We study the class of compact convex subsets of a topological vector space which admits a strictly convex and lower semicontinuous function. Dec 16, 2019 we give various relations between lower upper semicontinuous functions and cliquish functions in a generalized topological space. Error bounds for lower semicontinuous functions in normed. Pdf points of upper and lower semicontinuity for multivalued. We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds with exponents. Jul 19, 2011 in this paper, we establish the weak lower semicontinuity of variational functionals with variable growth in variable exponent sobolev spaces. The paper presents various results studying the asymptotic behavior of a sequence of lower semicontinuous functions on a metric space.
We realize that the pointwise convergence is a too strong assumption to analyze the upper and lower semicontinuous limit of functions, and as a result continuity. On lower and upper semicontinuous functions springerlink. I we are now going to expand our treatment of functions a bit and look at the class of lower semicontinuous functions and the class of convexfunctions. Those topological spaces for which each upper semicontinuous function is the pointwise limit of a.
Also, we answer the open problems raised by korczakkubiak et al. I the function jx clearly has an absolute minimum over function fx jxjdoes. Pdf lower semicontinuous functions semantic scholar. Compact convex sets that admit a lower semicontinuous. Pdf a lower semicontinuous regularization for setvalued. The specific features and the structure of the terms of the sequence are used to construct appropriate quantities relevant in the verification of palaissmale. Pdf a note on strongly lower semicontinuous functions and. Im looking for a reference for the theorem saying that a realvalued lower upper semicontinuous function on any metric space can be reached as a pointwise limit by a nondecreasing nonincreasing sequence of continuous functions. There seems to be a lot of examples of such upper lower semicontinuous functions in geometry counting certain discrete invariants, especially those related to stratifications of spaces. A proof of the minimax theorem for the game played over a pair of compact metric spaces, in which the payoff is an upper or lower semicontinuous function. We also give several equivalent characterizations of lower semicontinuity. Maximal classes for lower and upper semicontinuous strong. Find, read and cite all the research you need on researchgate. Decomposition of topologies which characterize the upper and.
This topological notion plays an important role also for multifunctions. Minimax theorem for upper and lower semicontinuous payoffs rand. Approximation of semicontinuous functions by continuous. Itll be great to have the list extended in mathoverflow. Minimax principles for lower semicontinuous functions and. Find out information about lower semicontinuous function. We also establish new formulas about distances to some subspaces of continuous functions that generalize some classical results. The concept of lower semicontinuity introduced for scalar functions. More upperlower semicontinuous functions in algebraic. For an interesting class dbsc differences of bounded semicontinuous see. Then is an upper semicontinuous function, is a lower semicontinuous function, and we have that. The weak lower semicontinuity is interesting by itself and can be applied to obtain the existence of an equilibrium solution in nonlinear elasticity.
A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Lower semicontinuous function an overview sciencedirect topics. Chapter 4 semicontinuities of multifunctions and functions the notion of the continuity of functions is certainly well known to the reader. Semicontinuous functions and convexity jordan bell jordan. The following theorem is the main result of this section. Semicon tinuous functions in this book we use semicontinuous functions when providing sufficient conditions for a function to achieve its maximum on a compact set. It is easily shown that the set e0 is equal to the set e e lfx g t. And the indicator function of a closed set is upper semicontinuous. A basic fact in real analysis is that every realvalued function f admits a lower semicontinuous regularization f, defined by means of the lower limit of f. Some characterizations of error bound for non lower. We prove that a function is both lower and upper semicontinuous if and only if it is continuous. Weak lower semicontinuity of variational functionals with. An extended real valued function f is upper respectively, lower semicontinuous.
Uscbax will indicate the collections of all real valued functions which are low er semicontinuous, upper semicontinuous, lower semicontinuous bounded. I these functions gives us some new insights into how we can try to nd extreme values of functions even when there is no compactness. For this, we introduce an index of upper semioscillation and lower semioscillation. In particular, different coercivity properties are obtained extending and refining previous results.
We consider general integral functionals on the sobolev spaces of multiple valued functions, introduced by almgren. A geometric algorithm for approximating semicontinuous function. The fundamental notion of convergence for lower semicontinuous functions proceeds from the identification of such a function with its epigraph 11. X r has the closed graph if and only if for each x. As a result, a lower semicontinuous convex function is the pointwise supremum of the affine functions that it majorizes. Lower semicontinuous function an overview sciencedirect. Lower semicontinuous functions archive of formal proofs. Weak lower semicontinuous envelope of functionals defined on a. Pdf on error bounds for lower semicontinuous functions.
Abstract in this paper, we characterize the maximal additive and multiplicative classes for lower and upper semicontinuous strong swiatkowsk functions and lower and upper semicontinuous extra strong swiatkowski functions. On error bounds for lower semicontinuous functions. What is an example of a lower semicontinuous function that is. We discuss several concepts of continuity, weaker than lower semicontinuity, but.
Given a bounded below, lower semicontinuous function. Coercivity properties for sequences of lower semicontinuous. Second, a closed convex set is the intersection of the closed half spaces that contain it. We define the notions of lower and upper semicontinuity for functions from a metric space to the extended real line. We say that fis lower semi continuous at x 0 if for every 0 there exists 0 so that fx fx 0 1 whenever kx 0 xk 2 but not on h1 0 fernando farroni, raffaella giova and franc. Upper semicontinuous functions and the stone approximation. When the functional 7 satisfies suitable lower semicontinuity and coercivity. The following proposition implies that the closed strong slope and the closed global slope of the function f is the strong slope and the global slope of the function clf, respectively.
Pdf we investigate joint upper and lower semicontinuity of twovariable set valued functions. Pdf a new class of functions called strongly lower semicontinuous slsc functions is defined and its properties. Banach space topological space convex subset lower semicontinuous inverse image these keywords were added by machine and not by the authors. Semicontinuous microemulsion polymerization pdf free download. In mathematical analysis, semicontinuity or semicontinuity is a property of extended realvalued functions that is weaker than continuity. Some properties of semicontinuous functions and quasiuniform. Pdf we investigate joint upper and lower semicontinuity of twovariable setvalued functions.
Pdf characterization of lower semicontinuous convex functions. It is shown that a radially lower semicontinuous function f. Every lower semicontinuous real valued function defined on a metric space is the pointwise limit of an increasing sequence of continuous functions. Semicontinuous function encyclopedia of mathematics. An extended realvalued function f is upper respectively, lower semicontinuous at a point x 0 if, roughly speaking, the function values for arguments near x 0 are not much higher respectively, lower than fx 0. Abstract pdf 236 kb 2010 equivalent extensions to caristikirks fixed point theorem, ekelands variational principle, and takahashis minimization theorem. The function jxjclearly has an absolute minimum over function fx jxjdoes. Find out information about upper semicontinuous function. We also give several equivalent characterizations of lower.
Chapter 4 semicontinuities of multifunctions and functions. In the sequel, we denote the lower semicontinuous functions on a by lx, and the real valued continuous functions on a by cx. In particular, we prove that a function is lower semicontinuous if and only if its epigraph is a closed set. Distance to spaces of semicontinuous and continuous functions. These functions gives us some new insights into how we can try to nd extreme values of functions even when there is no compactness.
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