×

zbMATH — the first resource for mathematics

Periodic solutions of a second order ordinary differential equation: A necessary and sufficient condition for nonresonance. (English) Zbl 0743.34045
The paper considers the periodic problem \(-x''=f(x)x'+g(x)+h(t)\) in \([0,2\pi]\); \(x(0)=x(2\pi)\), \(x'(0)=x'(2\pi)\); \(f\) and \(g\) continuous from \(\mathbb{R}\) to \(\mathbb{R}\) and \(h\in L^ \infty(0,2\pi)\). The author proves the existence of at least one solution if \(\lim\sup g(s)/s\) and \(\lim\sup(2/s^ 2)\int^ s_ 0g(r) dr\) are appropriately bounded for \(s\to\pm\infty\) and if there exist \(A\) and \(B\) in \(\mathbb{R}\) such that \(g(A)\leq-h(t)\leq g(B)\).

MSC:
34C25 Periodic solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adje, A, Sur et sous solutions dans LES équations différentielles discontinues avec conditions aux limites non linéaires, ()
[2] Amann, H; Ambrosetti, A; Mancini, G, Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z., 158, 179-194, (1978) · Zbl 0368.35032
[3] Bebernes, J; Martelli, M, Periodic solutions for Liénard systems, Boll. un. mat. ital., 16, 398-405, (1979) · Zbl 0425.34044
[4] Costa, D, Topicos EM analise nao-linear e aplicaçoes as equaçoes diferencials, ()
[5] Costa, D; Oliveira, A, Existence of solution for a class of semilinear elliptic problems at double resonance, Boll. soc. brasil. mat., 19, 21-37, (1988) · Zbl 0704.35048
[6] De Figueiredo, D, Lectures on the Ekeland variational principle with applications and détours, (1989), Tata Institute Springer · Zbl 0688.49011
[7] De Figueiredo, D; Gossez, J.-P, Conditions de non-résonance pour certains problèmes elliptiques semi-linéaires, C. R. acad. sci. Paris, 302, 543-545, (1986) · Zbl 0596.35049
[8] De Figueiredo, D; Gossez, J.-P, Nonresonance below the first eigenvalue for a semilinear elliptic problem, Math. ann., 281, 589-610, (1988) · Zbl 0637.35037
[9] {\scT. Ding, R. Iannacci, and F. Zanolin}, On periodic solutions of sublinear Duffing equations, J. Math. Anal. Appl., to appear. · Zbl 0727.34030
[10] {\scT. Ding, R. Iannacci, and F. Zanolin}, Existence and multiplicity results for periodic solutions of semilinear Duffing equations, to appear. · Zbl 0785.34033
[11] {\scA. Fonda and F. Zanolin}, On the use of time-maps for the solvability of nonlinear boundary value problems, to appear. · Zbl 0766.34012
[12] Gossez, J.-P; Omari, P, Nonresonance with respect to the fuc̆ik spectrum for periodic solutions of second order ordinary differential equations, Nonlinear anal. th. methods appl., 14, 1079-1104, (1990) · Zbl 0709.34037
[13] Habets, P; Sanchez, L, Periodic solutions of some Liénard equations with singularities, (), 1035-1044 · Zbl 0695.34036
[14] Invernizzi, S, (), 243-248
[15] Invernizzi, S; Zanolin, F, Periodic solutions of a differential delay equation of Rayleigh type, (), 25-37 · Zbl 0429.34064
[16] Mawhin, J, Compacité, monotonie et convexité dans l’étude de problèmes aux limites semi-linéaires, (1981), Univ. Sherbrooke · Zbl 0497.47033
[17] Mawhin, J, Nonlinear variational two-point boundary value problems, (), 209-218 · Zbl 0739.34026
[18] Mawhin, J; Willem, M, Multiple solutions of the periodic boundary value problems for some forced pendulum type equations, J. differential equations, 52, 264-287, (1984) · Zbl 0557.34036
[19] Mawhin, J; Willem, M, Critical point theory and Hamiltonian systems, () · Zbl 0678.35091
[20] Omari, P; Zanolin, F, A note on nonlinear oscillations at resonance, Acta math. sinica (N.S.), 3, 351-361, (1987) · Zbl 0648.34040
[21] Reissig, R, Schwingungssätze für die verallgemeinerte liénardsche differen-tialgleichung, (), 45-51 · Zbl 0323.34033
[22] Reuter, G.E, Boundedness theorems for nonlinear differential equations of the second order, J. London math. soc., 27, 48-58, (1952) · Zbl 0048.06901
[23] Rouche, N; Mawhin, J, ()
[24] Zanolin, F, Periodic solutions for differential systems of Rayleigh type, Rend. istit. mat. univ. trieste, 12, 69-77, (1980) · Zbl 0467.34030
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.