Rakotch type contractive maps, square roots and uniform convexity
- Antonio Jiménez-Melado^{1}Email author
https://doi.org/10.1186/s13663-015-0474-3
© Jiménez-Melado 2015
Received: 20 September 2015
Accepted: 25 November 2015
Published: 8 December 2015
Abstract
In this paper we introduce a family of weakly contractive maps on the space \(B(S)\) of bounded real valued functions and use it to show that a fundamental step in the proof of the well-known Stone-Weierstrass approximation theorem can be achieved via Rakotch’s fixed point theorem for weakly contractive maps. With the same technique, we obtain Zemanek’s theorem on the existence of square roots in certain Banach subalgebras of \(B(S)\), and, finally, in the context of abstract Banach algebras, we exhibit some relationship between weakly contractive maps on the closed unit ball and the geometry of the spheres.
Keywords
MSC
1 Introduction and preliminary results on weakly contractive maps
After this theorem, some authors tried to generalize it keeping the conclusions on the existence and uniqueness under more general hypotheses, and in many cases the maps satisfying the new conditions have been named weakly contractive. In this section we will be concerned with three of these conditions, due to Rakotch [3], Krasnosel’skii et al. [4] and Dugundji and Granas [5], which will be shown to be equivalent formulations of the same concept. There are some other conditions which are worth mentioning; for instance, the one introduced by Geraghty [6], which is still object of generalizations (see, for instance, Karapinar [7]), but they are not the subject of study in this paper.
Theorem 1
Let \((X,d)\) be a metric space, and D a nonempty subset of X. Then, for any map \(T:D\rightarrow X\), conditions (R), (K) and (D-G) are equivalent.
Proof
Observe also that inequality (R) is satisfied with the function α̅ because for any \(x,y\in D\), we have \(\alpha(x,y)\le\overline{\alpha}(d(x,y))\), and T satisfies condition (K) with the function α. Hence, T satisfies condition (R). □
The previous result allows us to say that \(T:D\subset X\rightarrow X\) is weakly contractive if it satisfies some (hence, all) of conditions (R), (K) or (D-G). Rakotch’s theorem [3] states that if \((X,d)\) is a complete metric space and \(T:X\rightarrow X\) is weakly contractive, then T has a unique fixed point in X, say \(x^{*}\), and for each \(x_{0}\in X\), the sequence of iterates \(\{ T^{n}(x_{0})\}\) converges to \(x^{*}\). Also, we can see that this theorem is a strict generalization of Banach-Caccioppoli’s theorem just considering the space \(X=[0,1]\) with the usual metric \(d(x,y)=\vert x-y\vert\) and the map \(T:X\rightarrow X\) given as \(T(x)=\sin(x)\), which is weakly contractive but not contractive.
In Section 2 we introduce a family of weakly contractive maps on the space \(B(S)\) of bounded real functions and show that Rakotch’s theorem can be used to prove a result of Zemanek [8] about the existence of square roots in some Banach subalgebras of \(B(S)\), a result which is a fundamental step in the proof of the well-known Stone-Weierstrass approximation theorem [9]. The use of Rakotch’s theorem in the proof of this theorem is also interesting because, for weakly contractive maps, it can be given a rate of convergence of the sequence of iterates (see [10]).
In Section 3 we show that there are some connections between weakly contractive maps, square roots in abstract algebras and the geometry of the unit ball.
2 Weakly contractive maps and square roots in \(B(S)\)
As we mentioned in the introduction, there are simple examples of maps that are weakly contractive and not contractive. In this section we provide a new example of this type, which is interesting because it existed before Rakotch’s theorem and appears in some proofs of the famous Stone-Weierstrass approximation theorem [9]. The aforementioned example is the map T defined on certain subset of the space \(\mathcal{C}([0,1])\) by \(T(f)(x)=\frac{1}{2}[1-x+f^{2}(x)]\), which is used in [11] to prove that the function \(\varphi(x)=\sqrt{x}\) can be uniformly approximated by polynomials (using Dini’s test for uniform convergence). In fact, a slight modification of this example will allow us to obtain the result of Zemanek [8] about the existence of square roots in certain Banach algebras. In the first place, we recall some notation and results that are necessary to understand the context which we shall work at.
If S is any nonempty set, then \(B(S)\) denotes the set of all bounded functions \(f:S\rightarrow\mathbb{R}\), which is known to be a Banach algebra with the pointwise multiplication and the norm \(\Vert\cdot \Vert_{\infty}\) (see, for instance, [11]). We also use the notation \(B^{+}(S)=\{ f\in B(S) : f(x)\ge0 \text{ for all } x\in S\}\) and, for a nonempty subset \(J\subset\mathbb{R}\), \(B(S,J)=\{ f\in B(S): f(x)\in J \text{ for all } x\in S\}\).
