Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 826851 | 21 pages | https://doi.org/10.1155/2011/826851

Fixed Point Theorems by Ways of Ultra-Asymptotic Centers

Academic Editor: Allan C. Peterson
Received24 Dec 2010
Revised23 Apr 2011
Accepted23 May 2011
Published21 Jul 2011

Abstract

We use an approach on ultra-asymptotic centers to obtain fixed point theorems for two classes of nonself multivalued mappings. The results extend and improve several known ones.

1. Introduction

Domínguez Benavides and Lorenzo Ramírez [13] introduced a new method to prove the existence of fixed points using asymptotic centers as main tools by comparing their asymptotic radii with various geometric moduli of Banach spaces. The method led Dhompongsa et al. [4] to define the (DL) condition and obtained a fixed point theorem by following the proof in [1].

Definition 1.1 (see [4, Definition  3.1]). A Banach space 𝑋 is said to satisfy the Domínguez-Lorenzo condition ((DL) condition, in short), if there exists 𝜆[0,1) such that for every weakly compact convex subset 𝐸 of 𝑋 and for every bounded sequence {𝑥𝑛} in 𝐸 which is regular relative to 𝐸, 𝑟𝐸𝐴𝑥𝐸,𝑛𝑥𝜆𝑟𝐸,𝑛.(1.1)

Theorem 1.2 (see [4, Theorem  3.3]). Let 𝑋 be a reflexive Banach space satisfying the (DL) condition, and let 𝐸 be a bounded closed and convex separable subset of 𝑋. If 𝑇𝐸𝐾𝐶(𝑋) is a nonexpansive and 1𝜒-contractive mapping such that 𝑇(E) is a bounded set and which satisfies the inwardness condition:𝑇𝑥𝐼𝐸(𝑥)forall𝑥𝐸, then 𝑇 has a fixed point.

Wiśnicki and Wośko [5] introduced an ultrafilter coefficient DL𝒰(𝑋) for a Banach space 𝑋 and presented a fixed point result under a stronger condition than the (DL) condition.

Definition 1.3 (see [5, Definition  5.2]). Let 𝒰 be a free ultrafilter defined on the set of natural numbers . The coefficient DL𝒰(𝑋) of a Banach space 𝑋 is defined as DL𝒰𝜒(𝑋)=sup𝐸𝐴𝒰𝑥𝐸,𝑛𝜒𝐸𝑥𝑛,(1.2) where the supremum is taken over all nonempty weakly compact convex subsets 𝐸 of 𝑋 and all weakly, not norm-convergent sequences {𝑥𝑛} in 𝐸 which are regular relative to 𝐸.

Theorem 1.4 (see [5, Theorem  5.3]). Let 𝐸 be a nonempty weakly compact convex subset of a Banach space 𝑋 with 𝐷𝐿𝒰(𝑋)<1. Assume that 𝑇𝐸𝐾𝐶(𝑋) is a nonexpansive and 1𝜒-contractive mapping such that 𝑇𝑥𝐼𝐸(𝑥) for all 𝑥𝐸. Then 𝑇 has a fixed point.

Observe that, unlike Theorem 1.2, it does not assume the “separability” condition on 𝐸 in Theorem 1.4. However, it should be mentioned that the idea of the proof came from the original one of Domínguez Benavides and Lorenzo Ramírez [1]. At the same time, with the same purpose, Garvira [6] introduced independently its counter part in terms of ultranets.

Definition 1.5 (see [6, Definition  3.1.2]). A Banach space 𝑋 is said to have the (DL) condition with respect to a topology 𝜏(𝜏(DL)𝛼 condition) if there exists 𝜆[0,1) such that for every 𝜏-compact convex subset 𝐸 of 𝑋 and for every bounded ultranet {𝑥𝛼} in 𝐸𝑟𝐸𝐴𝑥𝐸,𝛼𝑥𝜆𝑟𝐸,𝛼.(1.3) When 𝜏 is the weak topology 𝜔 we write (DL)𝛼 condition instead of 𝜔(DL)𝛼 condition.

It follows from [6, Proposition  3.1.1] that the (DL)𝛼 condition is stronger than the (DL) condition. The ultranet counterpart of Theorem 1.4 becomes:

Theorem 1.6 (special case of [6, Theorem  3.5.2]). Let 𝑋 be a Banach space satisfying the (𝐷𝐿)𝛼 condition. Let 𝐸 be a weakly compact convex subset of 𝑋. If 𝑇𝐸𝐾𝐶(𝑋) is a nonexpansive and 1𝜒-contractive mapping such that 𝑇𝑥𝐼𝐸(𝑥) for 𝑥𝐸, then 𝑇 has a fixed point.

In 2006, Dhompongsa et al. [7] introduced a property for a Banach space 𝑋.

Definition 1.7 (see [7, Definition  3.1]). A Banach space 𝑋 is said to have property (𝐷) if there exists 𝜆[0,1) such that for any nonempty weakly compact convex subset 𝐸 of 𝑋, any sequence {𝑥𝑛}𝐸 which is regular and asymptotically uniform relative to 𝐸, and any sequence {𝑦𝑛}𝐴(𝐸,{𝑥𝑛}) which is regular and asymptotically uniform relative to 𝐸 one has 𝑟𝑦𝐸,𝑛𝑥𝜆𝑟𝐸,𝑛.(1.4)

Theorem 1.8 (see [7, Theorem  3.6]). Let 𝐸 be a nonempty weakly compact convex subset of a Banach space 𝑋 which has property (𝐷). Assume that 𝑇𝐸𝐾𝐶(𝐸) is a nonexpansive mapping. Then 𝑇 has a fixed point.

The following definition is due to Butsan et al. [8].

Definition 1.9 (see [8, Definition  3.1]). Let 𝑇𝐸𝐸 be a mapping on a subset 𝐸 of a Banach space 𝑋. Then 𝑇 is said to satisfy condition () if (1)for each T-invariant subset 𝐾 of 𝐸, 𝑇 has an afps in 𝐾, (2)for each pair of 𝑇-invariant subsets 𝐾 and 𝑊 of 𝐸, 𝐴(𝑊,{𝑥𝑛}) is 𝑇-invariant for each afps {𝑥𝑛} in 𝐾.

The main fixed point theorem concerning condition () in [8] is Theorem 3.5 of which the correct statement should be stated as follows.

Theorem 1.10 (see [8, Theorem  3.5]). Let 𝑋 be a Banach space having property (𝐷), and let 𝐸 be a weakly compact convex subset of 𝑋. Let 𝑇𝐸𝐸 satisfy conditon (). If 𝑇 is continuous, then 𝑇 has a fixed point.

