Vol. 57, No. 1, 1975

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Sufficient conditions for the existence of convergent subsequences

Keith William Schrader and James Lewis Thornburg

Vol. 57 (1975), No. 1, 301–306

Let R be the real numbers, S R and E be an ordered topological vector space. Sufficient conditions are given that a sequence {yk},yh : S E, will have a subsequence {hk} such that for each i S,{hk(t)} is either eventually monotone or else is convergent. In case E is a Banach space, sufficient conditions are given that {yk} have a subsequence {hk} so that {hk(t)} converges for each t S. Finally, if E = R, the concept of {yk} being equioscillatory is defined and it is shown that a necessary and sufficient condition for {yk} to have a subsequence that converges at every point of S is that {yk} have a subsequence which is pointwise bounded and equioscillatory. An application of these results to differential equations is treated briefly.

Mathematical Subject Classification 2000
Primary: 46A40
Received: 7 June 1974
Revised: 12 December 1974
Published: 1 March 1975
Keith William Schrader
James Lewis Thornburg