Abstract
Some results extending the Goebel–Kirk fixed point theorem for asymptotically nonexpansive mappings are presented.
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1 Introduction
In 1965, Browder [1], Göhde [5] and Kirk [6], independently, using some nonconstructive arguments, proved that every nonexpansive self-mapping of a closed convex and bounded subset of a uniformly convex Banach space has a fixed point. A nice elementary proof was given by Goebel [2] (see also [4, 11, 12]).
Let F be a self-mapping of a nonempty bounded closed and convex subset C of uniformly convex normed space, and a function \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) be such that \(\lim _{t\rightarrow 0+}\frac{\beta \left( t\right) }{t}=1\). In a recent paper [10], two generalizations of the Browder–Göhde–Kirk fixed point theorem are proved. In view of the first one: ifF satisfies the nonlinear Lipschitz-type inequality
thenF has a fixed point; and, in view of the second: if F is continuous and for a sequence \(\left( t_{n}\right) \) of positive real numbers such that \( \lim _{n\rightarrow \infty }t_{n}=0\) the implication
holds true for all \(x,y\in C,\) and \(n\in {\mathbb {N}}\) , then F has a fixed point.
In 1972, Goebel and Kirk [3] extended the Browder–Göhde–Kirk theorem to a more general class of asymptotically nonexpansive mappings. Let C be a subset of a Banach space. A transformation \( F:C\rightarrow C\) is said to be asymptotically nonexpansive (in the Goebel–Kirk sense), if for all \(x,y\in C\),
where \(F^{i}\) is the \(i\,{\textrm{th}}\) iterate of F and \(\left( k_{i}:i\in {\mathbb {N}}\right) \) is a sequence of real numbers such that \( \lim _{i\rightarrow \infty }k_{i}=1\). They proved, among others, that if C is a nonempty, closed, convex and bounded subset of a uniformly convex Banach space X, and \(F:C\rightarrow C\) is asymptotically nonexpansive, then F has a fixed point.
In the present paper, we give two extensions of this result of Goebel and Kirk for the class of nonlinear asymptotically nonexpansive mappings. Theorem 1 in Sect. 3 says, in particular, that the above result remains true for a mapping F such that for every \(i\in {\mathbb {N}}\) there exists a function \(\beta _{i}:\left( 0,\infty \right) \rightarrow \left( 0,\infty \right) \) satisfying the conditions
the limits
exist, and \(\lim _{i\rightarrow \infty }k_{i}\le 1\).
In the next section, we show that the nonlinear asymptotical nonexpansivity condition in Theorem 1 can be considerably weakened. Namely, the result remains true if F is continuous and there exists a positive sequence \( \left( t_{n}:n\in {\mathbb {N}}\right) \) with \(\lim _{n\rightarrow \infty }t_{n}=0\) such that the implication
holds true for all \(x,y\in C,\) and \(n\in {\mathbb {N}}\) (Theorem 2).
The proofs are based on some properties of mappings satisfying the nonlinear-type Lipschitz conditions (Sect. 2), and the original result of Goebel–Kirk theorem on the fixed point theorem for asymptotically nonexpansive mappings.
2 Some lemmas on Lipschitz-type mappings
Let us quote the following lemma from a recent paper [10] (see also [7, 8]).
Lemma 1
Let X, Y be normed spaces, \(C\subset X\) a convex set, \(F:C\rightarrow Y\) a mapping, and \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) a real function such that
If
then
where
It turns out that, if in this lemma the map F is continuous, the first global nonlinear Lipschitz-type condition on F can be significantly weakened. To show it, let us quote the following
Lemma 2
[8, 10] Let X and Y be real normed spaces and \( C\subset X\) a convex set. Suppose that \(F:C\rightarrow Y\) is continuous. If there are a nonnegative real k and two positive sequences \(\left( t_{n}:n\in {\mathbb {N}}\right) ,\left( c_{n}:n\in {\mathbb {N}}\right) ,\)
such that for every \(n\in {\mathbb {N}}\) and for all \(x,y\in C\),
then F is Lipschitz continuous, and
From Lemma 2, we obtain the following
Lemma 3
Let X and Y be real normed spaces and \(C\subset X\) a convex set. Suppose that \(F:C\rightarrow Y\) is continuous. If there exist a function \(\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) and a sequence of positive real numbers \(\left( t_{n}\right) \), \( \lim _{n\rightarrow \infty }t_{n}=0\) satisfying the condition
such that for every \(n\in {\mathbb {N}}\) and for all \(x,y\in C,\)
then F is Lipschitz continuous, and
Proof
Setting \(c_{n}:=\frac{\beta \left( t_{n}\right) }{t_{n}}\), we have \( \lim _{n\rightarrow \infty }c_{n}=k\). Since for all \(n\in {\mathbb {N}}\) and \( x,y\in C\), if \(\left\| x-y\right\| =t_{n}\), then
the result follows from Lemma 2. \(\square \)
3 A fixed point theorem for nonlinear asymptotically nonexpansive mappings
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 [4]; see also [9]).