In fact, we can replace “continuity” by a weaker condition, namely “𝐼𝑇 is strongly demiclosed at 0”: for every sequence {𝑥𝑛} in 𝐸 strongly converges to 𝑧𝐸 and such that 𝑥𝑛𝑇𝑥𝑛0 we have 𝑧=𝑇𝑧 (cf. [9]).

Following the concept of 𝐷𝐿𝒰(𝑋), Dhompongsa and Inthakon [10] introduced the following coefficient.

Definition 1.11 (see [10, Definition  3.2]). Let 𝒰 be a free ultrafilter defined on . The coefficient 𝐷𝒰(𝑋) of a Banach space 𝑋 is defined as 𝐷𝒰𝜒(𝑋)=sup𝐸𝑦𝑛𝜒𝐸𝑥𝑛,(1.5) where the supremum is taken over all nonempty weakly compact convex subsets 𝐸 of 𝑋, all sequences {𝑥𝑛} in 𝐸 which are weakly, not norm-convergent and are regular relative to 𝐸 and all weakly, not norm-convergent sequences {𝑦𝑛}𝐴𝒰(𝐸,{𝑥𝑛}) which are regular relative to 𝐸.

A concept corresponding to the coefficient 𝐷𝒰(𝑋) is the following property.

Definition 1.12 (see [10, Definition  3.1]). A Banach space 𝑋 is said to have property (𝐷) if there exists 𝜆[0,1) such that for any nonempty weakly compact convex subset 𝐸 of 𝑋, any sequence {𝑥𝑛}𝐸 which is regular relative to 𝐸, and any sequence {𝑦𝑛}𝐴(𝐸,{𝑥𝑛}) which is regular relative to 𝐸 one has 𝑟𝑦𝐸,𝑛𝑥𝜆𝑟𝐸,𝑛.(1.6)

Obviously, DL𝒰(𝑋)<1(DL)(𝐷)(𝐷). Several well-known spaces have been proved to satisfy the (DL) condition and to have property (𝐷) (see, e.g., [1, 4, 7, 1116]). The class of spaces having property (𝐷) includes both spaces satisfying the (DL) condition as well as spaces satisfying the Kirk-Massa condition (see Section 3.2 for the definition of the condition). One of the main results in [10] is the following theorem.

Theorem 1.13 (see [10, Theorem  1.9]). Let 𝐸 be a nonempty weakly compact convex subset of a Banach space 𝑋, and let 𝑋 have property (𝐷). Assume that 𝑇𝐸𝐾𝐶(𝑋) is a nonexpansive and 1𝜒-contractive mapping such that 𝑇𝑥𝐼𝐸(𝑥) for every 𝑥𝐸. Then 𝑇 has a fixed point.

Due to Kuczumow and Prus [17], we can assume without loss of generality that 𝐸 in Theorem 1.8 is separable. Theorem 1.13 does not only extend Theorem 1.8 to nonself-mappings, but it can remove “separability” condition on the domains of the mappings in Theorem 1.2 without refering to ultrafilters or ultranets in its statement. The key to the proof of Theorem 1.13 is the following.

Theorem 1.14 (see [10, Theorem  3.4]). Let 𝐸 be a weakly compact convex subset of a Banach space 𝑋 and 𝒰 a free ultrafilter defined on . Then 𝐷𝒰(𝑋)<1 if and only if 𝑋 has property (𝐷).

Thus, unlike the coefficient DL𝒰(𝑋), we have, for any two free ultrafilters 𝒰 and 𝒱, 𝐷𝒰(𝑋)<1 if and only if 𝐷𝒱(𝑋)<1.

In Section 3.1, we extend Theorem 1.10 to multivalued nonself-mappings for spaces having property (𝐷). Examples of such mappings are given. To obtain a fixed point result for more mappings, a new class of mappings is introduced in Section 3.2. Some examples of those mappings are also given. Our results extend and improve several known results in [810, 1825] and many others (see Remark 3.4).

2. Preliminaries

Let 𝐸 be a nonempty closed and convex subset of a Banach space 𝑋. We will denote by 2𝑋 the family of all subsets of 𝑋, 𝐶𝐵(𝑋) the family of all nonempty bounded and closed subsets of 𝑋 and denote by 𝐾𝐶(𝑋) the family of all nonempty compact convex subsets of 𝑋. For a given mapping 𝑇𝐸𝐶𝐵(𝑋) the set of all fixed points of 𝑇 will be denoted by 𝐹(𝑇), that is, 𝐹(𝑇)={𝑥𝐸𝑥𝑇𝑥}. Let 𝐻(,) be the Hausdorff distance defined on 𝐶𝐵(𝑋), that is,𝐻(𝐴,𝐵)=maxsup𝑎𝐴dist(a,B),sup𝑏𝐵dist(𝑏,𝐴),𝐴,𝐵𝐶𝐵(𝑋),(2.1)

where dist(𝑎,𝐵)=inf{𝑎𝑏𝑏𝐵} is the distance from a point 𝑎 to a subset 𝐵. A multivalued mapping 𝑇𝐸𝐶𝐵(𝑋) is said to be nonexpansive if𝐻(𝑇𝑥,𝑇𝑦)𝑥𝑦,𝑥,𝑦𝐸,(2.2)

and 𝑇 is said to be a contraction if there exists a constant 𝑘<1 such that𝐻(𝑇𝑥,𝑇𝑦)𝑘𝑥𝑦,𝑥,𝑦𝐸.(2.3) A multivalued mapping 𝑇𝐸2𝑋 is called 𝜙-condensing (resp., 1𝜙-contractive), where 𝜙 is a measure of noncompactness, if for each bounded subset 𝐵 of 𝐸 with 𝜙(𝐵)>0, there holds the inequality𝜙(𝑇(𝐵))<𝜙(𝐵)(resp.𝜙(𝑇(𝐵))𝜙(𝐵)),(2.4)

where 𝑇(𝐵)=𝑥𝐵𝑇𝑥.

Recall that the inward set of 𝐸 at 𝑥𝐸 is defined by𝐼𝐸(𝑥)={𝑥+𝛼(𝑦𝑥)𝛼1,𝑦𝐸}.(2.5) A sequence {𝑥𝑛} in 𝐸 for which lim𝑛𝑥𝑛𝑇𝑥𝑛=0 for a mapping 𝑇𝐸𝐸 is called an approximate fixed point sequence (afps for short) for 𝑇. Analogously for a multivalued mapping 𝑇𝐸𝐶𝐵(𝑋), a sequence {𝑥𝑛} in 𝐸 of a Banach space 𝑋 for which lim𝑛dist(𝑥𝑛,𝑇𝑥𝑛)=0 is called an approximate fixed point sequence (afps for short) for 𝑇.

