Abstract
The following generalization of the Browder–Göhde–Kirk fixed point theorem is proved: if C is a nonempty bounded closed and convex subset of a uniformly convex normed space X and T is a self-mapping of C such that \(\left\| Tx-Ty\right\| \le \beta \left( \left\| x-y\right\| \right) \) for all \(x,y\in C,\) \(x\ne y,\) where a function \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) is such that \( \lim _{t\rightarrow 0+}\frac{\beta \left( t\right) }{t}=1,\) then T has a fixed point. Two modifications of this theorem as well as some accompanying results on Lipschitz-type mappings are given. An application in the theory of \(L^{p}\)-solutions of an iterative functional equation, and some refinements of the Radamacher theorem are proposed.
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1 Introduction
Basing on an observation that every continuous map of a convex set satisfying a restricted Lipschitz condition must be Lipschitz continuous (see [12]), we present some generalizations of Browder–Göhde–Kirk fixed point (Browder [1], Göhde [7], Kirk [9], see also [4, 6, 15, 16]), and propose their applications, including an extension of the classical Radamacher theorem.
Section 2 contains the auxiliary results characterizing the Lipschitz continuous functions with the aid of some weaker conditions.
In Sect. 3, we present two fixed point theorems. Let X be a uniformly convex Banach space, \(C\subset X\) a nonempty bounded convex closed set, and T a selfmapping of C. Theorem 1 says that T has a fixed point, if for some function \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) satisfying the conditions
we have
The second result, Theorem 2, says that the last inequality can be significantly weakened, namely, T has a fixed point, if T is continuous and, for a certain function \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) and a zero sequence \(\left( t_{n}\right) \) of positive numbers such that
we have, for all \(n\in N\) and for all \(x,y\in C,\)
In Sect. 4, we use Theorem 1 to get a result on the existence and uniqueness of \(L^{p}\)-solutions (\(1<p<+\infty \)) of the iterative functional equation
In Sect. 5, we give some refinements of the classical Radamacher theorem on the differentiability of the Lipschitz mappings.
2 Some auxiliary results on Lipschitz continuity
We begin with the following
Lemma 1
Let X, Y be normed spaces, \(C\subset X\) a convex set, \(T:C\rightarrow Y\) a mapping, and \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) a real function such that
If
then
where
Proof
Note that conditions (2) and (1) imply that T is continuous. Indeed, from (2) there are some real positive M and \(\delta \) such that \(\beta \left( t\right) \le Mt\) for all \(t\in \left( 0,\delta \right) ,\) and (1) implies that \(\left\| Tx-Ty\right\| \le M\left\| x-y\right\| \) for all \(x,y\in C\) such that \(\left\| x-y\right\| <\delta .\)
Take arbitrary \(x,y\in C,\) \(x\ne y.\) By (3), for every \(\varepsilon >0\) there is a \(t_{\varepsilon }>0\) and a unique \(n=n_{\varepsilon }\in {\mathbb {N}}_{0}\) such that
and
Put
Then, by the convexity of C,
clearly
and, by (5),
Hence, applying in turn: the triangle inequality, condition (1), some obvious identities, (4), (7) and (5), we get
that is
Since the continuity of T and the conditions (6) and (8) imply that
letting \(\varepsilon \rightarrow 0\) in the above inequality, we obtain
which completes the proof. \(\square \)
Remark 1
If \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) is subadditive, then
(see [8, p. 250, Theorem 7.11.1], also [11]). In this case, instead of (3) it is enough to assume that \(L<+\infty .\)
Remark 2
The reasoning in the proof of Lemma 1 simplifies, if
Indeed, if this condition holds, then for every \(\varepsilon >0\) there is a \( \delta >0\) such that
Take arbitrary \(x,y\in C,\) \(x\ne y,\) choose \(n\in {\mathbb {N}}\) such that
and put
Of course
and, by the convexity of C,
Applying the triangle inequality and (1), we hence get
that is \(\left\| Tx-Ty\right\| \le \left( L+\varepsilon \right) \left\| x-y\right\| .\) Since \(\varepsilon >0\) is chosen arbitrarily, letting \(\varepsilon \rightarrow 0,\) we conclude that T is L-Lipschitzian.
