Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients

Arun K. Tripathy; Shyam S. Santra

Mathematica Bohemica (2021)

  • Volume: 146, Issue: 2, page 185-197
  • ISSN: 0862-7959

Abstract

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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 ) ) γ ) ' + ∑ i = 1 m q i ( t ) x α i ( σ i ( t ) ) = 0 , t ≥ t 0 , where z ( t ) = x ( t ) + p ( t ) x ( τ ( t ) ) . Under the assumption ∫ ∞ ( r ( η ) ) - 1 / γ d η = ∞ , we consider two cases when γ > α i and γ < α i . Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.

How to cite

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Tripathy, Arun K., and Santra, Shyam S.. "Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients." Mathematica Bohemica 146.2 (2021): 185-197. <http://eudml.org/doc/297809>.

@article{Tripathy2021,
abstract = {In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type \[ (r(t)(z^\{\prime \}(t))^\gamma )^\{\prime \} +\sum \_\{i=1\}^m q\_i(t)x^\{\alpha \_i\}(\sigma \_i(t))=0, \quad t\ge t\_0, \] 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.},
author = {Tripathy, Arun K., Santra, Shyam S.},
journal = {Mathematica Bohemica},
keywords = {oscillation; non-oscillation; neutral; delay; Lebesgue's dominated convergence theorem},
language = {eng},
number = {2},
pages = {185-197},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients},
url = {http://eudml.org/doc/297809},
volume = {146},
year = {2021},
}

TY - JOUR
AU - Tripathy, Arun K.
AU - Santra, Shyam S.
TI - Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients
JO - Mathematica Bohemica
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 146
IS - 2
SP - 185
EP - 197
AB - In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type \[ (r(t)(z^{\prime }(t))^\gamma )^{\prime } +\sum _{i=1}^m q_i(t)x^{\alpha _i}(\sigma _i(t))=0, \quad t\ge t_0, \] 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.
LA - eng
KW - oscillation; non-oscillation; neutral; delay; Lebesgue's dominated convergence theorem
UR - http://eudml.org/doc/297809
ER -

References

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