International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1990 / Article

Open Access

Volume 13 |Article ID 363475 | https://doi.org/10.1155/S0161171290000382

Alan V. Lair, "A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities", International Journal of Mathematics and Mathematical Sciences, vol. 13, Article ID 363475, 18 pages, 1990. https://doi.org/10.1155/S0161171290000382

A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities

Received12 Dec 1988
Revised20 Jul 1989

Abstract

The author proves that the abstract differential inequality u(t)A(t)u(t)2γ[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)=t2ψ(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 01t1ψ(t)dt=. The author also shows that the second order differential inequality u(t)A(t)u(t)2γ[μ(t)+0tμ(η)dη] in which μ(t)=t4ψ0(t)u(t)2+t2ψ1(t)u(t)2 has a nontrivial solution near t=0 such that u(0)=u(0)=0 if and only if either 01t1ψ0(t)dt= or 01t1ψ1(t)dt=. Some mild restrictions are placed on the operators M and N. These results extend earlier uniqueness theorems of Hile and Protter.

Copyright © 1990 Hindawi Publishing Corporation. 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.


More related articles

 PDF Download Citation Citation
 Order printed copiesOrder
Views72
Downloads284
Citations

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.