Finite moments perturbations of 𝑦”=0 in Banach algebras

Spigler, Renato; Vianello, Marco

doi:10.1090/S0002-9939-1993-1181175-2

Finite moments perturbations of $y”=0$ in Banach algebras
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by Renato Spigler and Marco Vianello

Proc. Amer. Math. Soc. 119 (1993), 97-103

DOI: https://doi.org/10.1090/S0002-9939-1993-1181175-2

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Abstract:

Rigorous asymptotics for a basis of $y'' + g(x)y = 0,\;x \in [1, + \infty )$, is derived in the framework of Banach algebras. The key assumption is $\int _1^{ + \infty } {{x^k}} ||g(x)||dx < \infty$ for $k = 1$ or $k = 2$. Such results improve and generalize previous work on linear second-order matrix differential equations.

References

John H. Barrett, Matrix systems of second order differential equations, Portugal. Math. 14 (1956), 79–89. MR 80219
Philip Hartman, Self-adjoint, non-oscillatory systems of ordinary, second order, linear differential equations, Duke Math. J. 24 (1957), 25–35. MR 82591
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Einar Hille, Lectures on ordinary differential equations, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0249698
F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
Warren E. Shreve, Asymptotic behavior in a second order linear matrix differential equation, J. Differential Equations 9 (1971), 13–24. MR 271465, DOI 10.1016/0022-0396(70)90150-6
Donald R. Smith, Decoupling of recessive and nonrecessive solutions for a second-order system, J. Differential Equations 68 (1987), no. 3, 383–399. MR 891335, DOI 10.1016/0022-0396(87)90177-X
Renato Spigler and Marco Vianello, WKBJ-type approximation for finite moments perturbations of the differential equation $y''=0$ and the analogous difference equation, J. Math. Anal. Appl. 169 (1992), no. 2, 437–452. MR 1180902, DOI 10.1016/0022-247X(92)90089-V
Marco Vianello, A generalization of l’Hôpital’s rule via absolute continuity and Banach modules, Real Anal. Exchange 18 (1992/93), no. 2, 557–567. MR 1228425, DOI 10.2307/44152304