Summary: The work is devoted to the study of the solvability of boundary value problems for quasi-parabolic equations \[(-1)^pD^{2p+1}_tu-\frac{\partial}{\partial x}\left(a(x)u_x\right)+c(x,t)u=f(x ,t)\] \[((x,t)\in (0,1)\times (0,T), a(x)>0, D^k_t=\frac{\partial^k}{\partial t^k}, p>0 - \text{integer})\] with boundary conditions of one of the types \[u(0,t)-\beta u(1,t)=0, u_x(1,t)=0, t\in (0,T),\] or \[u_x(0,t)-\beta u_x(1,t)=0, u(1,t)=0, t\in (0,T).\] The problems under study can be treated as nonlocal problems with the generalized Samarskii-Ionkin condition in terms of spatial variable, for them we prove existence and uniqueness theorems for regular solutions – namely, solutions that have all generalized in the sense of S.L. Sobolev derivatives included in the corresponding equation.