A much weaker necessary and sufficient condition for a continuous map to be Lipschitz continuous gives the following
Lemma 2
Let X and Y be real normed spaces and \(C\subset X\) a bounded convex set. Suppose that \(T:C\rightarrow Y\) is continuous. If there are a nonnegative real L and two positive sequences \(\left( t_{n}\right) ,\left( c_{n}\right) ,\)
such that for every \(n\in {\mathbb {N}}\) and for all \(x,y\in C,\)
then T is Lipschitz continuous, and
Proof
Take arbitrary \(x,y\in C,\) \(x\ne y.\) For every \(n\in {\mathbb {N}}\), there is a unique \(m_{n}\in {\mathbb {N}}\cup \left\{ 0\right\} \) such that
Put
Since
and, for each \(k=0,1,\ldots ,m_{n},\)
the convexity of C implies that
Moreover, by (10),
and, for \(k=m_{n},\)
we have
so, by the triangle inequality,
whence, taking into account that \(m_{n}t_{n}\le \left\| x-y\right\| \), by (12), we get
Since, by (12), \(\left\| z_{m_{n}}-y\right\| <t_{n},\) we have
and, in view of the assumed continuity of T,
Hence, letting \(n\rightarrow \infty \) in (13), and taking into account that \( \lim _{n\rightarrow \infty }c_{n}=L,\) we conclude that
which was to be shown. \(\square \)
3 Fixed-point theorems
Recall that a real normed vector space \(\left( X,\left\| \cdot \right\| \right) \) is called uniformly convex, if for every \( \varepsilon \in \left( 0,2\right] \) there is some \(\delta >0\) such that for any two vectors \(x,y\in X\) with \(\left\| x\right\| =\left\| y\right\| =1,\) the condition \(\left\| x-y\right\| \ge \varepsilon \) implies that \(\left\| \frac{x+y}{2}\right\| \le 1-\delta \) (Goebel and Reich [6]; see also [13] for a generalization).
Applying Lemma 1 with \(L=1\) we obtain the following generalization of the Browder–Göhde–Kirk theorem.
Theorem 1
Let X be a uniformly convex Banach space, \(C\subset X\) a nonempty bounded convex closed set and T a selfmapping of C. If there exists a function \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) such that
and
then T has a fixed point in C.
Proof
Applying Lemma 1 with \(L=1,\) we get
that is T is nonexpansive, and the result follows from the original version of the Browder–Göhde–Kirk theorem. \(\square \)
In particular, the thesis of Browder–Göhde–Kirk theorem remains true, if the nonexpansivity of the mapping T is replaced for instance, by the inequality
Lemma 2 and the Browder–Göhde–Kirk theorem yield the following:
Proposition 1
Let X be a uniformly convex Banach space and \(C\subset X\) a nonempty bounded closed and convex set. Suppose that \(T:C\rightarrow C\) is continuous. If there exist two positive sequences \(\left( t_{n}\right) ,\left( c_{n}\right) ,\)
such that for every \(n\in {\mathbb {N}}\) and for all \(x,y\in C,\)
then T has a fixed point.
This proposition improves the relevant result in [11] where the uniform continuity of T is assumed.
The main result of this section reads as follows.
Theorem 2
Let X be a uniformly convex Banach space and \(C\subset X\) a nonempty bounded closed and convex set. Suppose that \(T:C\rightarrow C\) is continuous. If there exist a function \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) and a sequence of positive real \( \left( t_{n}\right) ,\) \(\lim _{n\rightarrow \infty }t_{n}=0\) satisfying the condition
such that for every \(n\in {\mathbb {N}}\) and for all \(x,y\in C,\)
then T has a fixed point.
Proof
Setting \(c_{n}:=\frac{\beta \left( t_{n}\right) }{t_{n}}\) we have \( \lim _{n\rightarrow \infty }c_{n}=1\) and for every \(n\in {\mathbb {N}}\) and for all \(x,y\in C,\) if \(\left\| x-y\right\| =t_{n},\) then
and the result follows from Proposition 1. \(\square \)
4 An application in the theory of iterative functional equations
For a measure space \(\left( \Omega ,\Sigma ,\mu \right) \) and a real \(p>1,\) denote by \(\left( L^{p}\left( \Omega \right) ,\left\| \cdot \right\| _{p}\right) \) the Banach space of all (equivalence classes with respect to the \(\mu \)-a.e. equality) of \(\Sigma \)-measurable functions \(\varphi :\Omega \rightarrow {\mathbb {R}}\) such that \(\left| \varphi \right| ^{p}\) is \( \mu \)-integrable, and
It is well known that \(\left( L^{p}\left( \Omega \right) ,\left\| \cdot \right\| _{p}\right) \) is a uniformly convex Banach space (Clarkson [3]).
