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Oscillation criteria for nonlinear Bianchi equations. (English) Zbl 0506.35005


MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
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[1] Bateman, H., Logarithmic solutions of Bianchi’s equation, Proc. natn. Acad. Sci. U.S.A., 19, 852-854 (1933) · Zbl 0007.35002
[2] Bianchi, L., Il metodo di Riemann esteso alla integrazione della equazione: \(∂^nu(∂x1∂x2…∂xn) = Mu\), Atti Accad. naz. Lincei Rc. Ser. 5, 4, 8-18 (1895) · JFM 26.0400.01
[3] Bianchi, H., Sulla estensione del metodo di Riemann alle equazioni lineari a derivate parziali d’ordine superiore, Atti Accad. naz. Lincei Rc. Ser. 5, 4, 133-142 (1895) · JFM 27.0282.01
[4] Conlan, J.; Diaz, J. B., Existence of solutions of an \(n\) th order hyperbolic partial differential equation, Contr. diff. Eqns, 2, 277-289 (1963)
[5] Ďurikovič, V., On the existence and uniqueness of solutions and on the convergence of successive approximations in the Darboux problem for certain differential equat ions of the type \(u_{x1…x_{n\)
[6] Fage, M. K., Cauchy problem for Bianchi equations, Mat. Sb., 45, 281-322 (1958), in Russian · Zbl 0082.09001
[7] Glick, I. I., On an analog of the Euler-Cauchy polygon method for the partial differential equation \(u_{x1…x_{n\)
[8] Hsiang, W.-T.; Kwong, M. K., A comparison theorem for the first nodal line of the solutions of quasilinear hyperbolic equations with non-increasing initial values, Proc. R. Soc. Edinb. Sect. A, 86, 139-151 (1980) · Zbl 0446.35007
[9] Kartsatos, A. G., On \(n\) th order differential inequalities, J. math. Analysis Applic., 52, 1-9 (1975) · Zbl 0327.34012
[10] Kiguradze, I. T., Oscillation properties of solutions of certain ordinary differential equations, Dokl. Akad. Nauk SSSR. Dokl. Akad. Nauk SSSR, Soviet Math. Dokl., 3, 649-652 (1962), in English · Zbl 0144.11201
[11] Kreith, K., Sturmian theorems for characteristic initial value problems, Atti Accad. naz. Lincei Rc., 47, 139-144 (1969) · Zbl 0194.41502
[12] Kreith, K., A Sturn theorem for partial equations of mixed type, Proc. Am. math. Soc., 81, 75-78 (1981) · Zbl 0481.35011
[13] Kreith, K., A class of comparison theorems for nonlinear hyperbolic initial value problems, Proc. R. Soc. Edinb. Sect. A, 87A, 189-191 (1981) · Zbl 0452.35003
[14] Kusano, T.; Yoshida, N., Oscillation criteria for a class of nonlinear partial differential equations, J. math. Analysis applic., 79, 236-243 (1981) · Zbl 0463.35008
[15] Lebedev, N. N., Special Functions and their Applications (1972), Dover Publications: Dover Publications New York · Zbl 0271.33001
[16] Ličko, I.; Švec, M., Le caractére oscillatoire des solutions de l’équation \(y^{(n)} + f(x)y^{α\) · Zbl 0123.28202
[17] Mikusiński, J., On Fite’s oscillation theorems, Colloqium math., 2, 34-39 (1951) · Zbl 0039.31302
[18] Okikiolu, G. O., Aspects of the Theory of Bounded Integral Operators in \(L^p\)-spaces (1971), Academic Press: Academic Press New York · Zbl 0219.44002
[19] Onose, H., A comparison theorem and the forced oscillation, Bull. Aust. Math. Soc., 13, 13-19 (1975) · Zbl 0307.34034
[20] Pagan, G., Oscillation theorems for characteristic initial value problems for linear hyperbolic equations, Atti Accad. naz. Lincei Rc., 55, 301-313 (1973) · Zbl 0298.35035
[21] Pagan, G., Existence of nodal lines for solutions of hyperbolic equations, Am. math. Mon., 83, 358-359 (1976) · Zbl 0349.35054
[22] Pagan, G., An oscillation theorem for characteristic initial value problems in linear hyperbolic equations, Proc. R. Soc. Edinb. Sect. A, 77, 265-271 (1977) · Zbl 0399.35073
[23] Pagan, G.; Stocks, D., Oscillation criteria for second order hyperbolic initial value problems, Proc. R. Soc. Edinb. Sect. A, 83, 239-244 (1979) · Zbl 0411.35060
[24] Travis, C. C.; Yoshida, N., Oscillation criteria for third order hyperbolic characteristic initial value problems, Proc. R. Soc. Edinb. Sect A, 88, 135-140 (1981) · Zbl 0472.35011
[25] Walter, W., Differential and Integral Inequalities (1970), Springer: Springer New York
[26] Wu, Zhueng-Hai, Theorems on differential inequalities for nonlinear Bianchi equations, Chin. Math., 4, 637-660 (1964) · Zbl 0171.07302
[27] Yoshida, N., An oscillation theorem for characteristic initial value problems for nonlinear hyperbolic equations, Proc. Am. math. Soc., 76, 95-100 (1979) · Zbl 0421.35003
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