A generalization of the Goebel–Kirk fixed point theorem for asymptotically nonexpansive mappings

Some results extending the Goebel–Kirk fixed point theorem for asymptotically nonexpansive mappings are presented.

Let F be a self-mapping of a nonempty bounded closed and convex subset C of uniformly convex normed space, and a function β : (0, ∞) → [0, ∞) be such that lim t→0+ β(t) 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: if F 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 (t n ) of positive real numbers such that lim n→∞ t n = 0 the implication holds true for all x, y ∈ C, and n ∈ 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 → C is said to be asymptotically nonexpansive (in the Goebel-Kirk sense), if for all x, y ∈ C, where F i is the i th iterate of F and (k i : i ∈ N) is a sequence of real numbers such that lim i→∞ 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 → 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 ∈ N there exists a function β i : (0, ∞) → (0, ∞) satisfying the conditions x,y ∈ C, x = y; the limits 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 (t n : n ∈ N) with lim n→∞ t n = 0 such that the implication holds true for all x, y ∈ C, and n ∈ 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.

Some lemmas on Lipschitz-type mappings
Let us quote the following lemma from a recent paper [10] (see also [7,8]). 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 ⊂ X a convex set. Suppose that F : C → Y is continuous. If there are a nonnegative real k and two positive sequences (t n : n ∈ N) , (c n : n ∈ N) , such that for every n ∈ N and for all x, y ∈ C, then F is Lipschitz continuous, and x, y ∈ C.
From Lemma 2, we obtain the following such that for every n ∈ N and for all x, y ∈ C, then F is Lipschitz continuous, and Proof. Setting c n := β(tn) tn , we have lim n→∞ c n = k. Since for all n ∈ N and the result follows from Lemma 2.

A fixed point theorem for nonlinear asymptotically nonexpansive mappings
Recall that a real normed vector space (X, · ) is called uniformly convex, if for every ε ∈ (0, 2] there is some δ > 0 such that for any two vectors x, y ∈ X with x = y = 1, the condition x − y ≥ ε implies that x+y 2 ≤ 1 − δ (Goebel and Reich [4]; see also [9]).
Applying Lemma 1 with k replaced by k i for i ∈ N, we obtain the following generalization of the Goebel-Kirk theorem. Theorem 1. Let X be a uniformly convex Banach space, C ⊂ 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 ∈ N, there exists a function β i : (0, ∞) → (0, ∞) such that the sequence (k i : i ∈ N) defined by converges and Then,

then F has a unique fixed point in C.
Proof. Applying Lemma 1 with F replaced by F i , the i th iterate of F , and k replaced by k i , for every i ∈ N, we get If lim i→∞ 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.

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. Assume that, for every i ∈ N there exists a function β i : (0, ∞) → (0, ∞) and a sequence (t i,n : n ∈ N) with lim n→∞ t i,n = 0 such that, is convergent, and for all x, y ∈ C, If k = lim i→∞ k i ≤ 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 ∈ 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.
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