## Periodic solutions of the equation $$u_{tt}+u_{xxxx}=\varepsilon f(.,.,u,u_t)$$.(English)Zbl 0263.35017

### MSC:

 35G20 Nonlinear higher-order PDEs 35B10 Periodic solutions to PDEs 35B20 Perturbations in context of PDEs
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### References:

 [1] Hall W. S.: On the existence of periodic solutions for the equations $$D_{tt}u+(-1)^{p}\,D_{x}^{2p}u=\varepsilon f(\cdot ,\cdot ,u)$$. Jour. of Diff. Equations, 7, 509-526, 1970. · Zbl 0198.14002 [2] Hall W. S.: Periodic solutions of a class of weakly nonlinear evolution equations. Arch. Rat. Mech. Anal., Vol. 39, 4, 294-322. · Zbl 0211.12704 [3] Lovicarová H.: Periodic solutions of a weakly nonlinear wave equation in one dimension. Czech. Math. J. 19, 1969, 324-342. · Zbl 0181.10901 [4] Krylová N.: Periodic solutions of hyperbolic partial differential equation with quadratic dissipative term. Czech. Math. J. 20, 1970, 375-405. · Zbl 0214.10402 [5] Krylová N., Vejvoda O.: A linear and weakly nonlinear equation of a beam: The boundary-value problem for free extremities and its periodic solutions. Czech. Math. J. 21, 1971, 535-566. · Zbl 0226.35008 [6] Moser J.: A rapidly convergent iteration method and nonlinear partial differential equations. Ann. Scuola Norm. Super. Pisa, Ser. 3, 20, 1956, 265-315. · Zbl 0144.18202 [7] Rabinowitz P. H.: Periodic solutions of nonlinear hyperbolic partial differential equations. Comm. Pure Appl. Math., 20, 1967, 145-205. · Zbl 0152.10003 [8] Rabinowitz P. H.: Time periodic solutions of nonlinear wave equations. Manuscripta math., 5, 1971, 165-194. · Zbl 0219.35062 [9] De Simon L., Torelli C.: Soluzioni periodiche di equazioni a derivate parziali di tipo iperbolico non lineari. Rend. Sem. Mat. Univ. Padova XL, 1968, 380-401. · Zbl 0198.13704 [10] Torelli G.: Soluzioni periodiche dell’equazione non lineare $$u_{tt}-u_{xx} + f(x, t, u) = 0$$. Rend. Ist. di Matem. Univ. Trieste 1, 1969, 123-137. · Zbl 0186.42903 [11] Petryshyn W. V.: On nonlinear $$P$$-compact operators in Banach space with applications to constructive fixed-point theorems. J. Math. Anal. Appl. 15, 1966, 228-242. · Zbl 0149.10602 [12] Bers L., John F., Schechter W.: Partial Differential Equations. New York, Interscience 1964. · Zbl 0126.00207 [13] Randol B.: A lattice-point problem II. TAMS 125, 1966, 101-113. · Zbl 0161.04902
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