Document Type

Article

Publication Date

1990

Abstract

The author proves that the abstract differential inequality ‖ u ′ ( t ) − A ( t ) u ( t ) ‖ 2 ≤ γ [ ω ( t ) + ∫ 0 t ω ( η ) d η ] in which the linear operator A ( t ) = M ( t ) + N ( t ) , M symmetric and N antisymmetric, is in general unbounded, ω ( t ) = t − 2 ψ ( 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 ∫ 0 1 t − 1 ψ ( t ) d t = ∞ . The author also shows that the second order differential inequality ‖ u ″ ( t ) − A ( t ) u ( t ) ‖ 2 ≤ γ [ μ ( t ) + ∫ 0 t μ ( η ) d η ] in which μ ( t ) = t − 4 ψ 0 ( t ) ‖ u ( t ) ‖ 2 + t − 2 ψ 1 ( t ) ‖ u ′ ( t ) ‖ 2 has a nontrivial solution near t = 0 such that u ( 0 ) = u ′ ( 0 ) = 0 if and only if either ∫ 0 1 t − 1 ψ 0 ( t ) d t = ∞ or ∫ 0 1 t − 1 ψ 1 ( t ) d t = ∞ . Some mild restrictions are placed on the operators M and N . These results extend earlier uniqueness theorems of Hile and Protter.

Comments

Sourced from the version of record at Hindawi:
Lair, A. V. (1990). A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities. International Journal of Mathematics and Mathematical Sciences, 13(2), 253–270. https://doi.org/10.1155/S0161171290000382. Article ID 363475.

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. CC BY 3.0 (http://creativecommons.org/licenses/by/3.0/)

DOI

10.1155/S0161171290000382

Source Publication

International Journal of Mathematics and Mathematical Sciences

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