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1.
In this paper, we investigate the continuities of the metric projection in a nonreflexive Banach space X, which improve the results in [X.N. Fang, J.H. Wang, Convexity and continuity of metric projection, Math. Appl. 14 (1) (2001) 47–51; P.D. Liu, Y.L. Hou, A convergence theorem of martingales in the limit, Northeast. Math. J. 6 (2) (1990) 227–234; H.J. Wang, Some results on the continuity of metric projections, Math. Appl. 8 (1) (1995) 80–84]. Under the assumption that X has some convexities, we discuss the relationship between approximative compactness of a subset A of X and continuity of the metric projection PA. We also give a representation theorem for the metric projection to a hyperplane in dual space X∗ and discuss its continuity. 相似文献
2.
Summary Under investigation are measures m defined on a σ-algebraA with range in a Banach space X having a Schauder basis {xn}. Utilizing the corresponding coefficient functionals and coefficient measures, we study the interaction between properties
for these measures and properties for the existing measure m. For the spaceCA(A, X) of all countably additive measures of finite variation fromA into X, we show that certain separable subspaces of it are isomorphic-isometric with certain separable subspaces of functions
of bounded variation on the interval [0, 1]. For {vn} inCA(A, X) such that {vn(A)} converges to v(A) for all A εA a certain ? control measure ? is constructed. An integral for set functions relative to a measure is defined and necessary
and sufficient conditions are given for weak convergence. Bounded operators on the space FAC(A,R) of finitelly additive scalar valued set functions onA which are absolutely continuous with respect to λ are considered. Necessary and sufficient conditions for continuity and
for compactness are given for Banach space valued operators T defined on such spaces. Weak compactness of T is also studied.
Finally, a different kind of representation is given for Banach space valued operators defined on the space FAv(A, X) of finitely additive set functions fromA into X.
Entrata in Redazione il 26 novembre 1976.
The first author expresses his gratitude to the Italian Consiglio Nazionale delle Ricerche and the Università degli Studi
di Param for the support of his work during his tenure as Visiting Professor of Mathematics. This work was supported by NATO
Research Grant No. 835. 相似文献
3.
In this paper we consider the operator equation in a real Banach space E with cone P:
where A = KF; here K is a e-positive, e-continuous and completely continuous operator, and F is a strictly increasing and continuous operator which is Fréchet differentiable at θ. Under certain conditions, we show that the operator equation has at least three solutions x1, x2, x3 such that x1 ∈ P, x2 ∈ (−P), x3 ∈ E\(P ∪ (−P)). Now since the third solution x3 ∈ E\(P ∪ (−P)), we call it a sign-changing solution. As an application of the main results, we investigate the existence of sign-changing
solutions for some three-point boundary value problem. 相似文献
4.
R. H. W. Hoppe 《Applied Mathematics and Optimization》1982,9(1):263-290
In this paper we are concerned with the approximate solution of time-optimal control problems in a nonreflexive Banach SpaceE by sequences of similar problems in Banach spacesE
n which are assumed to approximateE in a fairly general sense. The problems under consideration are such that the solution operator of the associated evolution equation is a strongly continuous holomorphic contraction semigroup and the class of controls is taken from the dual of the Phillips adjoint space with respect to the infinitesimal generator of that semigroup. The main object is to establish convergence of optimal controls, transition times and corresponding trajectories of the approximating control problems which can be done by means of some results from the theory of approximation of semigroups of operators. Finally, these abstract convergence results will be applied to time-optimal control problems arising from heat transfer and diffusion processes.Research supported in part by the Deutsche Forschungsgemeinschaft (DFG) 相似文献
5.
We prove versions of James' weak compactness theorem for polynomials and symmetric multilinear forms of finite type. We also
show that a Banach spaceX is reflexive if and only if it admits and equivalent norm such that there existsx
0≠0 inX and a weak-*-open subsetA of the dual space, satisfying thatx
*⊗x
0 attains its numerical radius. for eachx
* inA.
The first and third author were supported in part by D.G.E.S., project no. BFM 2000-1467. The second author was partially
supported by Junta de Andalucía Grant FQM0199. 相似文献
6.
