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

$$\begin{aligned} \left\| Fx-Fy\right\| \le \beta \left( \left\| x-y\right\| \right) \textit{for all}\,\, x,y\in C, \,x\ne y, \end{aligned}$$

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

$$\begin{aligned} \left\| x-y\right\| =t_{n}\Longrightarrow \left\| Fx-Fy\right\| \le \beta \left( t_{n}\right) \end{aligned}$$

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 \(x,y\in C\),

$$\begin{aligned} \left\| F^{i}x-F^{i}y\right\| \le k_{i}\left\| x-y\right\| \text {,} \end{aligned}$$

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

$$\begin{aligned} \left\| F^{i}x-F^{i}y\right\| \le \beta _{i}\left( \left\| x-y\right\| \right) ,\ \ \ \ \ x,y\in C,x\ne y; \end{aligned}$$

the limits

$$\begin{aligned} k_{i}:=\lim _{t\rightarrow 0+}\frac{\beta _{i}\left( t\right) }{t} \ge 1,\ \ \ \ \ i\in {\mathbb {N}}\text {,} \end{aligned}$$

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

$$\begin{aligned} \left\| x-y\right\| =t_{n}\Longrightarrow \left\| F^{i}x-F^{i}y\right\| \le \beta _{i}\left( t_{n}\right) \end{aligned}$$

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

$$\begin{aligned} \left\| Fx-Fy\right\| \le \beta \left( \left\| x-y\right\| \right) \text {, }\ \ \ \ \ x,y\in C\text {, }x\ne y. \end{aligned}$$

If

$$\begin{aligned} \limsup _{t\rightarrow 0+}\frac{\beta \left( t\right) }{t}<+\infty , \end{aligned}$$

then

$$\begin{aligned} \left\| Fx-Fy\right\| \le k\left\| x-y\right\| \text {,}\ \ \ \ \ x,y\in C, \end{aligned}$$

where

$$\begin{aligned} k:=\liminf _{t\rightarrow 0+}\frac{\beta \left( t\right) }{t}. \end{aligned}$$

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 \(\left( t_{n}:n\in {\mathbb {N}}\right) ,\left( c_{n}:n\in {\mathbb {N}}\right) ,\)

$$\begin{aligned} \lim _{n\rightarrow \infty }t_{n}=0\text {,} \lim _{n\rightarrow \infty }c_{n}=k, \end{aligned}$$

such that for every \(n\in {\mathbb {N}}\) and for all \(x,y\in C\),

$$\begin{aligned} \left\| x-y\right\| =t_{n}\Longrightarrow \left\| F\left( x\right) -F\left( y\right) \right\| \le c_{n}t_{n}\text {,} \end{aligned}$$

then F is Lipschitz continuous, and

$$\begin{aligned} \left\| Fx-Fy\right\| \le k\left\| x-y\right\| \text {, }\ \ \ \ \ x,y\in C\text {.} \end{aligned}$$

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

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\beta \left( t_{n}\right) }{t_{n}}=k, \end{aligned}$$

such that for every \(n\in {\mathbb {N}}\) and for all \(x,y\in C,\)

$$\begin{aligned} \left\| x-y\right\| =t_{n}\Longrightarrow \left\| Fx-Fy\right\| \le \beta \left( \left\| x-y\right\| \right) , \end{aligned}$$

then F is Lipschitz continuous, and

$$\begin{aligned} \left\| Fx-Fy\right\| \le k\left\| x-y\right\| \text {,}\ \ \ \ \ x,y\in C\text {.} \end{aligned}$$

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

$$\begin{aligned} \left\| Fx-Fy\right\| \le \beta \left( t_{n}\right) =c_{n}t_{n}, \end{aligned}$$

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

$$\begin{aligned} \left\| F^{i}x-F^{i}y\right\|\le & {} \beta _{i}\left( \left\| x-y\right\| \right) \text {, }\, \ \ \ \ \ x,y\in C,\,x\ne y\text {,}\\{} & {} \limsup _{t\rightarrow 0+}\frac{\beta _{i}\left( t\right) }{t}<+\infty \text {, } \end{aligned}$$

the sequence \(\left( k_{i}:i\in {\mathbb {N}}\right) \) defined by

$$\begin{aligned} k_{i}:=\liminf _{t\rightarrow 0+}\frac{\beta _{i}\left( t\right) }{t}\text {,}\ \ \ \ \ i\in {\mathbb {N}}\text {,} \end{aligned}$$

converges and

$$\begin{aligned} \lim _{i\rightarrow \infty }k_{i}\le 1. \end{aligned}$$

Then,

(i) if

$$\begin{aligned} \lim _{i\rightarrow \infty }k_{i}=1, \end{aligned}$$

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

$$\begin{aligned} \lim _{i\rightarrow \infty }k_{i}<1, \end{aligned}$$

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

$$\begin{aligned} \left\| F^{i}x-F^{i}y\right\| \le k_{i}\left\| x-y\right\| \text {,}\ \ \ \ \ x,y\in C\text {.} \end{aligned}$$

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,

$$\begin{aligned} \limsup _{t\rightarrow 0+}\frac{\beta _{i}\left( t\right) }{t}<+\infty , \end{aligned}$$

the sequence \(\left( k_{i}:i\in {\mathbb {N}}\right) ,\)

$$\begin{aligned} k_{i}:=\liminf _{t\rightarrow 0+}\frac{\beta _{i}\left( t\right) }{t}, \end{aligned}$$

is convergent, and for all \(x,y\in C,\)

$$\begin{aligned} \left\| x-y\right\| =t_{i,n}\Longrightarrow \left\| F^{i}x-F^{i}y\right\| \le \beta _{i}\left( \left\| x-y\right\| \right) \text {,}\,\,i\in {\mathbb {N}}\text {.} \end{aligned}$$

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

$$\begin{aligned} \left\| F^{i}x-F^{i}y\right\| \le k_{i}\left\| x-y\right\| \text {,}\ \ \ \ \ x,y\in C\text {.} \end{aligned}$$

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. \(\square \)