A dirichlet problem with asymptotically linear and changing by Lucia M., Magtone R., Zhou H.-S.

By Lucia M., Magtone R., Zhou H.-S.

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DeSpautz and R. A. Lerman, “Equations Equivalent to NonVol. 18, linear Differential Equations,” Proc. Amer. Math. , 1967, pp. 441-444. 22. Elimination of Middle Term Consider the general linear equation u” + p(s) u‘ + q(s) u = 0. 3) 0, as a direct calculation shows. 6) converge, we can apply the results of Sec. 7) 0 as s + co. 1), as we do below. 6) are in the 1 t-variable. Since s = fo a(tl) dt, , ds/dt = a(t), we see that they are < 00. 8) Exercises + 1. Determine the asymptotic behavior of the solutions of U" tau = 0, a > 0; u'' e"'u : 0, a > 0; U" (log t)"u = 0, a > 0.

Vol. 105, 1968, pp. 345-350. J. F. deSpautz and R. A. Lerman, “Equations Equivalent to NonVol. 18, linear Differential Equations,” Proc. Amer. Math. , 1967, pp. 441-444. 22. Elimination of Middle Term Consider the general linear equation u” + p(s) u‘ + q(s) u = 0. 3) 0, as a direct calculation shows. 6) converge, we can apply the results of Sec. 7) 0 as s + co. 1), as we do below. 6) are in the 1 t-variable. Since s = fo a(tl) dt, , ds/dt = a(t), we see that they are < 00. 8) Exercises + 1. Determine the asymptotic behavior of the solutions of U" tau = 0, a > 0; u'' e"'u : 0, a > 0; U" (log t)"u = 0, a > 0.

2. Hence, discuss the boundedness of the solutions of U“ for 0 + (1 + (cos t”)/tb) u = 0 < a, b < 1. See R. Bellman, “Boundedness of Solutions of Second Order Linear Differential Equations,” Duke Math. , Vol. 22, 1955, pp. 51 1514. 3. Show that if we set u = g exp[i J ( d t ) / ( p g 2 ) ]then , -(pu’)’ + qu = 0 is converted into ((pg’)’/g)- (I/pg4)= q. For applications of this result, see J. Walter, “Bemerkungen zu dem Grenzpunktfallkriterium von N. Levinson,” Math. , Vol. 105, 1968, pp. 345-350.

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