Nonoscillation theorems for a nonlinear differential equation
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- by H. E. Gollwitzer
- Proc. Amer. Math. Soc. 26 (1970), 78-84
- DOI: https://doi.org/10.1090/S0002-9939-1970-0259243-3
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Abstract:
This paper is concerned with the problem of specifying growth conditions on the positive function $q(t)$ which imply that all solutions of the nonlinear second order ordinary differential equation $y'' + q(t)|y{|^\alpha }\operatorname {sgn} y = 0,\alpha > 0$, are nonoscillatory on a half line. Several different results are given, and the usual explicit monotonicity condition on $q$ has been avoided to a certain degree.References
- F. V. Atkinson, On second-order non-linear oscillations, Pacific J. Math. 5 (1955), 643–647. MR 72316
- C. V. Coffman and D. F. Ullrich, On the continuation of solutions of a certain non-linear differential equation, Monatsh. Math. 71 (1967), 385–392. MR 227494, DOI 10.1007/BF01295129 C. V. Coffman and J. S. W. Wong, On a second order nonlinear oscillation problem, Carnegie-Mellon University Report 69-2. H. E. Gollwitzer, Nonlinear second order differential equations and Stieltjes integrals, University of Tennessee Report, April, 1969.
- Stuart P. Hastings, Boundary value problems in one differential equation with a discontinuity, J. Differential Equations 1 (1965), 346–369. MR 180723, DOI 10.1016/0022-0396(65)90013-6
- J. W. Heidel, A nonoscillation theorem for a nonlinear second order differential equation, Proc. Amer. Math. Soc. 22 (1969), 485–488. MR 248396, DOI 10.1090/S0002-9939-1969-0248396-0
- T. H. Hildebrandt, Introduction to the theory of integration, Pure and Applied Mathematics, Vol. XIII, Academic Press, New York-London, 1963. MR 0154957
- Einar Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948), 234–252. MR 27925, DOI 10.1090/S0002-9947-1948-0027925-7
- D. V. Izjumova and I. T. Kiguradze, Some remarks on the solutions of the equation $u^{n}+a(t)f(u)=0$, Differencial′nye Uravnenija 4 (1968), 589–605 (Russian). MR 0227544
- I. T. Kiguradze, On the conditions for oscillation of solutions of the differential equation $u^{\prime \prime }+a(t)|u|\ sp{n}\,\textrm {sgn}\, u=0$, Časopis Pěst. Mat. 87 (1962), 492–495 (Russian, with Czech and German summaries). MR 0181800
- Walter Leighton, On self-adjoint differential equations of second order, J. London Math. Soc. 27 (1952), 37–47. MR 46506, DOI 10.1112/jlms/s1-27.1.37
- Zeev Nehari, A nonlinear oscillation problem, J. Differential Equations 5 (1969), 452–460. MR 235203, DOI 10.1016/0022-0396(69)90085-0
Bibliographic Information
- © Copyright 1970 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 26 (1970), 78-84
- MSC: Primary 34.42
- DOI: https://doi.org/10.1090/S0002-9939-1970-0259243-3
- MathSciNet review: 0259243