We denote by 𝑥𝑛𝑥 to indicate that the sequence {𝑥𝑛} in 𝑋 converges to 𝑥𝑋.

Let 𝐸 be a nonempty closed and convex subset of a Banach space 𝑋 and {𝑥𝑛} a bounded sequence in 𝑋. For 𝑥𝑋, define the asymptotic radius of {𝑥𝑛} at 𝑥 as the number𝑟𝑥𝑥,𝑛=limsup𝑛𝑥𝑛𝑥.(2.6) Let𝑟𝑥𝐸,𝑛𝑟𝑥=inf𝑥,𝑛,𝐴𝑥𝑥𝐸𝐸,𝑛𝑥=𝑥𝐸𝑟𝑥,𝑛𝑥=𝑟𝐸,𝑛.(2.7) The number 𝑟(𝐸,{𝑥𝑛}) and the set 𝐴(𝐸,{𝑥𝑛}) are, respectively, called the asymptotic radius and asymptotic center of {𝑥𝑛} relative to 𝐸. The sequence {𝑥𝑛} is called regular relative to 𝐸 if 𝑟(𝐸,{𝑥𝑛})=𝑟(𝐸,{𝑥𝑛}) for each subsequence {𝑥𝑛} of {𝑥𝑛}. It was noted in [26] that if 𝐸 is nonempty and weakly compact, then 𝐴(𝐸,{𝑥𝑛}) is nonempty and weakly compact, and if 𝐸 is convex, then 𝐴(𝐸,{𝑥𝑛}) is convex.

Proposition 2.1 (see [27, Theorem  1]). Let {𝑥𝑛} and 𝐸 be as above. Then there exists a subsequence of {𝑥𝑛} which is regular relative to 𝐸.

We now present the formulation of an ultrapower of Banach spaces. Let 𝒰 be a free ultrafilter on . Recall ([26, 2830]) that the ultrapower (𝑋)𝒰 of a Banach space 𝑋 is the quotient space of𝑙𝑥(𝑋)=𝑛𝑥𝑛𝑥𝑋𝑛,𝑛=sup𝑛𝑥𝑛<(2.8) by𝑥ker𝒩=𝑛𝑙(𝑋)lim𝑛𝒰𝑥𝑛=0.(2.9) One can proof that 𝑋=(𝑋)𝒰 is a Banach space with the quotient norm given by {𝑥𝑛}𝒰=lim𝑛𝒰𝑥𝑛, where {𝑥𝑛}𝒰 is the equivalence class of {𝑥𝑛}. It is also clear that 𝑋 is isometric to a subspace of 𝑋 by the canonical embedding 𝑥{𝑥,𝑥,}𝒰. If 𝐸𝑋, we will use the symbols ̇𝐸 and ̇𝑥 to denote the image of 𝐸 and 𝑥 in 𝑋 under this isometry, respectively, and denote𝑥𝐸=̃𝑥𝑋𝑛𝑥suchthat̃𝑥=𝑛𝒰,𝑥𝑛𝐸𝑛.(2.10)

Thus ̇𝑥={𝑥,𝑥,}𝒰 and ̇𝐸={̇𝑥𝑋𝑥𝐸}.

If 𝑇𝐸𝐶𝐵(𝑋) is a multivalued mapping, we define a corresponding multivalued mapping 𝑇𝐸𝐶B(𝑋) by𝑢𝑇(̃𝑥)=̃𝑢𝑋𝑛𝑢suchthat̃𝑢=𝑛𝒰,𝑢𝑛𝑇𝑥𝑛𝑛,(2.11)

where ̃𝑥={𝑥𝑛}𝒰𝐸. Moreover the set 𝑇(̃𝑥) is bounded and closed (see [28, 29]). The Hausdorff metric on 𝐶𝐵(𝑋) will be denoted by 𝐻.

Proposition 2.2 (see [5, Proposition  3.1]). For every {𝑥𝑛}𝒰 and {𝑦𝑛}𝒰 in 𝐸, 𝐻𝑇𝑥𝑛𝒰,𝑇𝑦𝑛𝒰=lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑦𝑛.(2.12)

Proposition 2.3 (see [31, Page 37], [5, Proposition  3.2]). Let 𝐸 be a nonempty subset of a Banach space 𝑋 and 𝑇𝐸𝐶𝐵(𝑋). (i)If 𝑇 is convex-valued, then 𝑇 is convex-valued. (ii)If 𝑇 is compact-valued, then 𝑇 is compact-valued and ̇𝑇̇𝑥=(𝑇𝑥) for every 𝑥𝐸. (iii)If 𝑇 is nonexpansive, then 𝑇 is nonexpansive.

Let 𝒰 denote a free ultrafilter defined on . Wiśnicki and Wośko [5] defined the ultra-asymptotic radius 𝑟𝒰(𝐸,{𝑥𝑛}) and the ultra-asymptotic center 𝐴𝒰(𝐸,{𝑥𝑛}) of {𝑥𝑛} relative to 𝐸 by 𝑟𝒰𝑥𝐸,𝑛=inflim𝑛𝒰𝑥𝑛,𝐴𝑥𝑥𝐸𝒰𝑥𝐸,𝑛=𝑥𝐸lim𝑛𝒰𝑥𝑛𝑥=𝑟𝒰𝑥𝐸,𝑛.(2.13) It is not difficult to see that 𝐴𝒰(𝐸,{𝑥𝑛}) is a nonempty weakly compact convex set if 𝐸 is. Notice that the above notions have a natural interpretation in the ultrapower 𝑋 [5]:𝑟𝒰𝑥𝐸,𝑛=inf𝑥𝐸𝑥𝑛𝒰̇𝑥(2.14)

is the relative Chebyshev radius of {𝑥𝑛}𝒰, and𝐴𝒰̇𝑥𝐸,𝑛=̇𝐸𝐵𝑋𝑥𝑛𝒰,𝑟(2.15)

is the relative Chebyshev center of {𝑥𝑛}𝒰 relative to ̇𝐸 in the ultrapower 𝑋. (Here 𝐵𝑋({𝑥𝑛}𝒰,𝑟) denotes the ball in 𝑋 centered at {𝑥𝑛}𝒰 and of radius 𝑟=𝑟𝒰(𝐸,{𝑥𝑛}).) It should be noted that, in general, 𝐴(𝐸,{𝑥𝑛}) and 𝐴𝒰(𝐸,{𝑥𝑛}) may be different. The notion of the asymptotic radius is closely related to the notion of the relative Hausdorff measure of noncompactness defined by Domínguez Benavides and Lorenzo Ramírez [1] as𝜒𝐸(𝐴)=inf{𝜀>0𝐴canbecoveredbynitelymanyballsin𝐸ofradii𝜀}.(2.16)

