Fixed points for operators in a space of continuous functions and applications

This paper investigates the fixed points for self-maps of a closed set in a space of abstract continuous functions. Our main results essentially extend the Banach contracting mapping principle. An application to integro-differential equations is given.

     

Dispersion points and continuous functions

The aim of this paper is to answer the following question raised by J. Cobb and W. Voxman in 1980:If X is a connected space with a dispersion point p, and if ƒ: X→X is a nonconstant continuous function, then is ƒ(p)=p?The answer to this question is negative, and we give a counterexample along with three theorems on a space with a dispersion point.     

Characterizations of Steiner points

To each convex compact A in Euclidian space Eⁿ there corresponds a point S (A) from Eⁿ such that 1) S(x) = x for x Eⁿ, 2) S(A + B) = S(A) + S(B), 3) S (A_i) , if A_i converges in the Hausdorff metric to the unit sphere in Eⁿ, then S(A) is the Steiner point of the set A. The theorem improves certain earlier results on characterizations of the Steiner point.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 243–247, August, 1973.In conclusion, I wish to express my appreciation to E. M. Semenov for his constant help with this work.     

Haar ambivalent sets in the space of continuous functions

It is a well-known result of Solecki that every nonlocally compact Polish group with a two-sided invariant metric contains continuum many pairwise disjoint sets which are Haar ambivalent. Inspired by a recent result of Zajíček, we give such an explicit and natural collection in the space of continuous functions on the interval. Our collection involves functions which have infinite derivatives and knot points.     

Formally continuous functions on Baire space

A function from Baire space to the natural numbers is called formally continuous if it is induced by a morphism between the corresponding formal spaces. We compare formal continuity to two other notions of continuity on Baire space working in Bishop constructive mathematics: one is a function induced by a Brouwer‐operation (i.e., inductively defined neighbourhood function); the other is a function uniformly continuous near every compact image. We show that formal continuity is equivalent to the former while it is strictly stronger than the latter. The equivalence of formally continuous functions and those induced by Brouwer‐operations requires Countable Choice.     

On the density of the space of continuous and uniformly continuous functions

For X a metrizable space and (Y,ρ) a metric space, with Y pathwise connected, we compute the density of (C(X,(Y,ρ)),σ)—the space of all continuous functions from X to (Y,ρ), endowed with the supremum metric σ. Also, for (X,d) a metric space and (Y,‖⋅‖) a normed space, we compute the density of (UC((X,d),(Y,ρ)),σ) (the space of all uniformly continuous functions from (X,d) to (Y,ρ), where ρ is the metric induced on Y by ‖⋅‖). We also prove that the latter result extends only partially to the case where (Y,ρ) is an arbitrary pathwise connected metric space.To carry such an investigation out, the notions of generalized compact and generalized totally bounded metric space, introduced by the author and A. Barbati in a former paper, turn out to play a crucial rôle. Moreover, we show that the first-mentioned concept provides a precise characterization of those metrizable spaces which attain their extent.     

Diameters of classes of continuous functions in the Lp space

In the L_p(a, b) space the exact values of n-diameters (n=1, 2, ...) are found of the class H[a, b] of the functions f(x) such that ¦f(x)-f(x)¦ (¦x-x¦), where (t) is a given continuity module which is convex upwards.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 493–500, November, 1971.     

The mapping taking three points of a Banach space to their Steiner point

A mapping St taking any three points a, b, c of a Banach space X into a set St(a,b,c) of their Steiner points and the corresponding operator P_D of metric projection of a space X × X × X onto its diagonal subspace D = {(x,x,x): x ∈ X}, P_D(a,b,c) = {(s,s,s): s ∈ St(a,b,c)}, are considered. The linearity coefficient of an arbitrary selection from P_D is estimated depending on properties of the space X. Estimates for the Lipschitz constant of an arbitrary selection from the mapping St are obtained as a corollary.     

Supremally generating cones of the space of continuous functions

Translated from Matematicheskie Zametki, Vol. 46, No. 6, pp. 46–52, December, 1989.     

Fixed points of nonexpansive mappings in spaces of continuous functions

Let be a compact metrizable space and let be the Banach space of all real continuous functions defined on with the maximum norm. It is known that fails to have the weak fixed point property for nonexpansive mappings (w-FPP) when contains a perfect set. However the space , where and is the first infinite ordinal number, enjoys the w-FPP, and so also satisfies this property if . It is unknown if has the w-FPP when is a scattered set such that . In this paper we prove that certain subspaces of , with , satisfy the w-FPP. To prove this result we introduce the notion of -almost weak orthogonality and we prove that an -almost weakly orthogonal closed subspace of enjoys the w-FPP. We show an example of an -almost weakly orthogonal subspace of which is not contained in for any .

     

On some measures of noncompactness in the space of continuous functions

We consider some quantities in the space of functions continuous on a bounded interval, which are related to monotonicity of functions. Based on those quantities we construct a few measures of noncompactness in the mentioned function space. Several properties of those measures are established; among others it is shown that they are regular or “partly” regular measures and equivalent to the Hausdorff measure of noncompactness.     

Uniformly continuous superposition operators in the space of bounded variation functions

Let I, J ? ? be intervals. The main result says that if a superposition operator H generated by a function of two variables h: I × J → ?, H (φ)(x) ? h (x, φ (x)), maps the set BV (I, J) of all bounded variation functions, φ: I → J into the Banach space BV (I, ?) and is uniformly continuous with respect to the BV ‐norm, then h (x, y) = a (x)y + b (x), x ∈ I, y ∈ J, for some a, b ∈ BV (I, ?) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

One functional equation with displacement in the space of continuous functions

In the space of continuous functions defined on a simple continuous contour, we examine the functional equationa(t)(t)+b(t)[(t)]=g(t) A criterion for Eq. (1) being Noetherian is established under the condition that there exist a finite number of fixed points on the first multiplicity in the homeomorphism (t) of the contour onto itself.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 303–311, August, 1977.In conclusion, the author thanks G. S. Litvinchuk for guidance on the work.     

On unconditionally convergent basis expansions in the space of continuous functions

[0,1], - H .

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This paper was written during the author's scholarship at the State University of Odessa in the USSR.     

On superpositions of continuous functions defined on the baire space

For continuous functions specified on the Baire space, conditions for the representability of a function of several variables as a superposition of functions of a smaller number of variables are considered. With the use of linear functions of the form (1+α)t, a boundary value of the modulus of continuity separating the positive from the negative solution of the problem is found. For the case in which the problem has a negative solution, a constructive method for obtaining (n+1)-variable continuous functions with modulus of continuity ϕ(t) that are not representable as superposition ofn-variable continuous functions with the same modulus of continuity ϕ(t) is suggested. Translated fromMatermaticheskie Zametki, Vol. 66, No. 5, pp. 696–705, November, 1999.     

Combinatorial bounds on Hilbert functions of fat points in projective space

We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in , in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using nothing more than the multiplicities of the points and information about which subsets of the points are linearly dependent. When N=2, we give these bounds explicitly and we give a sufficient criterion for the upper and lower bounds to be equal. When this criterion is satisfied, we give both a simple formula for the Hilbert function and combinatorially defined upper and lower bounds on the graded Betti numbers for the ideal I_A defining A, generalizing results of Geramita et al. (2006) [16]. We obtain the exact Hilbert functions and graded Betti numbers for many families of examples, interesting combinatorially, geometrically, and algebraically. Our method works in any characteristic.