Volume 21, Issue 1

In this paper, we establish a result concerning the controllability of a mixed Volterra–Fredholm type integrodifferential third order dispersion equation. The result is obtained by using the theory of strongly continuous semigroups and the Banach fixed point theorem.

We study the behavior of small solutions depending on time of the generalized and regularized Benjamin–Ono equation in both continuous and periodic context. In particular, we prove that these solutions remain small for a time scale improving the natural time given by the local well-posedness. In the continuous case, the result becomes global-in-time.

The regularity theorem is a result stating that functions which have extremal growth or decrease in the given class display a regular behaviour. Such theorems for linearly invariant families of analytic functions are well known. We prove regularity theorems for some classes of harmonic functions. Many presented statements are new even in the analytic case.

We apply a fixed point theorem to prove that there exists a unique derivation close to an approximately generalized derivation in Lie C * -algebras. Also, we prove the hyperstability of generalized derivations. In other words, we find some conditions under which an approximately generalized derivation becomes a derivation.

Rothe's fixed-point theorem is applied to prove the interior approximate controllability of a semilinear impulsive strongly damped wave equation with Dirichlet boundary conditions in the space Z1/2=D((-Δ)1/2)×L2(Ω)${Z_{1/2} = D((-\Delta )^{1/2}) \times L^{2}(\Omega )}$, where Ω is a bounded domain in ℝ n ( n ≥ 1). Under some conditions we prove the following statement: For all open nonempty subsets ω of Ω the system is approximately controllable on [0,τ]${[0, \tau ]}$. Moreover, we exhibit a sequence of controls steering the nonlinear system from an initial state z 0 to a neighborhood of the final state z 1 at time τ>0${\tau >0}$.

The existence of additive selections of additive correspondences was investigated in [Ark. Mat. 4 (1960), 87–97], [Rev. Roumaine Math. Pures Appl. 28 (1983), 239–242.], [Math. Ser. Univ. Novi Sad 18 (1988), 143–148] and other papers. In this article, we find an existence theorem for additive selections of additive correspondences with convex compact values in a real normed linear space defined on an open convex cone of a real separable normed space.