Proposition 2.4 (see [5, Proposition  4.5]). If {𝑥𝑛} is a bounded sequence which is regular relative to 𝐸, then 𝑟𝑥𝐸,𝑛=𝑟𝒰𝑥𝐸,𝑛=𝜒𝐸𝑥𝑛.(2.17)

From Proposition 2.4, we have, for 𝑤𝐴(𝐸,{𝑥𝑛}),lim𝑛𝒰𝑥𝑛𝑤limsup𝑛𝑥𝑛𝑥𝑤=𝑟𝐸,𝑛=𝑟𝒰𝑥𝐸,𝑛.(2.18)

Therefore, 𝐴(𝐸,{𝑥𝑛})𝐴𝒰(𝐸,{𝑥𝑛}).

The following result plays an important role in our proofs.

Lemma 2.5 (see [10, Lemma  3.3]). Let 𝐸 be a nonempty closed and convex subset of a Banach space 𝑋 and {𝑥𝑛} a bounded sequence in 𝑋 which is regular relative to 𝐸. For each {𝑦𝑛}𝐴𝒰(𝐸,{𝑥𝑛}), there exists a subsequence {𝑥𝑛} of {𝑥𝑛} such that {𝑦𝑛}𝐴(𝐸,{𝑥𝑛}).

A direct consequence of Lemma 2.5 is as follows. If every center 𝐴(𝐸,{𝑥𝑛}) is compact for every bounded sequence {𝑥𝑛} in E which is regular relative to 𝐸, then 𝐴𝒰(𝐸,{𝑥𝑛}) is also compact for every bounded sequence {𝑥𝑛} in 𝐸 which is regular relative to 𝐸.

3. Main Results

3.1. Property (𝐷)

Lemma 3.1. Let 𝐸 be a nonempty subset of a Banach space 𝑋 and 𝑇𝐸𝐶𝐵(𝑋). Then (1)if 𝑇 is uniformly continuous, then 𝑇 is uniformly continuous; (2)if 𝑇 is continuous at 𝑧𝐸, then 𝑇 is continuous at ż.

Proof. (1) Let 𝜀>0. Since 𝑇 is uniformly continuous, there exists 𝛿>0 such that 𝐻(𝑇𝑥,𝑇𝑦)<𝜀 for each 𝑥,𝑦𝐸 with 𝑥𝑦<𝛿. Suppose {𝑥𝑛}𝒰,{𝑦𝑛}𝒰𝐸, and {𝑥𝑛}𝒰{𝑦𝑛}𝒰<𝛿. Let 𝐴={𝑛𝑥𝑛𝑦𝑛<𝛿} and 𝐵={𝑛𝐻(𝑇𝑥𝑛,𝑇𝑦𝑛)<𝜀}. Since 𝐴𝒰 and 𝐴𝐵, 𝐵𝒰. Thus, by Proposition 2.2𝐻(𝑇{𝑥𝑛}𝒰,𝑇{𝑦𝑛}𝒰)𝜀.
(2) Let 𝜀>0. Since 𝑇 is continuous at 𝑧, there exists 𝛿>0 such that 𝐻(𝑇𝑥,𝑇𝑧)<𝜀 for each 𝑥𝐸 with 𝑥𝑧<𝛿. If {𝑥𝑛}𝒰𝐸 such that {𝑥𝑛}𝒰̇𝑧<𝛿, then, letting 𝐴={𝑛𝑥𝑛𝑧<𝛿} and 𝐵={𝑛𝐻(𝑇𝑥𝑛,𝑇𝑧)<𝜀}, we see that 𝐴𝒰 and 𝐵𝒰. Thus, by Proposition 2.2𝐻(𝑇{𝑥𝑛}𝒰,𝑇̇𝑧)𝜀.

We now introduce condition () for multivalued mappings.

Definition 3.2. Let 𝐸 be a nonempty subset of a Banach space 𝑋. A mapping 𝑇𝐸𝐶𝐵(𝑋) is said to satisfy condition () if (1)𝑇 has an afps in 𝐸, (2)𝑇 has an afps in 𝐴(𝐸,{𝑥𝑛}) for some subsequence {𝑥𝑛} of any given afps {𝑥𝑛} for 𝑇 in 𝐸.

Theorem 3.3. Let 𝑋 be a Banach space having property (𝐷), and let 𝐸 be a weakly compact convex subset of 𝑋. Assume that 𝑇𝐸𝐾𝐶(𝑋) is a multivalued mapping satisfying condition (). If 𝑇 is continuous, then 𝑇 has a fixed point.

