We give several examples of Douglas algebras that do not have any maximal subalgebra. We find a condition on these algebras that guarantees that some do not have any minimal superalgebra. We also show that if $A$ is the only maximal subalgebra of a Douglas algebra $B$, then the algebra $A$ does not have any maximal subalgebra.

We consider the pair diffusion process which includes cluster reactions of high order. We are able to prove a local (in time) existence result in arbitrary space dimensions. The model includes a nonlinear system of reaction-drift-diffusion equations, a nonlinear system of ordinary differential equations in Banach spaces, and a nonlinear elliptic equation for the electrochemical potential. The local existence result is based on the fixed point theorem of Schauder.

We modify the definition of lopsided convergence of bivariate functionals to obtain stability results for the min/sup points of some control problems. In particular, we develop a scheme of finite dimensional approximations to a large class of non-convex control problems.

The nonlocal boundary value problem, , in an arbitrary Banach space $E$ with the strongly positive operator $A$, is considered. The coercive stability estimates in Hölder norms for the solution of this problem are proved. The exact Schauder’s estimates in Hölder norms of solutions of the boundary value problem on the range $\left\{0\le t\le 1,x{ℝ}^n\right\}$ for $2m$-order multidimensional parabolic equations are obtaine.