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...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
1990-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171290000382 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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. |
|---|---|
| ISSN: | 0161-1712 1687-0425 |