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1.
S. Simons 《Set-Valued Analysis》1996,4(3):271-282
Let E be a real Banach space with dual E
*. We associate with any nonempty subset H of E×E
* a certain compact convex subset of the first quadrant in 2, which we call the picture of H, (H). In general, (H) may be empty, but (M) is nonempty if M is a nonempty monotone subset of E×E
*. If E is reflexive and M is maximal monotone then (M) is a single point on the diagonal of the first quadrant of 2. On the other hand, we give an example (for E the nonreflexive space L
1[0,1]) of a maximal monotone subset M of E×E
* such that (0,1)(M) and (1,1)(M) but (1,0)(M). We show that the results for reflexive spaces can be recovered for general Banach spaces by using monotone operator of type (NI) — a class of multifunctions from E into E
* which includes the subdifferentials of all proper, convex, lower semicontinuous functions on E, all surjective operators and, if E is reflexive, all maximal monotone operators. Our results lead to a simple proof of Rockafellar's result that if E is reflexive and S is maximal monotone on E then S+J is surjective. Our main tool is a classical minimax theorem. 相似文献
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S. Simons 《Transactions of the American Mathematical Society》1998,350(7):2973-2980
This note is an addendum to Sum theorems for monotone operators and convex functions. In it, we prove some new results on convex functions and monotone operators, and use them to show that several of the constraint qualifications considered in the preceding paper are, in fact, equivalent.
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Prof. Dr. Heinz Bauer 《Mathematische Zeitschrift》1974,136(4):315-330
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Nazanin Azarnia 《Rendiconti del Circolo Matematico di Palermo》1985,34(1):105-110
It is shown that when domain of the modular operator associated with a center-valued weight satisfies certain density condition, the algebra is reduced to the product of a semi-finite algebra and a finite number of properly infinite factors. 相似文献
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《Optimization》2012,61(11):2071-2087
ABSTRACTIn this paper, we deal with three aspects of p-cyclically monotone operators. First, we introduce a notion of monotone polar adapted for p-cyclically monotone operators and study these kinds of operators with a unique maximal extension (called pre-maximal), and with a convex graph. We then deal with linear operators and provide characterizations of p-cyclical monotonicity and maximal p-cyclical monotonicity. Finally, we show that the Brézis-Browder theorem preserves p-cyclical monotonicity in reflexive Banach spaces. 相似文献
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Let X be a Banach space, X1 its dual, and Ω a measurable space. We study the solvability of nonlinear random equations involving operators of the form L + T, where L is a maximal monotone random operator from Ω × X into X1 and T : Ω × X → X1 a random operator of monotone type. 相似文献
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Muhammad Aslam Noor Khalida Inayat Noor Abdelouahed Hamdi Eman H. El-Shemas 《Optimization Letters》2009,3(3):329-335
In this paper, we introduce and consider the problem of finding zeroes of difference of two monotone operators in a Hilbert
space. Using the resolvent operator technique, we show that this problem is equivalent to the fixed point problem. This equivalence
is used to suggest and analyze an iterative method for finding a zero of difference of two monotone operators. We also discuss
the convergence of the iterative method under suitable conditions. Our method of proof is very simple as compared with other
techniques. 相似文献
10.
Eric Schechter 《Israel Journal of Mathematics》1982,43(1):49-61
We consideru′(t)+Au(t)∋f(t), whereA is maximal monotone in a Hilbert spaceH. AssumeA is continuous or A=ϱφ or intD(A)≠∅ or dimH<∞. For suitably boundedf′s, it is shown that the solution mapf→u is continuous, even if thef′s are topologized much more weakly than usual. As a corollary we obtain the existence of solutions ofu′(t)+Au(t)∋B(u(t)), whereB is a compact mapping inH.
An erratum to this article is available at . 相似文献
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Monotone operators are of central importance in modern optimization and nonlinear analysis. Their study has been revolutionized
lately, due to the systematic use of the Fitzpatrick function. Pioneered by Penot and Svaiter, a topic of recent interest
has been the representation of maximal monotone operators by so-called autoconjugate functions. Two explicit constructions
were proposed, the first by Penot and Zălinescu in 2005, and another by Bauschke and Wang in 2007. The former requires a mild
constraint qualification while the latter is based on the proximal average. We show that these two autoconjugate representers
must coincide for continuous linear monotone operators on reflexive spaces. The continuity and the linearity assumption are
both essential as examples of discontinuous linear operators and of subdifferential operators illustrate. Furthermore, we
also construct an infinite family of autoconjugate representers for the identity operator on the real line. 相似文献
14.
Yong-Zhuo Chen 《Journal of Mathematical Analysis and Applications》2004,291(1):282-291
In this paper, we use the monotone iterative method to prove the existence of the minimal and maximal fixed points of a discontinuous strongly monotone operator on an order interval in an ordered normed linear space. As an example of the application of our results, we show the existence of extremal solutions to a class of discontinuous initial value problems. 相似文献
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We prove some new results on the convergence of variational inequalities for monotone operators, when both the operator and
the obstacle are perturbed. 相似文献
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Summary Maximal monotone operatorsT:X→2
y such that {Tx}
xεx
is a finite family of sets are shown to be cyclically monotone. 相似文献
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《Optimization》2012,61(4):487-504
Maximally monotone operators play a key role in modern optimization and variational analysis. Two useful subclasses are rectangular (also known as star monotone) and paramonotone operators, which were introduced by Brezis and Haraux, and by Censor, Iusem and Zenios, respectively. The former class has a useful range of properties while the latter class is of importance for interior point methods and duality theory. Both notions are automatic for subdifferential operators and known to coincide for certain matrices; however, more precise relationships between rectangularity and paramonotonicity were not known. Our aim is to provide new results and examples concerning these notions. It is shown that rectangularity and paramonotonicity are actually independent. Moreover, for linear relations, rectangularity implies paramonotonicity but the converse implication requires additional assumptions. We also consider continuous linear monotone operators, and we point out that in the Hilbert space both notions are automatic for certain displacement mappings. 相似文献
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