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On the T-nature of differential inequalities. (English. Russian original) Zbl 0736.34020

Ukr. Math. J. 43, No. 6, 744-751 (1991); translation from Ukr. Mat. Zh. 43, No. 6, 795-802 (1991).
The paper deals with the properties of \(n\)-dimensional subspaces of \(C[a,b]\) spanned by the \(T_{n-1}\)-system of functions \(\{\varphi_ i(t)\}^{n-1}_{i=0}\) (a system of functions \(\{\varphi_ i(t)\}^{n- 1}_{i=0}\) is said to be a \(T_{n-1}\)-system if every linear combination of \(\varphi_ 0,\dots,\varphi_{n-1}\) has at most \(n-1\) different zeros on \([a,b]\)). It is shown that functions from such subspaces behave like solutions of \(n\)-th order disconjugate differential equations. The results presented in the paper complete and generalize some results given in the second author’s paper [Math. Notes 43, No. 5, 355-359 (1988); translation from Mat. Zametki 43, No. 5, 615-623 (1988; Zbl 0708.41007)].
Reviewer: O.Došlý (Brno)

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A40 Differential inequalities involving functions of a single real variable
41A10 Approximation by polynomials

Citations:

Zbl 0708.41007
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References:

[1] E. Beckenbach and R. Bellman, Inequalities [Russian translation], Mir, Moscow (1965). · Zbl 0128.27401
[2] A. Yu. Levin, ?Nonoscillatory solutions of the equation x(n) + p1(t)xn?1 + ... + pn(t)x = 0,? Usp. Matem. Nauk,26, No. 2, 43-96 (1965).
[3] Yu. V. Pokornyi, ?On the spectrum of an interpolational boundary-value problem,? Usp. Matem. Nauk,35, No. 6, 198-199 (1977).
[4] Yu. V. Pokornyi, ?On the distribution of T-continuations,? Dokl. Akad. Nauk SSSR,234, No. 6, 1298-1301 (1978).
[5] Yu. V. Pokornyi, ?On an oscillation theorem of Bernshtein’s,? Matem. Zam.,43, No. 5, 615-623 (1988).
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[7] F. R. Gantmakher, Matrix Theory [in Russian], Nauka, Moscow (1988).
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