R. Choukri A. El Kinani A. Oukhouya 《Rendiconti del Circolo Matematico di Palermo》2007,56(2):235-243
We characterize locally convex topological algebrasA satisfying: a sequence (x
n) inA converges to 0 if, and only if, (x
n
2) converges to 0. We also show that a real Banach algebra such thatx
n
2+y
n
2→0 if, and only if,x
n → 0 andy
n → 0, for every sequences (x
n) and (y
n) inA, is isomorphic to, whereX is a compact space.
相似文献
7.
Let E be a separable Banach space ordered by a reproducing cone with empty interior. We prove the existence of operator functions A : [0, ) P (P the cone of monotone increasing linear operators) and of initial values x0 such that the solution of x (t) = A(t) x(t), x(0) = x0, is dense in E.Received: 27 February 2004 相似文献
8.
Consider a real-valued bifunction f defined on C ×C, where C is a closed and convex subset of a Banach space X, which is concave in its first argument and convex in its second one. We study its subdifferential with respect to the second
argument, evaluated at pairs of the form (x,x), and the subdifferential of − f with respect to its first argument, evaluated at the same pairs. We prove that if f vanishes whenever both arguments coincide, these operators are maximal monotone, under rather undemanding continuity assumptions
on f. We also establish similar results under related assumptions on f, e.g. monotonicity and convexity in the second argument. These results were known for the case in which the Banach space
is reflexive and C = X. Here we use a different approach, based upon a recently established sufficient condition for maximal monotonicity of operators,
in order to cover the nonreflexive and constrained case (C ≠ X). Our results have consequences in terms of the reformulation of equilibrium problems as variational inequality ones. 相似文献
9.
In this paper we propose a modification of the von Neumann method of alternating projection x
k+1=P
A
P
B
x
k
where A,B are closed and convex subsets of a real Hilbert space ℋ. If Fix P
A
P
B
≠∅ then any sequence generated by the classical method converges weakly to a fixed point of the operator T=P
A
P
B
. If the distance δ=inf
x∈A,y∈B
‖
x−y
‖ is known then one can efficiently apply a modification of the von Neumann method, which has the form x
k+1=P
A
(x
k
+λ
k
(P
A
P
B
x
k
−x
k
)) for λ
k
>0 depending on x
k
(for details see: Cegielski and Suchocka, SIAM J. Optim. 19:1093–1106, 2008). Our paper contains a generalization of this modification, where we do not suppose that we know the value δ. Instead of δ we apply its approximation which is updated in each iteration. 相似文献
10.
Isaac Namioka conjectured that every nonreflexive Banach space can be renormed is such a way that, in the new norm, the set
of norm attaining functionals has an empty interior in the norm topology. We prove the rightness of this conjecture for spaces
containing an isomorphic copy of ℓ1. As a consequence, we prove also that the same result holds for a wide class of Banach spaces containing, for example, the
weakly compactly generated ones. 相似文献
11.
Summary We obtain several properties of the normed cone of semi-Lipschitz functions defined on a quasi-metric space (X,d) that vanish at a fixed point x0∈X. For instance, we prove that it is both bicomplete and right K-sequentially complete, and the unit ball is compact with respect to the topology of quasi-uniform convergence. Furthermore, it has a structure of a Banach space if and only if (X,d) is a metric space. 相似文献
12.
Kamil S. Kazimierski 《Computational Optimization and Applications》2011,48(2):309-324
For Tikhonov functionals of the form Ψ(x)=‖Ax−y‖
Y
r
+α‖x‖
X
q
we investigate a steepest descent method in the dual of the Banach space X. We show convergence rates for the proposed method and present numerical tests. 相似文献
13.
V. Yu. Protasov 《Functional Analysis and Its Applications》2011,45(1):46-55
We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x
1 + x
2) ⊂ φ(x
1) + φ(x
2). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: X → Y such that Ax ∈ φ(x), x ∈ X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is
proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces. 相似文献
14.