In this section, we consider solutions \(\varphi \in L^{p}\left( \Omega \right) \) of the iterative-type functional equation
We assume that \(\left( \Omega ,\Sigma ,\mu \right) \) the given functions f and h satisfy the following conditions:
-
(i)
\(k\in {\mathbb {N}}\); \(\Omega \subset {\mathbb {R}}^{k}\) is an open set; \(\mu \) is the Lebesgue measure, \(\mu \left( \Omega \right) =1;\) and \(f:\Omega \rightarrow \Omega ,\) \(f=\left( f_{1},\ldots ,f_{k}\right) ,\) is a locally Lipschitzian homeomorphic mapping;
-
(ii)
\(h:\Omega \times \mathbb {R\rightarrow R}\) is such that: for every \(y\in R\) the function \(\Omega \ni x\longmapsto h\left( x,y\right) \) is Lebesgue measurable, and \({\mathbb {R}}\ni y\longmapsto h\left( x,y\right) \) is continuous for almost all \( x\in \Omega \) (with respect to \(\mu );\)
-
(iii)
\(p\in {\mathbb {R}},\) \(p>1,\) and there are \(g_{1},g_{2}\in L^{p}\left( \Omega \right) ,\) \(g_{1}\le g_{2}\) a.e. in \(\Omega \) such that for all \(x\in \Omega \) and \(y\in {\mathbb {R}},\) the following implication holds true:
Applying Theorem 1, we prove the following:
Theorem 3
Let conditions (i)–(iii) be satisfied. Assume that there are a Lebesgue measurable function \(\alpha :\Omega \rightarrow \left[ 0,\infty \right) ,\) and a function \(\beta :\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) such that
where, for \(f=\left( f_{1},\ldots ,f_{k}\right) \) and \(x=\left( x_{1},\ldots ,x_{k}\right) ,\) the symbol \(J_{f}\left( x\right) :=\frac{\partial \left( f_{1},\ldots ,f_{k}\right) }{\partial \left( x_{1},\ldots ,x_{k}\right) }\) stands for the Jacobian of f;
the function
and
Then there exists \(\varphi \in L^{p}\left( \Omega \right) ,\) \(g_{1}\le \varphi \le g_{2}\) a.e. in \(\Omega ,\) such that
moreover, if \(\beta \left( t_{n}\right) \ne t_{n}\) for a sequence of \( t_{n}>0,\) \(\lim _{n\rightarrow \infty }t_{n}=0,\) then such \(\varphi \) is unique, and for any \(\varphi _{0}\in L^{p}\left( \Omega \right) ,\) \( g_{1}\le \varphi _{0}\le g_{2}\) a.e. in \(\Omega ,\) the sequence \(\left( \varphi _{n}\right) \) defined recursively by
converges to \(\varphi \) in the norm \(\left\| \cdot \right\| _{p}.\)
Proof
Put
It is easy to see that C is a nonempty, convex and closed subset of \( L^{p}\left( \Omega \right) .\) Define the mapping T on C by
Take an arbitrary \(\varphi \in C.\) Then, in view of Carathéodory theorem [2], conditions (ii) imply that the function \( T\left( \varphi \right) \) is Lebesgue measurable. Since \(g_{1}\le \varphi \le g_{2}\) a.e. in \(\Omega \) we have, for a.e. \(x\in \Omega \)
whence, in view of condition (iii),
for a.e. \(x\in \Omega ,\) that is \(T\left( \varphi \right) \in C,\) which proves that T maps C into itself.
