Oscillation of even order advanced type dynamic equations with mixed nonlinearities on time scales

2013 ◽  
Vol 44 (1-2) ◽  
pp. 357-377 ◽  
Author(s):  
Da-Xue Chen ◽  
Pei-Xin Qu
Keyword(s):  
Time Scales ◽  
Dynamic Equations ◽  
Mixed Nonlinearities ◽  
Even Order

scholarly journals Oscillation of even order nonlinear delay dynamic equations on time scales

2013 ◽  
Vol 63 (1) ◽  
pp. 265-279 ◽  
Author(s):  
Lynn Erbe ◽  
Raziye Mert ◽  
Allan Peterson ◽  
Ağacık Zafer
Keyword(s):  
Time Scales ◽  
Dynamic Equations ◽  
Even Order ◽  
Delay Dynamic Equations ◽  
Nonlinear Delay

scholarly journals Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

2020 ◽  
Vol 18 (1) ◽  
pp. 353-377 ◽  
Author(s):  
Zhien Li ◽  
Chao Wang
Keyword(s):  
Time Scales ◽  
Quaternion Algebra ◽  
Matrix Exponential ◽  
Dynamic Equations ◽  
Necessary Condition ◽  
Quaternion Matrix ◽  
Exponential Functions ◽  
Cauchy Matrix ◽  
Adjoint Matrix ◽  
Sufficient And Necessary Condition

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

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