Proof. The proof follows by adapting the proof of [10, Theorem  1.9]. By (1) of Definition 3.2, let {𝑥0𝑛} be an afps for 𝑇 in 𝐸. We can assume by Proposition 2.1 that {𝑥0𝑛} is regular relative to 𝐸. Condition (2) of Definition 3.2 gives us a subsequence {𝑥0𝑛0} of {𝑥0𝑛} so that the center 𝐴(𝐸,{𝑥0𝑛0}) contains an afps for 𝑇. Denote 𝐴0=𝐴(𝐸,{𝑥0𝑛0}), and let {𝑥1𝑛} be an afps in 𝐴0. Assume that {𝑥1𝑛} is regular relative to 𝐸. As before, 𝑇 has an afps in 𝐴(𝐸,{𝑥1𝑛1}) for some subsequence {𝑥1𝑛1} of {𝑥1𝑛}. Since 𝑋 has property (𝐷), put 𝜆=𝐷𝒰(𝑋)<1. Then, by Proposition 2.4 and Definition 1.11, 𝑟𝑥𝐸,1𝑛1=𝜒E𝑥1𝑛1𝜆𝜒𝐸𝑥0𝑛0𝑥=𝜆𝑟𝐸,0𝑛0.(3.1) Continue the procedure to obtain, for each 𝑚0, a regular sequence {𝑥𝑚𝑛𝑚} relative to 𝐸 in 𝐴𝑚1=𝐴(𝐸,{𝑥𝑛𝑚1𝑚1}) such that lim𝑛𝑥dist𝑚𝑛𝑚,𝑇𝑥𝑚𝑛𝑚=0,(3.2) and for all 𝑚1, 𝜒𝐸𝑥𝑚𝑛𝑚𝑥𝜆𝑟𝐸,𝑛𝑚1𝑚1.(3.3) Consequently, 𝑟𝑥𝐸,𝑚𝑛𝑚𝑥𝜆𝑟𝐸,𝑛𝑚1𝑚1𝜆𝑚𝑟𝑥𝐸,0𝑛0.(3.4) We show that {𝑥𝑚𝑛𝑚}𝒰 is a Cauchy sequence in 𝑋. Indeed, for each 𝑚1, take an element ̇𝑦𝑚̇𝐴𝑚1. Then ̇𝑥𝑚𝑛𝑚̇𝑦𝑚̇𝑥𝑚𝑛𝑚𝑥𝑛𝑚1𝑚1𝒰+𝑥𝑛𝑚1𝑚1𝒰̇𝑦𝑚𝑥2𝑟𝐸,𝑛𝑚1𝑚1(3.5) for all 𝑚1, and hence 𝑥𝑚𝑛𝑚𝒰𝑥𝑛𝑚1𝑚1𝒰𝑥𝑚𝑛𝑚𝒰̇𝑦𝑚+̇𝑦𝑚𝑥𝑛𝑚1𝑚1𝒰𝑥3𝑟𝐸,𝑛𝑚1𝑚1.(3.6) Thus 𝑥𝑚𝑛𝑚𝒰𝑥𝑛𝑚1𝑚1𝒰3𝜆𝑚1𝑟𝑥𝐸,0𝑛0,(3.7) implying that {𝑥𝑚𝑛𝑚}𝒰 is a Cauchy sequence and hence converges to some {𝑧𝑛}𝒰 in 𝐸 as 𝑚. Next, we will show that {𝑧𝑛}𝒰̇𝐸. For each 𝑚0, 𝑧dist𝑛𝒰,̇𝐸𝑧𝑛𝒰𝑥𝑚𝑛𝑚𝒰𝑥+𝑑𝑖𝑠t𝑚𝑛𝑚𝒰,̇𝐸{𝑧𝑛}𝒰𝑥𝑚𝑛𝑚𝒰+𝑥𝑚𝑛𝑚𝒰̇𝑥1𝑚+1𝑚+1=𝑧𝑛𝒰𝑥𝑚𝑛𝑚𝒰𝑥+𝑟𝐸,𝑚𝑛𝑚𝑧𝑛𝒰𝑥𝑚𝑛𝑚𝒰+𝜆𝑚𝑟𝑥𝐸,0𝑛0.(3.8) Taking 𝑚 we see that 𝑧dist𝑛𝒰,̇𝐸=0.(3.9) Thus, it follows that there exists 𝑧𝐸 such that {𝑧𝑛}𝒰=̇𝑧. By Lemma 3.1, 𝑇 is continuous at ż, and thus 𝐻(𝑇{𝑥𝑚𝑛𝑚}𝒰,𝑇̇𝑧)0 as 𝑚. For every 𝑚0, 𝑥disṫ𝑧,𝑇̇𝑧̇𝑧𝑚𝑛𝑚𝒰𝑥+dist𝑚𝑛𝑚𝒰,𝑇𝑥𝑚𝑛𝑚𝒰+𝐻𝑇𝑥𝑚𝑛𝑚𝒰,𝑇̇𝑧.(3.10) Taking 𝑚 we then obtain ̇𝑧𝑇̇𝑧. By Proposition 2.3, ̇𝑇̇𝑧=(𝑇𝑧), and therefore, 𝑧𝑇𝑧.

Remark 3.4. The proof presented here based on a standard proof appeared in a series of papers [1, 3, 5, 10]. However, we cannot follow its proof directly to be able to obtain a result for larger classes of spaces and mappings. We choose an ultralimit approach by using an ultra-asymptotic center 𝐴𝒰 as our main tool. As mentioned earlier, this powerful tool was introduced in [5] by Wiśnicki and Wośko. Thus our proof may not be totally new, but it significantly improves, generalizes, or extends many known results. (i)Theorem 3.3 (as well as Theorem 3.16) unifies many known theorems in one. Examples of mappings in both theorems are given throughout the rest of the paper. (ii)Theorem 3.3 improves condition () in [8, Definition  3.1] in which the mappings under consideration only are single valued and are self-mappings. Consequently [8, Theorem  3.5] is improved significantly. Obviously, [10, Theorem  1.9] is a special case of Theorem 3.3. (iii)In Remark 3.15(ii) below, we show the following implication:()+(𝐴)().(3.11)Thus results in [18, Corollaries  3.5 and  3.6], [19, Theorems  1 and 2], [20, Theorem  3.3], [9, Theorem  5], [21, Theorem  2.4], [22, Theorem  2], [23, Theorem  4.2, Corollary   4.3, Theorem  4.4], [24, Theorem  2.6], and [25, Theorem  2.3.1] are either improved, generalized, or extended. See Remark 3.17, Corollaries 3.18 and 3.19.See also Remark 3.24(iii) and (iv).

We now give some examples of mappings satisfying condition (). We will see that the ultracenter 𝐴𝒰(𝐸,{𝑥𝑛}) plays a significant role in verifying condition (2) of condition () for a given mapping.

