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Oscillations of the solutions of nonlinear hyperbolic equations of neutral type

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Publicacions Matemàtiques

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Publicacions Matemátiques, Vol 36 (1992), 3-18 . Abstract OSCILLATIONS OF THE SOLUTIONS OF NONLINEAR HYPERBOLIC EQUATIONS OF NEUTRAL TYPE D.P. MISHEV AND D.D . BAINOV In this paper nonlinear hyperbolic equations of neutral typé of the form 2 (~t2 [u(X' t) + a(t)u(x' t - T)] - [Au(x, t) + 11(t)AU(x, t - u)] + c(x, t, u) = f(X, t), (x, t) E 9 x (0, oo) - G, are considered, werre r, a = const > 0, with boundary conditions an +7(x, t)u = g(x, t), (x, t) E ast x [0, oo) or u= 0, (x, t) E asl x [O,oo) . Under certain constraints on the coefficients of the equation and the boundary conditions, sufficient conditions for oscillation of the solutions of the problems considered are obtained . 1 . Introduction In the last few years results related to the oscillatory properties and asymptotic behaviour of the solutions of some classes of hyperbolic equa- tions were published . We shall mention especially the work of K. Kreith, T . Kusano and N. Yoshida [4] in which suficient conditions for oscillation of the solutions of the nonlinear hyperbolic equation utt - Au + c(x, t, u) = f(x, t) considered in a cylindrical domain are obtained . Oscillatory properties of the solutions of hyperbolic differential equations with a deviating ar- gument were investigated in the works of D . Georgiou, K. Kreith [2], D . Georgiou [3] . Hyperbolic differential equations with maxima were inves- tigated in the work of D . Mishev [6] and some conditions for oscillation of the solutions of hyperbolic equations of neutral type were obtained by D. Mishev and D. Bainov in [7], [8] . The present investigation is supported by the Ministry of Culture, Science and Edu- cation of People's Republic of Bulgaria under Grant 61 . 4 D.P . MISHEv, D.D . BAINOV 2 . Preliminary notes In the present paper sufficient conditions for oscillation of the solutions of nonlinear hyperbolic equations of neutral type of the form a (1) eta [u(x, t) + A(t)u(x, t - T)] - [DU(x, t) + p,(t)DU

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