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1
A Kneser-type theorem for an integral equation in locally convex spaces
EN
We shall give suffcient conditions for the existence of solutions of the integral equation (1) in locally convex spaces. We also prove that the set of these solutions is a continuum.
EN
According to Mickael's selection theorem any surjective continuous linear operator from one Prechet space onto another has a continuous (not necessarily linear) right inverse. Using this theorem Herzog and Lemmert proved that if E is a Frechet space and T : E -> E is a continuous linear operator such that the Cauchy problem x = T x, x(0) = X0 is solvable in [0,1] for any X06 E, then for anyf zawiera się C([0, 1],E), there exists a continues map S : [0,1] x E -> E, (t x) ->o StX such that for any X0 zawiera się w E, the function x(t) = StX0 is a solution of the Cauchy problem x(t) = Tx(t) +- f(t), x(0) = X0 (they call S a fundamental system of solutions of the equation x = Tx + f). We prove the same theorem, replacing "continuous" by "sequentially continuous" for locally convex spaces from a class which contains strict inductive limits of Frechet spaces and strong duals of Frechet-Schwarz spaces and is closed with respect to finite products and sequentially closed subspaces. The key-point of the proof is an extension of the theorem on existence of a sequentially continuous right inverse of any surjective sequentially continuous linear operator to some class of non-metrizable locally convex spaces.
3
Fourier analysis on locally convex spaces of distributions, II
EN
This is the second in a series of papers, extending the theory of Fourier analysis to locally convex spaces of distributions (LCD-spaces). In this paper, LCD-spaces admitting conjugation and multiplier operators on LCD-spaces are discussed. It is also shown that if E is an LCD-space having C°° as a dense subset, then E*, endowed with the topology of precompact convergence, is an LCD-space having C°° as a dense subset.
4
Fourier analysis on locally convex spaces of distributions, I
EN
In this series of papers, many results of Fourier analysis which are known for Lp (1 < p < oo) , C (the space of continuous functions) and other Banach spaces of functions have been generalized to locally convex spaces of distributions. Also, in this paper, the (C, 1)-complementary space E' of a locally convex space of distributions E is defined and it is shown that E' , as a subspace of E* with strong* topology, is a locally convex space of distributions.
EN
It is proved that there exist common fixed points for monotone operators in locally convex spaces with a cone.
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