New comparison and oscillation theorems for second-order half-linear dynamic equations on time scales

✍ Scribed by Baoguo Jia; Lynn Erbe; Allan Peterson

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 338 KB
• Volume: 56
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/j.camwa.2008.05.014

✦ Synopsis

a b s t r a c t Let T be a time scale (i.e., a closed nonempty subset of R) with sup T = +∞. Consider the second-order half-linear dynamic equation

where r(t) > 0, p(t) are continuous,

In particular, no explicit sign assumptions are made with respect to the coefficient p(t). We give conditions under which every positive solution of the equations is strictly increasing. For α = 1, T = R, the result improves the original theorem [see: [Lynn Erbe, Oscillation theorems for second-order linear differential equation, Pacific J. Math. 35 (2) (1970) 337-343]]. As applications, we get two comparison theorems and an oscillation theorem for half-linear dynamic equations which improve and extend earlier results. Some examples are given to illustrate our theorems.