By Yu. Mitropolskii, G. Khoma, M. Gromyak (auth.)

The thought of partial differential equations is a large and quickly constructing department of latest arithmetic. difficulties concerning partial differential equations of order greater than one are so different common concept can rarely be equipped up. There are numerous basically other forms of differential equations known as elliptic, hyperbolic, and parabolic. concerning the development of ideas of Cauchy, combined and boundary price difficulties, every one type of equation shows solely assorted homes. Cauchy difficulties for hyperbolic equations and platforms with variable coefficients were studied in classical works of Petrovskii, Leret, Courant, Gording. combined difficulties for hyperbolic equations have been thought of through Vishik, Ladyzhenskaya, and that for basic dimensional equations have been investigated by means of Bitsadze, Vishik, Gol'dberg, Ladyzhenskaya, Myshkis, and others. In final decade the speculation of solvability in general of boundary price difficulties for nonlinear differential equations has acquired extensive improvement. major effects for nonlinear elliptic and parabolic equations of moment order have been got in works of Gvazava, Ladyzhenskaya, Nakhushev, Oleinik, Skripnik, and others. in regards to the solvability generally of nonlinear hyperbolic equations, that are hooked up to the speculation of neighborhood and nonlocal boundary price difficulties for hyperbolic equations, there are just partial effects acquired through Bronshtein, Pokhozhev, Nakhushev.

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E. 28), ufO, t) = u(rr, t) = O. Proof. :31), we have t {F[u, 'II" u,,)(O - T, T) + F[u, 'Ill, ux]( -(0- T 1 -T),T)}dO-4" j'I';) It. )(rr + 0 - T, T) t t dT + F[u, Ut, u,,]( -rr- T -0 + T, T) }dO == O. This completes the proof. 1. 8, for the boundary condition uta, t) u(rr, t) 0 to be fulfilled it is necessary that both the right-hand side f(x, t, u, Ut, u x ) of equation (3,27) and its solution u(:t, t) be 2rr-periociic in variable x. 10. 1 be fulfilled. "11), thc function f(x, t, u(x, t), Ut(x, t), ux(a:, t)) can be expanded into the FO'U1'iel' sCI'ics of the f01'1l!

XI (0,0) = and uXI (1I',t) = O'Vt : ~ t ~ 11'. Let now (x, t) E X 2 (see Fig. 2). 85) is then equivalent to the system of integral equations ° ° J x UI (x, t) = -U2(0, t + x - 211') - E *J[u, UI, U2] (71, t + x - 211' + 11) d7)+ o +E J" ,h[U,'lLI> U 2] (Il,t+ a: -71) dll, x J :I: U2(X, t) = U2(0, t - :1:) + E 'h [u, 1£1, '1£2] (71, t - x + II) dl], o u(x, t) = u(x, 211' - :I:) + ~ J t {Ul(X, r) + U2(X, rn dr. 92) 211'-x By virtue of the notation U2 = Ut -Ux and condition UI(O, t) = 0, the function U2(0, t) == -ux(O, t) = -~(t) is known and the value of U(X, 211' - x) is defined by the function UA (x, 211' - x). *

12. 99) be a 2rr-system of the second class in the /'cgion -a ~ 11 ~ (I, -0 ~ 1i ~ b. s state be isolated and of 7l0nzc1'O index. As an example, consider the Van der Pol equation e eo ji + y = e(l - y 2 l1i. c:l + 1 c .. , '2t 1 c:l .. OS ,- 0 , ~(2VO - eV6)6 + ~ (el - ~er) sin 2t + ~erSin 4t = O. 104) Averaging this system, we obtain vo(2 - €Vo) = o. 10:3), the only periodic solution of this equation is a 2rr-periodic one corresponding to e~ = 2, 6 = 0, Vo = O. ,ymptotic methods [14, 74]. CHAPTER 3 PERIODIC SOLUTIONS OF THE FIRST CLASS SYSTEMS This chapter singles out some new classes of periodic boundary value problems for the second order wave differential equations of hyperbolic type (T-systems of the first class, i.