Let (Omega,Sigma,mu) be a sigma-finite measure space, 1 Z be a bounded bilinear map. We say that an X-valued function is p-integrable with respect to B whenever sup{integral(Omega) parallel to B(f(w), y parallel to(p...
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Let (Omega,Sigma,mu) be a sigma-finite measure space, 1 <= p < infinity, X be a Banach space X and B : X x Y -> Z be a bounded bilinear map. We say that an X-valued function is p-integrable with respect to B whenever sup{integral(Omega) parallel to B(f(w), y parallel to(p) d mu : parallel to y parallel to = 1} is finite. We identify the spaces of functions integrable with respect to the bilinear maps arising from Holder's and Young's inequalities. We apply the theory to give conditions on X-valued kernels for the boundedness of integral operators TB(f)(w) = integral(Omega') B(k(w, w'), f(w'))d mu'(w') from L-p(Y) into L-p(Z), extending the results known in the operator-valued case, corresponding to B : L(X, Y) x X -> Y given by B(T, x) = Tx.
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]
We extend the well-known Peano Kernel Theorem to a class of linear operators L C-n+1([a, b];X)--> X, X being a Banach space, which vanish on abstract polynomials of degree less than or equal to n. We then recover, ...
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We extend the well-known Peano Kernel Theorem to a class of linear operators L C-n+1([a, b];X)--> X, X being a Banach space, which vanish on abstract polynomials of degree less than or equal to n. We then recover, in the abstract setting, classical estimates of remainders in polynomial interpolation and quadrature formulas. Finally, we present an application to the error analysis of the trapezoidal time discretization scheme for parabolic evolution equations.
作者：
SHI, DSLING, CInstructor
Department of Mathematical Economics Zhejiang Institute of Finance and Economics Hangzhou China. Professor
Department of Mathematical Economics Zhejiang Institute of Finance and Economics Hangzhou China.
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.
Improving a recent result by the authors, a vector-valued version of Peano's Kernel Theorem is proposed, which gives an integral estimate for a class of linear operators L : Cn+1([a, b];X) --> X, with X general...
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Improving a recent result by the authors, a vector-valued version of Peano's Kernel Theorem is proposed, which gives an integral estimate for a class of linear operators L : Cn+1([a, b];X) --> X, with X general normed space, vanishing on all abstract polynomials of degree less than or equal to n. Continuity and derivatives are intended in the weak sense. When the space is complete, the usual integral representation is retrieved. We show that all usual remainder estimates for polynomial, piecewise polynomial, and spline polynomial interpolation, numerical differentiation and numerical quadrature, can be readily transferred in the vector-valued setting.
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 spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with classes of locally convex topologies, which include and generalize various known ones such as weighted space-...
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We consider spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with classes of locally convex topologies, which include and generalize various known ones such as weighted space- or inductive limit-type topologies. The main result states that every continuous linear functional on such a function space can be expressed as an integral with respect to some canonical (dual space-valued) vector measure.
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