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Determining bounds for the first conjugate point

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Summary

The purpose of this paper is to provide relatively simple methods for the determination of useful upper and lower bounds for the first zero greater than a of solutions of systems y″+p(x)y=0, y(a)=0, where p(x) is positive and continuous on an interval [a, b] and is either convex or concave there. On occasion, p(x) is required to be of class C′ or of class C″.

References

  1. [1]

    A. M. Fink, The functional\(T\int\limits_0^T R \) and the zeroes of a second order linear differential equation, Jour. de Math. pures et appl., 46 (1967), pp. 1–9.

  2. [2]

    —— ——,On the zeroes of y″+py= 0with linear, convex and concave p, Bour. de Math. pures et appl., 46 (1967), pp. 1–10.

  3. [3]

    A S. Galbraith,On the zeros of solutions of ordinary differential equations of second order, Proc. Amer. Math. Soc., 17 (1966), pp. 333–337.

  4. [4]

    Walter Leigthon,On the zeros of solutions of a second-order linear differential equation, Jour. de Math. pures et appl., 44 (1965), pp. 297–310.

  5. [5]

    —— ——,Some elementary Sturm Theory, Jour. of Diff-Equations, 4 (1968), pp. 187–193.

  6. [6]

    -- --,Bounds for conjugate points, to appear in Journal für die reine und angewandte Mathematik.

  7. [7]

    —— ——,The conjugacy function, Proc. Amer. Math. Soc., vol. 24 (1970), pp. 870–873.

  8. [8]

    A. Samanich Skidmore andWalter Leighton,On the oscillation of solutions of a second-order linear differential equation, Rend. Mat. di Palermo, Serie II, 14 (1965), pp. 1–8.

  9. [9]

    G. N. Watson,A treatise on the theory of Bessel functions, Cambridge Univ. Press, 2nd edit., 1966.

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Additional information

This will acknowledge the partial support of the authors by the U.S. Army Research Office (Durham) under Grant numbered DA-ARO-D-31-124-G1007.

Entrata in Redazione il 4 novembre 1969.

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Leighton, W., Oo Kian Ke, W. Determining bounds for the first conjugate point. Annali di Matematica 86, 99–114 (1970). https://doi.org/10.1007/BF02415713

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Keywords

  • Lower Bound
  • Conjugate Point