Nonlinear differential equations of fourth-order: Qualitative properties of the solutions

Omar Bazighifan; Omar Bazighifan

doi:10.3934/math.2020414

AIMS Mathematics

2020, Volume 5, Issue 6: 6436-6447. doi: 10.3934/math.2020414

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Nonlinear differential equations of fourth-order: Qualitative properties of the solutions

Omar Bazighifan ^{1,2
,
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1.
Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout 50512, Yemen
2.
Department of Mathematics, Faculty of Education, Seiyun University, Seiyun, Yemen

Received: 21 June 2020 Accepted: 09 August 2020 Published: 17 August 2020
MSC : 34C10, 34K11

In this paper, we study the oscillation of solutions for a fourth-order neutral nonlinear differential equation, driven by a $p$-Laplace differential operator of the form $ \begin{equation*} \begin{cases} \left( r\left( t\right) \Phi _{p_{1}}[w^{\prime \prime \prime }\left( t\right) ]\right) ^{\prime }+q\left( t\right) \Phi _{p_{2}}\left( u\left( \vartheta \left( t\right) \right) \right) = 0, & \\ r\left( t\right) \gt 0,\ r^{\prime }\left( t\right) \geq 0,\ t\geq t_{0} \gt 0, & \end{cases} \end{equation*} $ The oscillation criteria for these equations have been obtained. Furthermore, some examples are given to illustrate the criteria.
- fourth-order differential equations,
- neutral delay,
- oscillation
Citation: Omar Bazighifan. Nonlinear differential equations of fourth-order: Qualitative properties of the solutions[J]. AIMS Mathematics, 2020, 5(6): 6436-6447. doi: 10.3934/math.2020414

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Abstract

In this paper, we study the oscillation of solutions for a fourth-order neutral nonlinear differential equation, driven by a $p$-Laplace differential operator of the form $ \begin{equation*} \begin{cases} \left( r\left( t\right) \Phi _{p_{1}}[w^{\prime \prime \prime }\left( t\right) ]\right) ^{\prime }+q\left( t\right) \Phi _{p_{2}}\left( u\left( \vartheta \left( t\right) \right) \right) = 0, & \\ r\left( t\right) \gt 0,\ r^{\prime }\left( t\right) \geq 0,\ t\geq t_{0} \gt 0, & \end{cases} \end{equation*} $ The oscillation criteria for these equations have been obtained. Furthermore, some examples are given to illustrate the criteria.

References

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