Comparison theorems for second-order nonlinear differential equations

Eliason, Stanley B.

doi:10.1090/qam/402177

Author: Stanley B. Eliason
Journal: Quart. Appl. Math. 29 (1971), 391-402
MSC: Primary 34C10; Secondary 34B15
DOI: https://doi.org/10.1090/qam/402177
MathSciNet review: 402177
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Abstract: Comparison theorems for a nonlinear eigenvalue problem as well as a Lyapunov type of inequality are derived. They are used to establish upper and lower bounds for various integral functionals associated with real solutions of the nonlinear boundary value problem $y” + p\left ( x \right ){y^{2n + 1}} = 0, y\left ( a \right ) = y’\left ( b \right ) = 0$, where $a < b$ are real, $n$ is a positive integer and $p$ is positive and continuous on $\left [ {a,b} \right ]$. Some of the results are analogues of a distance between zeros problem for the linear case of $n = 0$.

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
Dallas Banks, An integral inequality, Proc. Amer. Math. Soc. 14 (1963), 823–828. MR 153806, DOI https://doi.org/10.1090/S0002-9939-1963-0153806-8
David C. Barnes, Some isoperimetric inequalities for the eigenvalues of vibrating strings, Pacific J. Math. 29 (1969), 43–61. MR 244545
P. R. Beesack and Binyamin Schwarz, On the zeros of solutions of second-order linear differential equations, Canadian J. Math. 8 (1956), 504–515. MR 80218, DOI https://doi.org/10.4153/CJM-1956-057-6
Stanley B. Eliason, A Lyapunov inequality for a certain second order non-linear differential equation, J. London Math. Soc. (2) 2 (1970), 461–466. MR 267191, DOI https://doi.org/10.1112/jlms/2.Part_3.461
A. M. Fink, Comparison theorems for eigenvalues, Quart. Appl. Math. 28 (1970), 289–292. MR 264153, DOI https://doi.org/10.1090/S0033-569X-1970-0264153-6
John W. Hooker, Existence and oscillation theorems for a class of nonlinear second order differential equations, J. Differential Equations 5 (1969), 283–306. MR 236463, DOI https://doi.org/10.1016/0022-0396%2869%2990044-8
Richard A. Moore and Zeev Nehari, Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc. 93 (1959), 30–52. MR 111897, DOI https://doi.org/10.1090/S0002-9947-1959-0111897-8
R. M. Moroney, A class of characteristic-value problems, Trans. Amer. Math. Soc. 102 (1962), 446–470. MR 148982, DOI https://doi.org/10.1090/S0002-9947-1962-0148982-0
R. M. Moroney, Note on a theorem of Nehari, Proc. Amer. Math. Soc. 13 (1962), 407–410. MR 148983, DOI https://doi.org/10.1090/S0002-9939-1962-0148983-8
Zeev Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc. 85 (1957), 428–445. MR 87816, DOI https://doi.org/10.1090/S0002-9947-1957-0087816-8
Zeev Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960), 101–123. MR 111898, DOI https://doi.org/10.1090/S0002-9947-1960-0111898-8
Zeev Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math. 105 (1961), 141–175. MR 123775, DOI https://doi.org/10.1007/BF02559588
Donald F. St. Mary, Some oscillation and comparison theorems for $(r(t)y^{\prime } )^{\prime } +p(t)y=0$, J. Differential Equations 5 (1969), 314–323. MR 233009, DOI https://doi.org/10.1016/0022-0396%2869%2990046-1