Graef, John R.; Spikes, Paul W. On the nonlinear limit-point/limit-circle problem. (English) Zbl 0535.34023 Nonlinear Anal., Theory Methods Appl. 7, 851-871 (1983). A perturbed second order nonlinear equation \((a(t)x')'+q(t)f(x)=r(t,x)\) is defined to be of the limit circle type if, for any solution x(t), either \(\int^{\infty}x(u)f(x(u))du<\infty\) or \(\int^{\infty}F(x(u))du<\infty\), where \(F(v)=\int^{v}_{0}f(u)du\) (this is a generalization of H. Weyl’s [Math. Ann. 68, 220-269 (1910)] classification of second order linear differential equations \((a(t)x')'+q(t)x=0)\). The authors give sufficient conditions that such equations are of the limit circle type. Moreover, they discuss the relationships between the above property and the boundedness, oscillation and convergence to zero of the solution of the above equation. Reviewer: M.Boudourides Cited in 2 ReviewsCited in 10 Documents MSC: 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 34C11 Growth and boundedness of solutions to ordinary differential equations Keywords:limit cycle; limit circle; limit point; boundedness; oscillation PDF BibTeX XML Cite \textit{J. R. Graef} and \textit{P. W. Spikes}, Nonlinear Anal., Theory Methods Appl. 7, 851--871 (1983; Zbl 0535.34023) Full Text: DOI References: [1] Atkinson, F. V., Nonlinear extensions of limit-point criteria, Math. Z., 130, 297-312 (1973) · Zbl 0273.34008 [2] Burlak, J., On the non-existence of \(L_2\)-solutions of nonlinear differential equations, Proc. Edinb. math. Soc., 14, 257-268 (1965) · Zbl 0149.04403 [3] Burton, T. A.; Patula, W. T., Limit circle results for second order equations, Mh. Math., 81, 185-194 (1976) · Zbl 0339.34028 [4] Detki, J., The solvability of a certain second order nonlinear ordinary differential equation in \(L^p(0, ∞)\), Math. Balk., 4, 115-119 (1974) [5] Dunford, N.; Schwartz, J. T., Linear operators; Part II: Spectral Theory (1963), Interscience: Interscience New York [6] Graef, J. R., Limit circle criteria and related properties for nonlinear equations, J. diff. Eqns, 35, 319-338 (1980) · Zbl 0441.34024 [7] Graef, J. R., Limit Circle type results for sublinear equations, Pacif. J. Math., 104, 85-94 (1983) · Zbl 0535.34024 [8] Graef, J. R.; Spikes, P. W., Asymptotic properties of solutions of a second order nonlinear differential equation, Publ math. Debrecen, 24, 39-51 (1977) · Zbl 0379.34036 [9] Hallam, T. G., On the non-existence of \(L^p\)-solutions of certain nonlinear differential equations, Glasg. math. J., 8, 133-138 (1967) · Zbl 0163.10602 [10] Kauffman, R. M.; Read, T. T.; Zettl, A., The Deficiency Index Problem for Powers of Ordinary Differential Expressions, (Lecture Notes in Mathematics, 621 (1977), Springer: Springer New York) · Zbl 0367.34014 [11] Spikes, P. W., Some stability type results for a nonlinear differential equation, Rc. Mat., 9, 6, 259-271 (1976) · Zbl 0346.34034 [12] Spikes, P. W., On the integrability of solutions of perturbed non-linear differential equations, Proc. R. Soc. Edinb., 77, 309-318 (1977), Section A · Zbl 0384.34035 [13] Spikes, P. W., Criteria of limit circle type for nonlinear differential equations, SIAM J. Math Analysis, 10, 456-462 (1979) · Zbl 0413.34033 [14] Suyemoto, L.; Waltman, P., Extension of a theorem of A. Wintner, Proc. Am. math. Soc., 14, 970-971 (1963) · Zbl 0127.31102 [15] Weyl, H., Uber gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörige Entwicklung willkürlicher Funktionen, Math. Annln, 68, 220-269 (1910) · JFM 41.0343.01 [16] Winter, A., A criterion for the non-existence of \((L^2)\)-solutions of a nonoscillatory differential equation, J. London math. Soc., 25, 347-351 (1950) [17] Wong, J. S.W., Remark on a theorem of A. Wintner, Enseign. math., 13, 2, 103-106 (1967) · Zbl 0173.09903 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.