Applying the methods of characteristics and contractive mappings, the existence and uniqueness of global generahzed continuous solution of mixed problem for hyperbohc system of the first order semilinear equations with horizontal characteristics are established.

[1] Rozhdestvensky B. L. Systems of quasilinear equations and their applications to gas dynamics / B. L. Rozhdestvensky, H. H. Yanenko. - M. : Nauka, 1978.  - 592 p.

[2] Riemann B.  On the propagation of waves of finite amplitude [essays] / B. Riemann. - M.; L., 1948. - P.  376-395.

[3] Schauder J. Cauchy’sches Problem für partielle Diffеrеnialgleichungen erster Ordnung.  Anwendung  einiger  sich  auf die  Absolutbeträge  der  Lösungen  bezi­ehenden  Abschätzungen / J.  Schauder/ / Commen. Mathem. Helvetici. - 1936­-1937. - Bd. 9. - S. 263-282.

[4] Courant R. On nonlinear partial di­fferential equations with two independent variables / R. Courant, P. Lax / / Comm. Pure. Appl. Math. - 1949. - Vol. 2. - P. 255-273.

[5] Kirilich V. M.  M. Generalised continuous solvability of the problem with unknown boundaries for singular hyperbolic systems of quasilinear equations / V. M. Kirilich, A. M. Filimonov // Matem. studii. - 2008. - Т.  30.  - № 1. - P.  42-60.

[6] Ishlinsky A. Yu. Longitudinal vibrations of a rod in the presence of a linear law of aftereffect and relaxation / A. Yu. Ishlinsky / / Prikl. matematika i mechanika.  - 1940.  - Vol.4. - Issue 1.  - P.  79-92.

[7] Shashkov A. G. Wave phenomena of heat conduction: system-structural approach / A. G. Shashkov, V. A. Bubnov, S. Y. Yanovsky.  - M.: Unitorial Urss, 2004.  - 296 p.

[8] Uflyand Ya.  С.   To the question of propagation of oscillations in composite electric lines / Ya. S. Uflyand / / Inzh.-phys. zhurn. - 1964.  - Т.  7.  - № 1.- P.  89-92.

[9] Friedman A. The Stefan problem for a hyperbolic heat equation / A. Friedman, B. Hu / / Math. Anal. and Appl. - 1989. - Vol. 138. - N 1. - P. 249-279.

[10] Florescu  D.  Asupra  existentei  solutiilor unor  sisteme  hiperbolice  de  tip  special./ D.  Florescu / /  Studii si cerc. Mat. - 1978. - T. 30, N. 3, P. 279-285.

[11] Trenogin V. A. Functional analysis / V. A. Trenogin.  A. Functional analysis / V. A. Trenogin. - M.: Nauka, 1980. - 325 p.

[12] Maulenov O. On the solvability of the mixed problem for a degenerate semilinear hyperbolic system on a segment / O. Maulenov, A. D. Myshkis/ / Izv. of the Academy of Sciences of the Kazakh SSR.  Ser. phys.-mat. - 1981.  - №5.  - P.  25-29.