Nonexpansive Mappings
We will show by following the proof of Theorem  5.3 in [5] that if 𝑇𝐸𝐾𝐶(𝑋) is nonexpansive and 1𝜒-contractive such that 𝑇𝑥𝐼𝐸(𝑥) for every 𝑥𝐸. Then 𝑇 satisfies condition (). The main tools are Lemma 2.5 and the following result.
Theorem 3.5 (see [32, Theorem  11.5]). Let 𝐸 be a nonempty bounded closed and convex subset of a Banach space 𝑋 and 𝐹𝐸𝐾𝐶(𝑋) an upper semicontinuous and 𝜒-condensing mapping. If 𝐹(𝑥)𝐼𝐸(𝑥) for all 𝑥𝐸, then 𝐹 has a fixed point.
Proposition 3.6. Let 𝐸 be a nonempty weakly compact convex subset of a Banach space 𝑋. Assume that 𝑇𝐸𝐾𝐶(𝑋) is nonexpansive and 1𝜒-contractive such that 𝑇𝑥𝐼𝐸(𝑥) for every 𝑥𝐸. Then 𝑇 satisfies condition ().
Proof. First, we will show that 𝑇 has an afps in 𝐸. Let 𝑦0𝐸, and consider, for each 𝑛1, the contraction 𝑇𝑛𝐸𝐾𝐶(𝑋) defined by 𝑇𝑛1(𝑥)=𝑛𝑦0+11𝑛𝑇𝑥,𝑥𝐸.(3.12) It is not difficult to see that 𝑇𝑛(𝑥)𝐼𝐸(𝑥) for every 𝑥𝐸. Since 𝑇 is 1𝜒-contractive, 𝑇𝑛 is (1(1/𝑛))𝜒-contractive, and by Theorem 3.5, there exists a fixed point 𝑥𝑛 of 𝑇𝑛. Clearly, {𝑥𝑛} is an afps for 𝑇 in 𝐸.
Next, let us see that 𝑇 has an afps in 𝐴(𝐸,{𝑥𝑛}) for some subsequence {𝑥𝑛} of an afps {𝑥𝑛} for 𝑇 in 𝐸. Let {𝑥𝑛} be an afps in 𝐸. By Proposition 2.1, we can assume that {𝑥𝑛} is weakly convergent and regular relative to 𝐸. Let 𝐴𝒰=𝐴𝒰(𝐸,{𝑥𝑛}). We show that 𝑇𝑥𝐼𝐴𝒰(𝑥)forevery𝑥𝐴𝒰.(3.13) Let 𝑥𝐴𝒰. Observe first that {𝑥𝑛}𝒰𝑇{𝑥𝑛}𝒰. By Proposition 2.3, ̇𝑇̇𝑥=(𝑇𝑥) is compact, and hence there exists 𝑢𝑇𝑥 such that 𝑥𝑛𝒰=𝐻𝑇𝑥̇𝑢𝑛𝒰,𝑥𝑇̇𝑥𝑛𝒰̇𝑥=𝑟𝒰𝑥𝐸,𝑛.(3.14) Since 𝑢𝑇𝑥𝐼𝐸(𝑥), there exists 𝛼1 and 𝑦𝐸 such that 𝑢=𝑥+𝛼(𝑦𝑥). If 𝛼=1 then 𝑢=𝑦𝐸, and it follows from (3.14) that 𝑢𝐴𝒰. If 𝛼>1 then 𝑦=(1/𝛼)𝑢+(11/𝛼)𝑥, and therefore, we have 𝑥𝑛𝒰1̇𝑦𝛼𝑥𝑛𝒰+1̇𝑢1𝛼𝑥𝑛𝒰̇𝑥𝑟𝒰𝑥𝐸,𝑛.(3.15) Hence 𝑦𝐴𝒰 and consequently 𝑢𝐼𝐴𝒰(𝑥). Thus (3.13) is justified.
Fixed 𝑦0𝐴𝒰, and consider for each 𝑛1, the contraction 𝑇𝑛𝐴𝒰𝐾𝐶(𝑋) defined by 𝑇𝑛1(𝑥)=𝑛𝑦0+11𝑛𝑇𝑥,𝑥𝐴𝒰.(3.16) As before, 𝑇𝑛 is (11/𝑛)𝜒-contractive, and by Theorem 3.5, there exists a fixed point 𝑧𝑛𝐴𝒰 of 𝑇𝑛. Again, as above, {𝑧𝑛} is an afps for 𝑇 in 𝐴𝒰. By Lemma 2.5, there exists a subsequence {𝑥𝑛} of {𝑥𝑛} such that {𝑧𝑛}𝐴(𝐸,{𝑥𝑛}).

