一类非线性变系数波方程解的存在性The Existence of a Class of Nonlinear Wave Equations with Variable Coefficients

>This paper is devoted to the existence results for the one dimensional wave equation with x-de- pendent coefficients when resonance occurs at the eigenvalue rN . By using the Mawhins continuation theo- rem, the authors get a result which is similar to the literature.

[1] L. Cesari, R. Kannan. Solutions of nonlinear hyperbolic equations at resonace. Nonlinear Analysis, 1982, 6(8): 751-805.

[2] M. Schechter. Bounded resonance problems for semilinear elliptic equations. Nonlinear Analysis, 1995, 24(10): 1471-1482.

[3] S. Solimini. On the solvability of some elliptic partial differential equations with the linear part at resonance. Journal of Mathematical Analy- sis and Applications, 1986, 117: 138-152.

[4] M. Willem. Periodic solutions of wave equations with jumping nonlinearities. Journal of Differential Equations, 1980, 36(1): 20-27.

[5] J. Berkovits, V. Mustonen. On nonresonance for systems of semilinear wave equations. Nonlinear Analysis, 1997, 29(6): 627-638.

[6] S. Rybicki. Periodic solutions of vibrating strings. Degree theory approach. Annali di Matematica Pura ed Applicata, 2001, 179(1): 197-214.

[7] J. M. Coron. Periodic solutions of a nonlinear wave equation without assumption of monotonicity. Mathematische Annalen, 1983, 262(2): 273-285.

[8] P. L. Felmer, R. F. Manasevich. Periodic solutions of a coupled system of telegraph-wave equations. Journal of Mathematical Analy- sis and Applications, 1986, 116(1): 10-21.

[9] R. Iannacci, M. N. Nkashama. Nonlinear boundary value problems at resonance. Nonlinear Analysis, 1987, 11(4): 455-473.

[10] R. Iannacci, M. N. Nkashama. Nonlinear elliptic partial differential equations at resonance: Higher eigenvalues. Nonlinear Analysis, 1995, 25(5): 455-471.

[11] M. R. Grossinho, M. N. Nkashama. Periodic solutions of parabolic and telegraph equations with asymmetric nonlinearities. Nonlinear Analysis, 1998, 33(2): 187-210.

[12] J. K. Kim, N. H. Pavel. Existence and regularity of weak periodic solutions of the 2-D wave equation. Nonlinear Analysis, 1998, 32(7): 861-870.

[13] J. K. Kim, N. H. Pavel. $L^\infty$-optimal control of the periodic 1-D wave equation with x-dependent coefficients. Nonlinear Analysis, 1998, 33(1): 25-39.

[14] V. Barbu, N. H. Pavel. Periodic solutions to nonlinear one dimensional wave equation with x-dependent coefficients. Transactions of the Am- erican Mathematical Society, 1997, 349(5): 2035-2048.

[15] J. Mawhin. Compacite, monotonie et convexite dans l’Etude des problems aux limites semi-lineaires. Quebec: Universite de Sherbrooke, 1981.

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