Nonlinear boundary-value problems for second-order systems with one-sided bounds on the growth of the right-hand side in terms of the first derivatives

Abstract

For system of integrodifferential equation

$$\begin{gathered} u''_i + Q_i (t)u'_i + R_i (t)u_i = f_i (t, u_1 , ..., u_n , u'_1 , ..., u'_n , \int\limits_0^1 {K_i (t, s,} \hfill \\ u_1 (s), ..., u_n (s))ds) (i = 1, ..., n) \hfill \\ \end{gathered} $$

we establish existence theorems for the solutions of the problem with boundary conditions

$$\begin{gathered} a_i u_i (0) - b_i u'_i (0) = g_i \varphi _i (u_1 (0), ..., u_n (0), u_1 (1), ..., u_n (1), \int\limits_0^1 {l_i } (s, \hfill \\ u_1 (s), ..., u_n (s))ds); c_i u_i (1) + d_i u'_i (1) = h_i \Psi _i (u_i (0), ..., u_n (0), \hfill \\ u_1 (1), ..., u_n (1),\int\limits_0^1 {M_i (s, u_1 (s), ..., u_n (s))ds)} (i = 1, ..., n). \hfill \\ \end{gathered} $$