Diametrically Contractive Mappings
In [33] Istratescu introduced a new class of mappings.
Definition 3.7 (see [33]). A mapping 𝑇 defined on a complete metric space (𝑋,𝑑) is said to be diametrically contractive if 𝛿(𝑇𝐾)<𝛿(𝐾) for all closed subsets 𝐾 with 0<𝛿(𝐾)<. (Here 𝛿(𝐾)=sup{𝑑(𝑥,𝑦)𝑥,𝑦𝐾} denotes the diameter of 𝐾𝑋.)
Xu [34] proved the fixed point theorem for a diametrically contractive mapping in the framework of Banach spaces.
Theorem 3.8 (see [34, Theorem  2.3]). Let 𝐸 be a weakly compact subset of a Banach space 𝑋, and let 𝑇𝐸𝐸 be a diametrically contractive mapping. Then 𝑇 has a fixed point.
Dhompongsa and Yingtaweesittikul [35] defined a multivalued version of mappings in Theorem 3.8 which is weaker than the condition required in Definition 3.7. Recall that 𝑇𝐾=𝑘𝐾𝑇𝑘 and 𝐸 is said to be 𝑇-invariant if 𝑇𝑥𝐸 for all 𝑥𝐸.
Theorem 3.9 (see [35, Theorem  2.2]). Let 𝐸 be a weakly compact subset of a Banach space 𝑋, and let 𝑇𝐸𝐾𝐶(𝑋) be a multivalued mapping such that 𝛿(𝑇𝐾𝐾)<𝛿(𝐾) for all closed sets 𝐾 with 𝛿(𝐾)>0 and 𝐸 is invariant under 𝑇. Then 𝑇 has a unique fixed point.
The following result extends Theorem 3.9 partially.
Proposition 3.10. Let 𝐸 be a nonempty weakly compact convex subset of a Banach space 𝑋, and let 𝑇𝐸𝐾𝐶(𝑋) be a multivalued mapping such that 𝛿(𝑇𝐾)𝛿(𝐾) for all closed sets 𝐾 with 𝛿(𝐾)>0 and 𝐸 is invariant under 𝑇. Then 𝑇 satisfies condition ().
Proof. First, we will see that 𝑇 has an afps in 𝐸. Let 𝑦0𝐸, and consider, for each 𝑛1, the contraction 𝑇𝑛𝐸𝐾𝐶(𝑋) defined by 𝑇𝑛1(𝑥)=𝑛𝑦0+11𝑛𝑇𝑥,𝑥𝐸.(3.17) For 𝑥𝐸, let 𝑎𝑇𝑥𝐸. Thus (1/𝑛)𝑦0+(11/𝑛)𝑎𝑇𝑛𝑥𝐸, and therefore, 𝑇𝑛𝑥𝐸 for every 𝑥𝐸. We show that 𝛿(𝑇𝑛𝐾)<𝛿(𝐾)forallclosedsets𝐾with𝛿(𝐾)>0. Let 𝐾 be a closed subset of 𝐸 with 𝛿(𝐾)>0. For 𝑥,𝑦𝑇𝑛𝐾, there exist 𝑥,𝑦𝑇𝐾 such that 1𝑥=𝑛𝑦0+11𝑛1𝑥,𝑦=𝑛𝑦0+11𝑛𝑦,(3.18) and this entails 𝑥𝑦=(11/𝑛)𝑥𝑦(11/𝑛)𝛿(𝑇𝐾). Hence 𝛿(𝑇𝑛𝐾)(11/𝑛)𝛿(𝑇𝐾)<𝛿(𝐾). By Theorem 3.9, there exists a fixed point 𝑥𝑛 of 𝑇𝑛, and thus the sequence {𝑥𝑛} forms an afps for 𝑇 in 𝐸.
Next, let us see that 𝑇 has an afps in 𝐴(𝐸,{𝑥𝑛}) for some subsequence {𝑥𝑛} of an afps {𝑥𝑛} for 𝑇 in 𝐸. Let {𝑥𝑛} be an afps in 𝐸. We can assume that {𝑥𝑛} is weakly convergent and regular relative to 𝐸. Let 𝐴𝒰=𝐴𝒰(𝐸,{𝑥𝑛}). First, we show that 𝐴𝒰𝑇𝑥,forevery𝑥𝐴𝒰.(3.19) Let 𝑥𝐴𝒰, and for each 𝑛1, we see that 𝐻(𝑇𝑥𝑛,𝑇𝑥)𝛿(𝑇{𝑥𝑛,𝑥})𝛿({𝑥𝑛,𝑥})=𝑥𝑛𝑥. Take 𝑦𝑛𝑇𝑥𝑛 so that 𝑥𝑛𝑦𝑛𝑥=dist𝑛,𝑇𝑥𝑛,(3.20) and select 𝑧𝑛𝑇𝑥 for each 𝑛 such that 𝑧𝑛𝑦𝑛𝑦=dist𝑛,𝑇𝑥.(3.21) Let lim𝑛𝒰𝑧𝑛=𝑧𝑇𝑥. Note that 𝑥𝑛𝑥𝑧𝑛𝑦𝑛+𝑦𝑛𝑧𝑛+𝑧𝑛𝑧.(3.22) We obtain lim𝑛𝒰𝑥𝑛𝑧lim𝑛𝒰𝑦𝑛𝑧𝑛=lim𝑛𝒰𝑦dist𝑛,𝑇𝑥lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑥lim𝑛𝒰𝑥𝑛𝑥=𝑟𝒰𝑥𝐸,𝑛(3.23) proving that 𝑧𝐴𝒰. Thus (3.19) is satisfied. Fix 𝑦0𝐴𝒰, and consider, for each 𝑛1, the contraction 𝑇𝑛𝐴𝒰𝐾𝐶(𝑋) defined by 𝑇𝑛1(𝑥)=𝑛𝑦0+11𝑛𝑇𝑥,𝑥𝐴𝒰.(3.24) For 𝑥𝐸, let 𝑎𝐴𝒰𝑇𝑥. Thus (1/𝑛)𝑦0+(11/𝑛)𝑎𝐴𝒰𝑇𝑛𝑥. Therefore, 𝐴𝒰𝑇𝑛𝑥 for every 𝑥𝐴𝒰. Let 𝐾 be a closed subset of 𝐸 with 𝛿(𝐾)>0. As before, 𝛿(𝑇𝑛𝐾)(11/𝑛)𝛿(𝑇𝐾)<𝛿(𝐾). By Theorem 3.9 (or we can apply Theorem 3.5), there exists a fixed point 𝑧𝑛 of 𝑇𝑛. Again, as above, {𝑧𝑛} is an afps for 𝑇 in 𝐴𝒰. Finally, by Lemma 2.5, there exists a subsequence {𝑥𝑛} of {𝑥𝑛} such that {𝑧𝑛}𝐴(𝐸,{𝑥𝑛}).

3.2. Kirk-Massa Condition

In 1990, Kirk and Massa [22] generalized Lim's Theorem [27] using asymptotic centers of sequences and nets and obtained the following result.

Theorem 3.11 (Kirk and Massa theorem). Let 𝐸 be a nonempty bounded closed and convex subset of a Banach space 𝑋 and 𝑇𝐸𝐾𝐶(𝐸) a nonexpansive mapping. Suppose that the asymptotic center in 𝐸 of each bounded sequence of 𝑋 is nonempty and compact. Then 𝑇 has a fixed point.

We call the assumption in Kirk and Massa theorem the Kirk-Massa condition. Xu [25] extended Kirk and Massa theorem to nonexpansive nonself-mappings.

Theorem 3.12 ([25, Theorem  2.3.1]). Let 𝑋 be a Banach space satisfying the Kirk-Massa condition and let 𝐸 be a nonempty bounded closed and convex subset of 𝑋. Let 𝑇𝐸𝐾𝐶(𝑋) be a nonexpansive mapping which satisfies the inwardness condition. Then 𝑇 has a fixed point.

Remark 3.13. Obviously, every space that satisfies the Kirk-Massa condition always has property (𝐷). Thus, particularly, the fixed point result in Section 3.1 holds for uniform convex Banach spaces, uniformly convex in every direction (UCED) and spaces satisfying the Opial condition.

We aim to extend Xu's result to a wider class of mappings. Thus, the domains of mappings are more general than the ones in Section 3.1.

Definition 3.14. Let 𝒰 be a free ultrafilter defined on . Let 𝐸 be a bounded closed and convex subset of a Banach space 𝑋. A mapping 𝑇𝐸𝐶𝐵(𝑋) is said to satisfy condition () if it fulfills the following conditions. (1)𝑇 has an afps in 𝐸; (2)if {𝑥𝑛} is an apfs for 𝑇 in 𝐸 and 𝑥𝐸, then lim𝑛𝒰𝐻(𝑇𝑥𝑛,𝑇𝑥)lim𝑛𝒰𝑥𝑛𝑥.

Remark 3.15. (i)  Let 𝐸 be a bounded closed and convex subset of a Banach space 𝑋, and let a mapping 𝑇𝐸𝐾𝐶(𝑋) satisfy condition (). If in addition, 𝑇 satisfies the following:
     (A)  every 𝑇-invariant, closed, and convex subset possesses an afps,
then 𝑇 satisfies condition ().

