hacettepe Journal of Mathematics and Statistics. 2018;47:1248–1267.
MLA Başcı, Yasemin. “Kamenev-Type Oscillation Criteria for Second Order Matrix Differential Systems With Damping”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 5, 2018, pp. 1248-67.
Vancouver Başcı Y. Kamenev-type oscillation criteria for second order matrix differential systems with damping. Hacettepe Journal of Mathematics and Statistics. 2018;47(5):1248-67.

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