Applying Lemma 1 with k replaced by \(k_{i}\) for \(i\in {\mathbb {N}}\), we obtain the following generalization of the Goebel–Kirk theorem.
Theorem 1
Let X be a uniformly convex Banach space, \(C\subset X\) a nonempty bounded convex closed set and\(\ F\) a self-mapping of C. Assume that F is nonlinear asymptotically nonexpansive, i.e., that for every \(i\in {\mathbb {N}}\), there exists a function \(\beta _{i}:\left( 0,\infty \right) \rightarrow \left( 0,\infty \right) \) such that
the sequence \(\left( k_{i}:i\in {\mathbb {N}}\right) \) defined by
converges and
Then,
(i) if
then F has a fixed point in C and the set of all fixed points of F is a closed convex subset of C;
(ii) if
then F has a unique fixed point in C.
Proof
Applying Lemma 1 with F replaced by \(F^{i}\), the \(i\,{\textrm{th}}\) iterate of F, and k replaced by \(k_{i}\), for every \(i\in {\mathbb {N}}\), we get
If \(\lim _{i\rightarrow \infty }k_{i}=1\), then the transformation F is asymptotically nonexpansive in the sense of Goebel and Kirk [3] and, in view of their principal Theorems 1 and 2, F has a fixed point in C and the set of all fixed points is closed and convex.
In the case (ii), for i large enough, the transformation \(F^{i}\) is a contraction, and the result follows from the Banach principle. \(\square \)
4 A fixed point theorem for continuous mappings satisfying a weaker nonlinear asymptotical nonexpansivity condition
In this section, we show that Theorem 1 remains valid if the nonlinear asymptotical nonexpansivity of the mapping is replaced by a much weaker condition.
Theorem 2
Let C be a nonempty, closed, convex and bounded subset of a uniformly convex Banach space, and let a mapping \(F:C\rightarrow C\) be continuous. Assume that, for every \(i\in {\mathbb {N}}\) there exists a function \(\beta _{i}:\left( 0,\infty \right) \rightarrow \left( 0,\infty \right) \) and a sequence \(\left( t_{i,n}:n\in {\mathbb {N}}\right) \) with \(\lim _{n\rightarrow \infty }t_{i,n}=0\) such that,
the sequence \(\left( k_{i}:i\in {\mathbb {N}}\right) ,\)
is convergent, and for all \(x,y\in C,\)
If \(k=\lim _{i\rightarrow \infty }k_{i}\) \(\le 1\), then F has a fixed point in C; if moreover \(k<1\), then F has a unique fixed point.
Proof
In view of Lemma 3, for every \(i\in {\mathbb {N}}\), the mapping \(F^{i}\) is Lipschitz continuous and
Thus, the transformation F is asymptotically nonexpansive and, in view of the result of Goebel and Kirk [3] (Theorem 2 or Theorem 3), F has a fixed point in C. The uniqueness of the fixed point in the case when \(k<1\) is obvious. \(\square \)
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Matkowski, J. A generalization of the Goebel–Kirk fixed point theorem for asymptotically nonexpansive mappings. J. Fixed Point Theory Appl. 25, 69 (2023). https://doi.org/10.1007/s11784-023-01072-w
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DOI: https://doi.org/10.1007/s11784-023-01072-w