This work is devoted to solving some classes of operator equations, based on the solution of auxiliary one-parameter family of equations, which is obtained fromthe original operator equation by formal replacement of t...
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This work is devoted to solving some classes of operator equations, based on the solution of auxiliary one-parameter family of equations, which is obtained fromthe original operator equation by formal replacement of the operator of the integrated parameter. Solutions are vector-valued functions represented by power series or integral. We investigate some properties of these solutions, namely, growth characteristics, the domain of analyticity. The investigation is realized by means of order and type of operator, operator order and operator type of the vector relative to the operator.
We deal with the minimax problem relative to a vector-valued function f: X0xY0--> V, where a partial ordering in the topological vector space V is induced by a closed and convex cone C. In Ref. 1, under suitable hy...
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We deal with the minimax problem relative to a vector-valued function f: X0xY0--> V, where a partial ordering in the topological vector space V is induced by a closed and convex cone C. In Ref. 1, under suitable hypotheses, we proved that [GRAPHICS] the exact meaning of the symbols is given in Section 2. In this work, we prove that, under a reasonable setting of hypotheses, the previous inclusion holds and also we have that [GRAPHICS]
This paper is concerned with minimax theorems in vector-valued optimization. A class of vector-valued functions which includes separated functions f(x, y) = u(x) + v(y) as its proper subset is introduced. Minimax theo...
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This paper is concerned with minimax theorems in vector-valued optimization. A class of vector-valued functions which includes separated functions f(x, y) = u(x) + v(y) as its proper subset is introduced. Minimax theorems and cone saddle-point theorems for this class of functions are investigated.
We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset Omega subset of R-N and a Banach space V, we compare the classical Sobolev space W-1,W- p(Omega, V) with the so-called Sobo...
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We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset Omega subset of R-N and a Banach space V, we compare the classical Sobolev space W-1,W- p(Omega, V) with the so-called Sobolev-Reshetnyak space R-1,R-p(Omega, V). We see that, in general, W-1,W-p(Omega, V) is a closed subspace of R-1,R- p(Omega, . As a main result, weobtain that W-1,W- p(Omega, V) = R-1,R- p(Omega, V) if, and only if, the Banach space V has the Radon-Nikodym property
We establish the following theorems: (i) an existence theorem for weak type generalized saddle points;(ii) an existence theorem for strong type generalized saddle points;(iii). a generalized minimax theorem for a vect...
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We establish the following theorems: (i) an existence theorem for weak type generalized saddle points;(ii) an existence theorem for strong type generalized saddle points;(iii). a generalized minimax theorem for a vector-valued function. These theorems are generalizations and extensions of the author's recent results. For such extensions, we propose new concepts of convexity and continuity of vector-valued functions, which are weaker than ordinary ones. Some of the proofs are bawd on a few key observations and also on the Browder coincidence. theorem or the Tychonoff fixed-point theorem. Also, the minimax theorem follows from the existence theorem for weak type generalized saddle points. The main spaces with mathematical structures considered are real locally convex spaces and real ordered topological vector spaces.
This paper provides new Farkas-type results characterizing the inclusion of a given set, called contained set, into a second given set, called container set, both of them are subsets of some locally convex space, call...
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This paper provides new Farkas-type results characterizing the inclusion of a given set, called contained set, into a second given set, called container set, both of them are subsets of some locally convex space, called decision space. The contained and the container sets are described here by means of vectorfunctions from the decision space to other two locally convex spaces which are equipped with the partial ordering associated with given convex cones. These new Farkas lemmas are obtained via the complete characterization of the conic epigraphs of certain conjugate mappings which constitute the core of our approach. In contrast with a previous paper of three of the authors (Dinh et al. in J Optim Theory Appl 173:357-390, 2017), the aimed characterizations of the containment are expressed here in terms of the data.
We consider ordered spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with appropriate locally convex topologies. Using suitable sets of such functions as test systems a Korovk...
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We consider ordered spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with appropriate locally convex topologies. Using suitable sets of such functions as test systems a Korovkin type approximation theorem for equicontinuous nets of positive operators is established. As in the classical theory, the Korovkin closure is characterized both through envelopes of functions and through measure theoretical conditions. (C) 2013 Elsevier Inc. All rights reserved.
We introduce a new class FV(Omega, E) of weighted spaces of functions on a set Omega with values in a locally convex Hausdorff space E which covers many classical spaces of vector-valued functions like continuous, smo...
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We introduce a new class FV(Omega, E) of weighted spaces of functions on a set Omega with values in a locally convex Hausdorff space E which covers many classical spaces of vector-valued functions like continuous, smooth, holomorphic or harmonic functions. Then we exploit the construction of FV(Omega, E) to derive sufficient conditions such that FV(Omega, E) can be linearised, i.e. that FV(Omega, E) is topologically isomorphic to the epsilon-product FV(Omega)epsilon E where FV(Omega) := FV(Omega, K) and K is the scalar field of E.
Abstract: Representations of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces are given in terms of operator-valued measures. Then spaces whose duals are Banach fu...
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Abstract: Representations of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces are given in terms of operator-valued measures. Then spaces whose duals are Banach function spaces are characterized. With this last information, reflexivity of this type of space is discussed. Finally, the structure of compact operators on these spaces is studied, and an observation is made on the approximation problem in this context.
Abstract: The Banach space ${L^1}(0,T;X)$ is retopologized by $|||f||| = \max ||\int _a^bfdt||$, $0 \leqslant a \leqslant b \leqslant T$, where $||.||$ is the norm in the given Banach space $X$. It is shown he...
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Abstract: The Banach space ${L^1}(0,T;X)$ is retopologized by $|||f||| = \max ||\int _a^bfdt||$, $0 \leqslant a \leqslant b \leqslant T$, where $||.||$ is the norm in the given Banach space $X$. It is shown here that this topology coincides with the usual weak topology of ${L^1}(0,T;X)$ on a wide class of weakly compact subsets.
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