Second-order half-linear delay differential equations: Oscillation tests

Authors : O BAZİGHİFAN, Shyam Sundar SANTRA

Pages : 385-393

Doi:10.31197/atnaa.751034

View : 12 | Download : 9

Publication Date : 2020-12-30

Article Type : Research Paper

Abstract :In this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the form $ \biglinsert ignore into journalissuearticles values(rinsert ignore into journalissuearticles values(y^{\prime});^\gamma\bigr);^{\prime}insert ignore into journalissuearticles values(t);+ qinsert ignore into journalissuearticles values(t);y^\alphainsert ignore into journalissuearticles values(\tauinsert ignore into journalissuearticles values(t););=0\,.$ We study this equation under the assumption $\int^{\infty}\biginsert ignore into journalissuearticles values(rinsert ignore into journalissuearticles values(\eta);\big);^{-1/\gamma} d\eta=\infty$ and consider two cases when $\gamma > \alpha$ and $\gamma < \alpha$. We provide examples, illustrating the results and state an open problem.
Keywords : Oscillation, nonoscillation, delay, half linear, Lebesgue`s dominated convergence theorem