Solution with periodic second derivative of a certain third order differential equation. (English) Zbl 0629.34048

Sufficient conditions for the existence of periodic solutions of the third kind, i.e. those with the second periodic derivative, to the equation \(x\prime''+ax''+g(t,x')+cx=p(t)\), where a, c are constants, \(g(t,y+w)\equiv g(t+T,y)\equiv g(t,y)\in {\mathfrak C}^ 1(R^ 2)\) and p(t)\(\in {\mathfrak C}^ 1({\mathbb{R}}^ 1)\) is of the special (necessary) form, are given using a topological degree argument. This problem is solvable for \(0\neq | c| <| T|^{-3}\).


34C25 Periodic solutions to ordinary differential equations
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[1] FARKAS M.: Controllably periodic perturbations of autonomous systems. Acta Math. Acad. Sci. Hungar., 22, 1971, 337-348. · Zbl 0239.34016
[2] FARKAS M.: Determination of controllably periodic perturbed solutions by Poincaré’s method. Stud. Sci. Math. Hungar., 7, 1972, 257-266. · Zbl 0275.34038
[3] FARKAS M.: Personal communication.
[4] ANDRES J.: Periodic boundary value problem for certain nonlinear differential equations of the third order. Math. Slovaca, 3, 35, 1985, 305-309. · Zbl 0591.34037
[5] ANDRES J., VORÁČEK J.: Periodic solutions to a nonlinear parametric differential equation of the third order. Atti Accad. Naz. Lincei, 3-4, 77, 81-86.
[6] ANDRES J.: Periodic derivative of solutions to nonlinear differential equations. to appear in Czech. Math. J. · Zbl 0727.34029
[7] MAWHIN J.: Topological degree methods in nonlinear boundary value problems. CBMS 40, AMS, Providence 1979. · Zbl 0414.34025
[8] ANDRES. J., VORÁČEK J.: Periodic solutions of a certain parametric third order differential equation. Kniž. odb. věd. sp. VUT v Brné, B94, 1983, 7-11.
[9] REISSIG R.: Continua of periodic solutions of the Liénard equation. Constr. meth. nonlin. BVPs and nonlin. oscill. (ed. J. Albrecht, L. Collatz, K. Kirchgässner), Birkhäuser, Basel 1979, pp. 126-133. · Zbl 0416.34045
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