International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 2, Pages 253-270
doi:10.1155/S0161171290000382
A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities
Department of Mathematics and Computer Science, Air Force Institute of Technology, Wright-Patterson AFB 45433, OH, USA
Received 12 December 1988; Revised 20 July 1989
Copyright © 1990 Alan V. Lair. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.