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A linear and weakly nonlinear equation of a beam: the boundary-value problem for free extremities and its periodic solutions. (English) Zbl 0226.35008

##### MSC:
 35B10 Periodic solutions to PDEs 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 35G15 Boundary value problems for linear higher-order PDEs
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##### References:
 [1] N. Krylová O. Vejvoda: Periodic solutions to partial diff. equations, especially to a biharmonic wave equation. Atti del Convegno sui problemi di evoluzione. Maggio, 1970. Roma. · Zbl 0226.35009 [2] W. S. Hall: On the Existence of Periodic solutions for the Equations $$D_{tt} u + (- 1)^p D^{2p}_x u = \varepsilon f(., ., m)$$. Journal of Differential Equations, Vol. 7, No 3 May 1970, 509-526. · Zbl 0198.14002 [3] W. S. Hall: Periodic Solutions of a class of Weakly Nonlinear Evolution Equations. Archive for Rational Mechanics and Analysis, Vol. 39, N. 4, 1970, 294-322. · Zbl 0211.12704 [4] H. Petceltová: Periodic solutions of the equation $$u_{tt} + u_{xxxx} = \varepsilon f(., ., u)$$. to appear. [5] O. Vejvoda: Periodic solutions of a linear and weakly nonlinear wave equation in one dimension. I, Czech. Math. J., 14 (89), 1964, 341-382. · Zbl 0178.45302 [6] D. Ch. Karimov: On the periodical solutions of nonlinear equations of the fourth order. Dokl. Akad. Nauk SSSR, 49 (1945), 618-621. · Zbl 0061.22203 [7] D. Ch. Karimov: On the periodical solutions of nonlinear equations of the fourth order. (Russian), Dokl. Akad. Nauk SSSR, 57 (1947), 651-653. [8] D. Ch. Karimov: On the periodical solutions of nonlinear equations of the fourth order. (Russian), Dokl. Akad. Nauk Uz. SSR, 1949, No 8, 3-7. [9] A. P. Mitrjakov: On the periodic solution of nonlinear partial differential equations of higher order. (Russian), Trudy Uzb. Gos. Univ., 1956, No 65, 31 - 44. [10] P. V. Solovieff: Sur les solutions périodique de certaines équations non-linéaires du quatrième ordre. Dokl. Akad. Nauk SSSR, 25 (1939), 731-734. · Zbl 0022.34902
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