No abstract

No abstract

Abstract. A necessary and sufficient condition to extend a continuous linear real functionals which is positive with respect to a semi-group defined on a subspace of a linear space is discussed in this paper. The case of a closed subspace of a Banach space is also discussed.

This paper discusses some regularity of almost periodic solutions of the Poisson equation −∆ = in R , where is an almost periodic function. It wasproved by Sibuya [Almost periodic solutions of Poisson†TM s equation. Proc. Amer. Math. Soc., 28:195-198, 1971.] that if is a bounded continuous function and solves the Poisson equation in the distribution sense, then is an almost periodic function. In this work, we weaken the assumption of the usual boundedness to boundedness in the sense of distribution, which we refer to as a bounded generalized function. The set of bounded generalized functions are wider than the set of usual bounded functions. Then, assuming that is a bounded generalized function and solves the Poisson equation in the distribution sense, we prove that this solution is bounded in the usual sense, continuous and almost periodic. Moreover, we show that the first partial derivatives of the solution / , = 1,. .. , , are also continuous, bounded and almost periodic functions. The technique is based on extending a representation formula using Green function for Poisson equation for solutions in the distribution sense. Some useful properties of distributions are also shown that can be used in studying other elliptic problems.

Abstract. We study the existence and uniqueness of a solution bounded in the entire space for a class of higher order linear partial differential equations. We prove the theorem on the necessary and sufficient condition for the existence and uniqueness of a bounded solution for a studied class of equations. This theorem is an analogue of the Bohl theorem known in the theory of ordinary differential equations. In a partial case the unique solvability conditions are expressed in terms of the coefficients of the equation and we provide the integral representation for the bounded solution.