If S is a compact topological space, then it is also known that the space of continuous real functions \(\mathcal{C}(S)\) with its usual norm \(\Vert\cdot\Vert_{\infty}\) is a Banach subalgebra of \(B(S)\). Notice that, by elementary rules of calculus, the subalgebra \(\mathcal{C}(S)\) has the following property: for any \(f\in B^{+}(S)\cap\mathcal {C}(S)\), we have \(\sqrt{f} \in\mathcal{C}(S)\). The question of whether this property could be extended to abstract subalgebras of \(B(S)\) was answered by Zemanek [8] in 1977 by showing that the aforementioned property is satisfied by those subalgebras \(\mathcal{A}\) of \(B(S)\) that are complete and contain the constant functions. We obtain this result in Theorem 2 by using Rakotch’s theorem, and this will provide us with an extra information about how to obtain \(\sqrt{f}\) as a uniform limit of functions of \(\mathcal{A}\).
As a first step, in Lemma 1 we introduce a family of weakly contractive maps that will be used in Theorem 2.
Lemma 1
- (a)T is weakly contractive on D if, and only if, there exists a function \(\alpha:J\times J\rightarrow[0,1]\) compactly less than 1 such thatfor every \(u,v\in J\) and every \(x\in S\);$$ \bigl\vert G(x,u)-G(x,v)\bigr\vert \le\alpha(u,v)\vert u-v \vert $$(2)
- (b)T is contractive if, and only if, there exists \(\alpha \in[0,1)\) such thatfor every \(u,v\in J\) and every \(x\in S\).$$ \bigl\vert G(x,u)-G(x,v)\bigr\vert \le\alpha \vert u-v\vert $$
Proof
Hence, we have proved that \(\Vert T(f)-T(g)\Vert_{\infty}\le\beta (f,g)\Vert f-g\Vert_{\infty}\).
Theorem 2
- (a)
\(\sqrt{ g^{*}}\in\mathcal{A}\), and
- (b)the sequence of functions \(\{ g_{n}\}\) given as \(g_{0}(x)=0\) andis a sequence in \(\mathcal{A}\) that converges uniformly on S to \(\sqrt{g^{*}}\).$$ g_{n}(x)= g_{n-1}(x)+\frac{1}{2\sqrt{M}} \bigl[ g^{*}(x)-g_{n-1}^{2}(x) \bigr],\quad n\ge1 , $$
Proof
Suppose that \(g^{*}\in B^{+}(S)\cap\mathcal{A}\) with \(M=\Vert g^{*}\Vert_{\infty}>0\). Observe that since \(\mathcal{A}\) contains the constant functions, \(\sqrt {g^{*}}\) belongs to A if, and only if, the function \(f^{*}\) given as \(f^{*}(x)=\sqrt{M} -\sqrt{g^{*}(x)}\) is in \(\mathcal{A}\). It is also obvious that the sequence \(\{ g_{n}\}\) is in \(\mathcal{A}\) and converges uniformly to \(\sqrt{g^{*}}\) if, and only if, the sequence \(\{ f_{n}\}\) defined as \(f_{n}=\sqrt{M} - g_{n}\) is in \(\mathcal{A}\) and converges uniformly to \(f^{*}\).
Hence, by Rakotch’s theorem [3], T has a unique fixed point in K, and, for any \(h\in K\), the sequence of iterates \(\{ T^{n}(h)\}\) converges uniformly on S to this fixed point. Since \(K\subset B(S,J)\) and \(f^{*}\) is the unique fixed point for T in \(B(S,J)\), we conclude that \(f^{*}\in K\). Also, starting with \(h=f_{0}\in K\), the sequence \(\{ T^{n}(h)\}\) is just \(\{ f_{n}\}\), and so it converges uniformly to \(f^{*}\). □
Notice that Theorem 2 generalizes the following well-known result, which our operator T is inspired on.
Corollary 1
(Dieudonné [11])
The map \(T:K\rightarrow B(S)\) appearing in the previous theorem is of the form \(T(f)(x)=a(x)+F(f(x))\), where in our case \(a(x)=\frac {M-g^{*}(x)}{2\sqrt{M}}\) and \(F(u)=\frac{1}{2\sqrt{M}}u^{2}\). Applying Lemma 1, we obtained that T is weakly contractive on K if, and only if, F is weakly contractive on the interval J, which suggests the following question.
