Mathematica Bohemica, Vol. 146, No. 2, pp. 185-197, 2021
Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients
Arun K. Tripathy, Shyam S. Santra
Received April 30, 2019. Published online August 12, 2020.
Abstract: In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type $r(t)(z'(t))^\gamma)' +\sum_{i=1}^m q_i(t)x^{\alpha_i}(\sigma_i(t))=0, \quad t\geq t_0$,
where $z(t)=x(t)+p(t)x(\tau(t))$. Under the assumption $\int^{\infty}(r(\eta))^{-1/\gamma} {\rm d}\eta=\infty$, we consider two cases when $\gamma>\alpha_i$ and $\gamma<\alpha_i$. Our main tool is Lebesgue's dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.
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Affiliations: Arun K. Tripathy, Department of Mathematics, Sambalpur University, Jyoti Vihar, Burla, Sambalpur, Odisha-768019, India, e-mail: arun_tripathy70@rediffmail.com; Shyam S. Santra, (corresponding author), Department of Mathematics, JIS College of Engineering, Barrackpore-Kalyani Expy, Phase III, Block A, Kalyani, Nadia 741235, Indie e-mail: shyam01.math@gmail.com
