Nonlinear boundary value problems for second order systems with one-sided growth constraints of the right side with respect to the first derivatives

For system of integrodifferential equation $u_i^{\prime\prime}+Q_i(t)u^\prime_i+R_i(t)u_i=f_i(t,u_1,\dots,u_n,u_1^\prime,\dots,u_n^\prime,\int_0^1k_i(t,s,u_1(s),\dots,u_n(s))ds)$ $(i=1,\dots,n)$ we establish existence theorems for the solutions of the problem with boundary conditions $a_iu_i(0)-b_iu_i^\prime(0)=q_i\varphi_i(u_1(0),\dots,u_n(0),u_1(1),\dots,u_n(1),\int_0^1\ell_i(s,u_1(s),\dots,u_n(s))ds)$; $ c_iu_i(1)+d_iu_1^\prime(1)=h_i\Psi_i(u_1(0),\dots,u_n(0),u_1(1),\dots,u_n(1),\int_0^1M_i(s,u_1(s),\dots,u_n(s))ds)$ $(i=1,\dots,n)$.