Take arbitrary \(\varphi _{1},\varphi _{2}\in C.\) Making use in turn of: the definition of T; (14) (we use here the measurability of \(\alpha \) and \(\beta \)); (15); the theorem on change of the variables under integral (see Łojasiewicz [10]), the inclusion \(f\left( \Omega \right) \subset \Omega ;\) an obvious equality; the assumption that the Lebesgue measure of \(\Omega \) is 1 and the Jensen integral inequality for the concave function (16); and the definition of the norm \(\left\| \cdot \right\| _{p},\) we obtain
which, taking into account (17), proves that T satisfies the conditions of Theorem 1. Since \(\left( L^{p}\left( \Omega \right) ,\left\| \cdot \right\| _{p}\right) \) is a uniformly convex Banach space, all the assumptions of Theorem 1 are satisfied. Consequently, there is a function \( \varphi \in C\) such that \(\varphi =T\left( \varphi \right) .\)
If \(\beta \left( t_{n}\right) \ne t_{n}\) for a sequence of \(t_{n}>0\) such that \(\lim _{n\rightarrow \infty }t_{n}=0,\) then, by the concavity of \(\beta ,\) it is increasing and
Since \(\lim _{n\rightarrow \infty }\beta ^{n}\left( t\right) =0\) for every \( t>0,\) the “moreover” result follows from Theorem 1.2 in [14]. \(\square \)
In one-dimensional case, if \(\beta =\hbox {id}|_{\left[ 0,\infty \right) }\) and \(p\ge 1,\) the theory of \(L^{p}\)-solutions of the considered functional equation simplifies. Namely, from [14], Corollary 3.1 and Theorem 3.2, we have the following
Remark 3
[14] Let \(\Omega =\left( 0,a\right) \) where \(0<a\le \infty ,\) and \(p\ge 1.\) Assume that:
\(f:\Omega \rightarrow \Omega \) is absolutely continuous and
\(h:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is such that for every \(y\in {\mathbb {R}}\) the function \(\Omega \ni x\longmapsto h\left( x,y\right) \) is Lebesgue measurable, the function \({\mathbb {R}}\ni y\longmapsto h\left( x,y\right) \) is continuous for almost all \(x\in \Omega ;\)
moreover, for some \(x_{0}\in \Omega \) and a function \(\alpha :\left( 0,x_{0}\right) \rightarrow \left[ 0,\infty \right) \); we have
Then
-
(a)
if for some \(x_{0}\in \Omega \) we have
$$\begin{aligned} \alpha ^{p}\left( x\right) \le f^{\prime }\left( x\right) \quad \text {a.e. in }\left( 0,x_{0}\right) , \end{aligned}$$then there exists at most one solution \(\varphi \in L^{p}\left( \Omega \right) \) of (18);
-
(b)
if for some \(x_{0}\in \Omega \) and \(c\in \left[ 0,1\right) \) we have
$$\begin{aligned} \alpha ^{p}\left( x\right) \le cf^{\prime }\left( x\right) \quad \text {a.e. in }\left( 0,x_{0}\right) , \end{aligned}$$then there exists exactly one solution \(\varphi \in L^{p}\left( \Omega \right) \) of Eq. (18).
5 A refinement of Radamacher’s theorem
Applying Lemma 1 we obtain the following refinement of the classical Radamacher’s theorem (see, for instance [5, Theorem 3.1.6], or [10, p.161]).
Theorem 4
Let \(\Omega \subset {\mathbb {R}}^{k}\) be an open convex set and \(f:\Omega \rightarrow {\mathbb {R}}^{m}\) for some \(k,m\in {\mathbb {N}}.\) If there is a function \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) such that
and
where \(\left\| \cdot \right\| \) denotes the respective Euclidean norm, then f is differentiable almost everywhere in \(\Omega ;\) that is, the points in \(\Omega \) at which f is not differentiable form a set of Lebesgue measure zero.
Proof
Assume first that \(\Omega \) is convex. By Lemma 1, we have
where
In view of Radamacher’s theorem, the function f is differentiable almost everywhere in \(\Omega .\)
To end the proof, it is enough to note that every open set \(\Omega \subset {\mathbb {R}}^{k}\) is a countable sum of convex sets. \(\square \)
Similarly, making use of Lemma 2, we obtain the following improvement of Radamacher’s theorem.
Theorem 5
Let \(k,m\in {\mathbb {N}},\) \(\Omega \subset {\mathbb {R}}^{k}\) be an open convex set and \(f:\Omega \rightarrow {\mathbb {R}}^{m}\) be continuous. If for some function \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) there is a positive sequence \(\left( t_{n}\right) \) with \( \lim _{n\rightarrow \infty }t_{n}=0,\) such that
the function f is such that for every \(n\in {\mathbb {N}}\) and for all \( x,y\in \Omega ,\)
where \(\left\| \cdot \right\| \) denotes the Euclidean norm, then f is differentiable almost everywhere in \(\Omega .\)
The results of this section show that condition (i) in Theorem 3 can be replaced by a significantly weaker one.
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Matkowski, J. A refinement of the Browder–Göhde–Kirk fixed point theorem and some applications. J. Fixed Point Theory Appl. 24, 70 (2022). https://doi.org/10.1007/s11784-022-00985-2
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DOI: https://doi.org/10.1007/s11784-022-00985-2