A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities
The author proves that the abstract differential inequality ‖u′(t)−A(t)u(t)‖2≤γ[ω(t)+∫0tω(η)dη] in which the linear operator A(t)=M(t)+N(t), M symmetric and N antisymmetric, is in general unbounded, ω(t)=t−2ψ(t)‖u(t)‖2+‖M(t)u(t)‖‖u(t)‖ and γ is a positive constant has a nontrivial solution near t=0...
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| Language: | English |
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Wiley
1990-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171290000382 |
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| author | Alan V. Lair |
| author_facet | Alan V. Lair |
| author_sort | Alan V. Lair |
| collection | DOAJ |
| description | The author proves that the abstract differential inequality ‖u′(t)−A(t)u(t)‖2≤γ[ω(t)+∫0tω(η)dη] in which the linear operator A(t)=M(t)+N(t), M symmetric and N antisymmetric, is in general unbounded, ω(t)=t−2ψ(t)‖u(t)‖2+‖M(t)u(t)‖‖u(t)‖ and γ is a positive constant has a nontrivial solution near t=0 which vanishes at t=0 if and only if ∫01t−1ψ(t)dt=∞. The author also shows that the second order differential inequality ‖u″(t)−A(t)u(t)‖2≤γ[μ(t)+∫0tμ(η)dη] in which μ(t)=t−4ψ0(t)‖u(t)‖2+t−2ψ1(t)‖u′(t)‖2 has a nontrivial solution near t=0 such that u(0)=u′(0)=0 if and only if either ∫01t−1ψ0(t)dt=∞ or ∫01t−1ψ1(t)dt=∞. Some mild restrictions are placed on the operators M and N. These results extend earlier uniqueness theorems of Hile and Protter. |
| format | Article |
| id | doaj-art-bc8ca7a16af74692b542018ef70c398d |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1990-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-bc8ca7a16af74692b542018ef70c398d2025-08-20T02:18:51ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251990-01-0113225327010.1155/S0161171290000382A necessary and sufficient condition for uniqueness of solutions of singular differential inequalitiesAlan V. Lair0Department of Mathematics and Computer Science, Air Force Institute of Technology, Wright-Patterson AFB 45433, OH, USAThe author proves that the abstract differential inequality ‖u′(t)−A(t)u(t)‖2≤γ[ω(t)+∫0tω(η)dη] in which the linear operator A(t)=M(t)+N(t), M symmetric and N antisymmetric, is in general unbounded, ω(t)=t−2ψ(t)‖u(t)‖2+‖M(t)u(t)‖‖u(t)‖ and γ is a positive constant has a nontrivial solution near t=0 which vanishes at t=0 if and only if ∫01t−1ψ(t)dt=∞. The author also shows that the second order differential inequality ‖u″(t)−A(t)u(t)‖2≤γ[μ(t)+∫0tμ(η)dη] in which μ(t)=t−4ψ0(t)‖u(t)‖2+t−2ψ1(t)‖u′(t)‖2 has a nontrivial solution near t=0 such that u(0)=u′(0)=0 if and only if either ∫01t−1ψ0(t)dt=∞ or ∫01t−1ψ1(t)dt=∞. Some mild restrictions are placed on the operators M and N. These results extend earlier uniqueness theorems of Hile and Protter.http://dx.doi.org/10.1155/S0161171290000382uniqueness of solutionsingular differential inequalitysingular equation. |
| spellingShingle | Alan V. Lair A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities International Journal of Mathematics and Mathematical Sciences uniqueness of solution singular differential inequality singular equation. |
| title | A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities |
| title_full | A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities |
| title_fullStr | A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities |
| title_full_unstemmed | A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities |
| title_short | A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities |
| title_sort | necessary and sufficient condition for uniqueness of solutions of singular differential inequalities |
| topic | uniqueness of solution singular differential inequality singular equation. |
| url | http://dx.doi.org/10.1155/S0161171290000382 |
| work_keys_str_mv | AT alanvlair anecessaryandsufficientconditionforuniquenessofsolutionsofsingulardifferentialinequalities AT alanvlair necessaryandsufficientconditionforuniquenessofsolutionsofsingulardifferentialinequalities |