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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 Mathematics
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 (October 2018), 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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