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Article

Tripathy, Arun K. ; Santra, Shyam S.
Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients. (English). Mathematica Bohemica, vol. 146 (2021), issue 2, pp. 185-197
MSC: 34C10, 34K11 | DOI: 10.21136/MB.2020.0063-19
Keywords:
oscillation; non-oscillation; neutral; delay; Lebesgue's dominated convergence theorem
Summary:
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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