Geodesic
Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients
Mathematica Bohemica, Tome 146 (2021) no. 2, pp. 185-197
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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.
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.
DOI : 10.21136/MB.2020.0063-19
Classification : 34C10, 34K11
Keywords: oscillation; non-oscillation; neutral; delay; Lebesgue's dominated convergence theorem 
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Tripathy, Arun K.; Santra, Shyam S. Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients. Mathematica Bohemica, Tome 146 (2021) no. 2, pp. 185-197. doi: 10.21136/MB.2020.0063-19

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