Many interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationPK(x) to anyxXfrom the setKC∩A−1(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andbY. The main point of this paper is to show thatPK(x)isidenticaltoPC(x+A*y)—the best approximationto a certain perturbationx+A*yofx—from the convexsetCor from a certain convex extremal subsetCbofC. Thelatter best approximation is generally much easier to computethan the former. Prior to this, the result had been known onlyin the case of a convex cone or forspecialdata sets associatedwith a closed convex set. In fact, we give anintrinsic characterizationof those pairs of setsCandA−1(b) for which this canalways be done. Finally, in many cases, the best approximationPC(x+A*y) can be obtained numerically from existingalgorithms or from modifications to existing algorithms. Wegive such an algorithm and prove its convergence 相似文献
15.
Huang Jianfeng Wang Yuanheng 《高校应用数学学报(英文版)》2007,22(3):311-315
This paper studies the convergence of the sequence defined by x0∈C,xn 1=αnu (1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results. 相似文献
16.
Let X be a real Banach space and let (f(n)) be a positive nondecreasing sequence. We consider systems of unit vectors (xi)∞i=1 in X which satisfy ∑iA±xi|A|−f(|A|), for all finite A
and for all choices of signs. We identify the spaces which contain such systems for bounded (f(n)) and for all unbounded (f(n)). For arbitrary unbounded (f(n)), we give examples of systems for which [xi] is H.I., and we exhibit systems in all isomorphs of ℓ1 which are not equivalent to the unit vector basis of ℓ1. We also prove that certain lacunary Haar systems in L1 are quasi-greedy basic sequences. 相似文献
17.
LetA be anm-accretive operator in a Banach spaceE. Suppose thatA
−10 is not empty and that bothE andE
* are uniformly convex. We study a general condition onA that guarantees the strong convergence of the semigroup generated by—A and of related implicit and explicit iterative schemes to a zero ofA. Rates of convergence are also obtained. In Hilbert space this condition has been recently introduced by A. Pazy. We also
establish strong convergence under the assumption that the interior ofA
−10 is not empty. In Hilbert space this result is due to H. Brezis.
Sponsored by the United States Army under Contract No. DAAG29-75-C-0024. 相似文献
18.
Jan A. van Casteren 《Journal of Evolution Equations》2011,11(2):457-476
Let
(tj)j ? \mathbbN{\left(\tau_j\right)_{j\in\mathbb{N}}} be a sequence of strictly positive real numbers, and let A be the generator of a bounded analytic semigroup in a Banach space X. Put
An=?j=1n(I+\frac12 tjA) (I-\frac12 tjA)-1{A_n=\prod_{j=1}^n\left(I+\frac{1}{2} \tau_jA\right) \left(I-\frac{1}{2} \tau_jA\right)^{-1}}, and let x ? X{x\in X}. Define the sequence
(xn)n ? \mathbbN ì X{\left(x_n\right)_{n\in\mathbb{N}}\subset X} by the Crank–Nicolson scheme: x
n
= A
n
x. In this paper, it is proved that the Crank–Nicolson scheme is stable in the sense that
supn ? \mathbbN||Anx|| < ¥{\sup_{n\in\mathbb{N}}\left\Vert A_nx\right\Vert<\infty}. Some convergence results are also given. 相似文献
19.
Guoxiang Chen Meiying Wang 《分析论及其应用》2007,23(3):266-273
For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family. 相似文献
20.
Erik Talvila 《Czechoslovak Mathematical Journal》2012,62(1):77-104
Let B
c
denote the real-valued functions continuous on the extended real line and vanishing at −∞. Let B
r
denote the functions that are left continuous, have a right limit at each point and vanish at −∞. Define A
c
n
to be the space of tempered distributions that are the nth distributional derivative of a unique function in B
c
. Similarly with A
r
n
from B
r
. A type of integral is defined on distributions in A
c
n
and A
r
n
. The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ, the spaces A
c
n
and A
r
n
are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to B
c
and B
r
, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable.
The dual space is isometrically isomorphic to the functions of bounded variation. The space A
c
1 is the completion of the L
1 functions in the Alexiewicz norm. The space A
r
1 contains all finite signed Borel measures. Many of the usual properties of integrals hold: H?lder inequality, second mean
value theorem, continuity in norm, linear change of variables, a convergence theorem. 相似文献