Question 1
Find an easy-to-check characterization of weakly contractive maps on the real line.
3 Weakly contractive maps and the geometry of spheres in abstract algebras
As we mentioned before, Zemanek [8] proved that if \(\mathcal{A}\) is a complete subalgebra of \(B(S)\) that contains the constant functions, then for any \(g\in B^{+}(S)\cap\mathcal{A}\), we have \(\sqrt{g} \in\mathcal{A}\). In his proof, Zemanek used the Banach-Caccioppoli theorem and was inspired by a previous paper by Bonsall and Stirling [12] about the existence of square roots in abstract algebras. Specifically, the following result was proved in [12]: if \(\mathcal{A}\) is a real or complex Banach algebra, then for each \(a\in \mathcal{A}\) with \(\Vert a\Vert<1\), there exists a unique \(x\in \mathcal{A}\) such that \(\Vert x\Vert<1\) and \(x=\frac{1}{2}(a+x^{2})\). The proof of this fact is easy if we additionally assume that \(\mathcal{A}\) is commutative: just consider a positive number d with \(\Vert a\Vert\le d<1\) and check that the map \(T:\overline{B}(0,d)\rightarrow \overline{B}(0,d)\) given as \(T(x)=\frac{1}{2}(a+x^{2})\) is contractive.
In view of the above, it is natural to ask whether T is weakly contractive. In the next theorem we show that the answer is yes if \(\mathcal{A}\) is commutative and the norm on \(\mathcal{A}\) is a uniformly convex norm (that is, for each \(\varepsilon\in(0,2]\), there exists \(\delta>0\) such that for any \(x,y\in\overline{B}(0,1)\) with \(\Vert x-y\Vert\ge\varepsilon\), we have \(\Vert\frac{1}{2}(x+y)\Vert\le 1-\delta\)). Before that, we shall show that the notion of a function compactly less than 1 can be used to characterize the uniform convexity of a Banach space.
Lemma 2
A Banach space \((X,\Vert\cdot\Vert)\) is uniformly convex if, and only if, the map \(\alpha: \overline{B}(0,1)\times\overline{B}(0,1)\rightarrow[0,1]\) given as \(\alpha(x,y)=\Vert \frac {x+y}{2}\Vert \) is compactly less than 1.
Proof
Theorem 3
If \(\mathcal{A}\) is a commutative Banach algebra and the norm on \(\mathcal{A}\) is a uniformly convex norm, then for each \(a\in\overline{B}(0,1)\), the map \(T:\overline{B}(0,1)\rightarrow\overline{B}(0,1)\) given as \(T(x)=\frac{1}{2}(a+x^{2})\) is weakly contractive.
Proof
Finally, α is compactly less than 1 by Lemma 2. □
The converse of the theorem is not true, even under the presence of commutativity, as the following example shows.
Example 1
Consider the algebra \(\mathcal{A}=\mathcal{C}([0,1])\) with the pointwise multiplication and its usual norm \(\Vert\cdot\Vert _{\infty}\). Then, \(\mathcal{A}\) is a commutative Banach algebra whose norm is not uniformly convex, but, for any \(f_{0}\in\mathcal{A}\), the map \(T:\overline{B}(0,1)\rightarrow\mathcal{A}\) given as \(T(f)=\frac {1}{2}(f_{0}+f^{2})\) is weakly contractive.
Then, we have that \(\vert T(f)(x)-T(g)(x)\vert\le\beta(f,g)\Vert f-g\Vert_{\infty}\), and since x is arbitrary in \([0,1]\), (5) is satisfied.
Remark 1
Notice that, although the map T in the example is weakly contractive, the function \(\alpha(f,g)=\Vert \frac{1}{2}(f+g)\Vert \) is not compactly less than 1 on \(\overline{B}(0,1)\times\overline{B}(0,1)\), as it can be seen either directly or using Lemma 2.
We end this paper with some natural questions related to Theorem 3:
Question 2
Can we remove the commutativity assumption?
Question 3
Can we replace the uniform convexity by a weaker geometrical property?
Question 4
Which consequences (geometrical or not) can we obtain from the fact that the map T is weakly contractive?
Question 5
Can we extend Theorem 3 to the context of Banach spaces (using a different map T)?
Declarations
Acknowledgements
The work was partially supported by the Spanish (Grant MTM2014-52865-P) and regional Andalusian (Grants FQM210 and P06-FQM01504) Governments.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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