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The limit-4 case of fourth-order self-adjoint differential equations

  • M. S. P. Eastham (a1)


A new method is developed for identifying real-valued coefficients r(x), p(x), and q(x) for which all solutions of the fourth-order differential equation

are L2(0, ∞). The results are compared with those derived from the asymptotic theory of Devinatz, Walker, Kogan and Rofe-Beketov.



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1Atkinson, F. V.. Limit-n criteria of integral type. Proc. Roy. Soc. Edinburgh Sect. A 73 (1975), 167198.
2Devinatz, A.. The deficiency index of certain fourth-order ordinary self-adjoint differential operators. Quart. J. Math. Oxford Ser. 23 (1972), 267286.
3Devinatz, A.. On limit-2 fourth-order differential operators. J. London Math. Soc. 7 (1973), 135146.
4Eastham, M. S. P.. The limit-3 case of self-adjoint differential expressions of the fourth order with oscillating coefficients. J. London Math. Soc. 8 (1974), 427437.
5Eastham, M. S. P.. Square-integrable solutions of the differential equation y(4)+a(qy')' + (bq2 + q”)y =0. Nieuw Arch. Wisk. 24 (1976), 256269.
6Everitt, W. N.. On the limit-point classification of fourth-order differential equations. J. London Math. Soc. 44 (1969), 273281.
7Everitt, W. N.. Integrable-square solutions of ordinary differential equations. Nieuw Arch. Wisk., to appear.
8Kogan, V. I. and Rofe-Beketov, F. S.. On the question of the deficiency indices of differential operators with complex coefficients. Proc. Roy. Soc. Edinburgh Sect. A. 72 (1975), 281298.
9Kuptsov, N. P.. An estimate for solutions of a system of linear differential equations.Uspehi Mat. Nauk. 18(1) (1963), 159164.
10Walker, P. W.. Asymptotics for a class of fourth-order differential equations. J. Differential Equations 11 (1972), 321–334.


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