Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations
We present new oscillation criteria for the differential equation of the form [𝑟(𝑡)𝑈(𝑡)]+𝑝(𝑡)𝑘2(𝑥(𝑡),𝑥(𝑡))|𝑥(𝑡)|𝜈𝑈(𝑡)+𝑞(𝑡)𝜙(𝑥(𝑔1(𝑡)),𝑥(𝑔2(𝑡)))𝑓(𝑥(𝑡))=0, where 𝑈(𝑡)=𝑘1(𝑥(𝑡),𝑥(𝑡))|𝑥(𝑡)|𝛼−1𝑥(𝑡), 𝛼≤𝛽,𝜈=(𝛽−𝛼)/(𝛼+1). Our research is different from most known ones in the sense that H function is not...
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| Format: | Article |
| Language: | English |
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Wiley
2009-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2009/714357 |
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| author | Hui-Zeng Qin Yongsheng Ren |
| author_facet | Hui-Zeng Qin Yongsheng Ren |
| author_sort | Hui-Zeng Qin |
| collection | DOAJ |
| description | We present new oscillation criteria for the differential equation of the form [𝑟(𝑡)𝑈(𝑡)]+𝑝(𝑡)𝑘2(𝑥(𝑡),𝑥(𝑡))|𝑥(𝑡)|𝜈𝑈(𝑡)+𝑞(𝑡)𝜙(𝑥(𝑔1(𝑡)),𝑥(𝑔2(𝑡)))𝑓(𝑥(𝑡))=0, where 𝑈(𝑡)=𝑘1(𝑥(𝑡),𝑥(𝑡))|𝑥(𝑡)|𝛼−1𝑥(𝑡), 𝛼≤𝛽,𝜈=(𝛽−𝛼)/(𝛼+1). Our research is different from most known ones in the sense that H function is not employed in our results, though Riccati's substitution and its generalized forms are used. Our criteria which are established under quite general assumptions are an extension for previous results. In particular, by taking 𝛽=𝛼, the above-mentioned equation can be reduced into the various types of equations concerned by people currently. |
| format | Article |
| id | doaj-art-daf7cf928fa342318500ccdd6dc54835 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2009-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-daf7cf928fa342318500ccdd6dc548352025-02-03T06:01:54ZengWileyInternational Journal of Differential Equations1687-96431687-96512009-01-01200910.1155/2009/714357714357Oscillation Theorems for Second-Order Damped Nonlinear Differential EquationsHui-Zeng Qin0Yongsheng Ren1Institute of Applied Mathematics, Shandong University of Technology, Zibo, Shandong 255049, ChinaCollage of Mechanical and Electronic Engineering, Shandong University of Science and Technology, Qindao 266510, ChinaWe present new oscillation criteria for the differential equation of the form [𝑟(𝑡)𝑈(𝑡)]+𝑝(𝑡)𝑘2(𝑥(𝑡),𝑥(𝑡))|𝑥(𝑡)|𝜈𝑈(𝑡)+𝑞(𝑡)𝜙(𝑥(𝑔1(𝑡)),𝑥(𝑔2(𝑡)))𝑓(𝑥(𝑡))=0, where 𝑈(𝑡)=𝑘1(𝑥(𝑡),𝑥(𝑡))|𝑥(𝑡)|𝛼−1𝑥(𝑡), 𝛼≤𝛽,𝜈=(𝛽−𝛼)/(𝛼+1). Our research is different from most known ones in the sense that H function is not employed in our results, though Riccati's substitution and its generalized forms are used. Our criteria which are established under quite general assumptions are an extension for previous results. In particular, by taking 𝛽=𝛼, the above-mentioned equation can be reduced into the various types of equations concerned by people currently.http://dx.doi.org/10.1155/2009/714357 |
| spellingShingle | Hui-Zeng Qin Yongsheng Ren Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations International Journal of Differential Equations |
| title | Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations |
| title_full | Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations |
| title_fullStr | Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations |
| title_full_unstemmed | Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations |
| title_short | Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations |
| title_sort | oscillation theorems for second order damped nonlinear differential equations |
| url | http://dx.doi.org/10.1155/2009/714357 |
| work_keys_str_mv | AT huizengqin oscillationtheoremsforsecondorderdampednonlineardifferentialequations AT yongshengren oscillationtheoremsforsecondorderdampednonlineardifferentialequations |