Proof. By (1) of condition (), let {𝑥𝑛} be an apfs for 𝑇 in 𝐸. From Proposition 2.1 by passing through a subsequence, we may assume that {𝑥𝑛} is regular relative to 𝐸. Let 𝐴𝒰=𝐴𝒰(𝐸,{𝑥𝑛}) and 𝑥𝐴𝒰. The compactness of 𝑇𝑥𝑛 implies that for each 𝑛 we can take 𝑦𝑛𝑇𝑥𝑛 so that 𝑥𝑛𝑦𝑛𝑥=dist𝑛,𝑇𝑥𝑛.(3.25) Since 𝑇𝑥 is compact, select 𝑧𝑛𝑇𝑥 for each 𝑛 such that 𝑧𝑛𝑦𝑛𝑦=dist𝑛,𝑇𝑥.(3.26) Let lim𝑛𝒰𝑧𝑛=𝑧𝑇𝑥. Note that 𝑥𝑛𝑥𝑧𝑛𝑦𝑛+𝑦𝑛𝑧𝑛+𝑧𝑛𝑧.(3.27) We obtain lim𝑛𝒰𝑥𝑛𝑧lim𝑛𝒰𝑦𝑛𝑧𝑛=lim𝑛𝒰𝑦dist𝑛,𝑇𝑥lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑥lim𝑛𝒰𝑥𝑛𝑥=𝑟𝒰𝑥𝐸,𝑛(3.28) proving that 𝑧𝐴𝒰 and hence 𝐴𝒰𝑇𝑥 for all 𝑥𝐴𝒰, that is, 𝐴𝒰 is 𝑇-invariant. By assumption, there exists an afps in 𝐴𝒰. By Lemma 2.5, there exists a subsequence {𝑥𝑛} of {𝑥𝑛} such that {𝑧𝑛}𝐴(𝐸,{𝑥𝑛}). Thus, 𝑇 satisfies condition ().

We wonder if we can drop condition (A) in proving the implication: ()(). An example of a mapping satisfies condition () but not condition () is given in Remark 3.24(i).

(ii)  In [23, Definition  3.1] the following concept of mappings is defined: a mapping 𝑇𝐸𝐸 satisfies condition (𝐿) on 𝐸 provided that it fulfills the following two conditions. (1)If a set 𝐷𝐸 is nonempty, closed, convex, and 𝑇-invariant, then there exists an afps for 𝑇 in 𝐷. (2)For any afps {𝑥𝑛} of 𝑇 in 𝐸 and each 𝑥𝐸,limsup𝑛𝑥𝑛𝑇𝑥limsup𝑛𝑥𝑛𝑥.(3.29)

Therefore, (i) shows that the class of mappings satisfying condition () contains and extends mappings satisfying condition (𝐿) as a multivalued nonself version.

The main idea of the proof of the following theorem is originated from Kirk and Massa [22].

Theorem 3.16. Let 𝑋 be a Banach space satisfying the Kirk-Massa condition, and let 𝐸 be a nonempty bounded closed and convex subset of 𝑋. Let 𝑇𝐸𝐾𝐶(𝑋) be a multivalued mapping satisfying condition (). If 𝑇 is an upper semicontinuous mapping, then 𝑇 has a fixed point.

Proof. Let {𝑥𝑛} be an afps for 𝑇 in 𝐸. From Proposition 2.1 by passing through a subsequence, we may assume that {𝑥𝑛} is regular relative to 𝐸. Let 𝐴𝒰=𝐴𝒰(𝐸,{𝑥𝑛}). The compactness of 𝑇𝑥𝑛 implies that for each 𝑛 we can take 𝑦𝑛𝑇𝑥𝑛 such that 𝑥𝑛𝑦𝑛𝑥=dist𝑛,𝑇𝑥𝑛.(3.30) If 𝑥𝐴𝒰, since 𝑇𝑥 is compact, select 𝑧𝑛𝑇𝑥 for each 𝑛 such that 𝑧𝑛𝑦𝑛𝑦=dist𝑛,𝑇𝑥.(3.31) Let lim𝑛𝒰𝑧𝑛=𝑧𝑇𝑥. Note that 𝑥𝑛𝑥𝑧𝑛𝑦𝑛+𝑦𝑛𝑧𝑛+𝑧𝑛𝑧.(3.32) Thus lim𝑛𝒰𝑥𝑛𝑧lim𝑛𝒰𝑦𝑛𝑧𝑛=lim𝑛𝒰𝑦dist𝑛,𝑇𝑥lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑥lim𝑛𝒰𝑥𝑛𝑥=𝑟𝒰𝑥𝐸,𝑛(3.33) proving that 𝑧𝐴𝒰 and hence 𝐴𝒰𝑇𝑥 for all 𝑥𝐴𝒰. By assumption, 𝐴(𝐸,{𝑥𝑛}) is nonempty and compact which implies that 𝐴𝒰 is also nonempty and compact. Now define a mapping 𝐹𝐴𝒰𝐾𝐶(𝐴𝒰) by 𝐹𝑥=𝐴𝒰𝑇𝑥 for all 𝑥𝐴𝒰. Thus 𝐹 is upper semicontinuous. Indeed, let {𝑢𝑛}𝐴 be such that lim𝑛𝑢𝑛=𝑢, and let 𝑣𝑛𝐹𝑢𝑛 be such that lim𝑛𝑣𝑛=𝑣. Since 𝑇 is upper semicontinuous and 𝐴𝒰 is compact, we have 𝑣𝑇𝑢 and 𝑣𝐴𝒰, that is 𝑣𝐹𝑢. By the Bohnenblust-Karlin fixed point theorem [36], 𝐹 and hence 𝑇, have a fixed point in 𝐴𝒰.

Remark 3.17. If, in addition, mappings in Theorem 3.16 also satisfy condition (𝐴), then the condition on “upper semicontinuity” can be dropped. This is because an afps in a compact set can be chosen so that its asymptotic center is only a singleton, and a fixed point can be easily derived. Consequently, Theorem 3.3 can be extended to a bigger class of domains, namely, the bounded, closed, and convex ones. And the following results are immediate.

Corollary 3.18 (see [23, Theorem  4.2]). Let 𝐸 be a nonempty compact convex subset of a Banach space 𝑋 and 𝑇𝐸𝐸 a mapping satisfying condition (𝐿). Then, 𝑇 has a fixed point.

Corollary 3.19 (see [23, Corollary  4.3]). Let 𝐸 be a nonempty compact convex subset of a Banach space 𝑋 and 𝑇𝐸𝐸 a mapping satisfying condition (𝐿). Suppose that the asymptotic center in 𝐸 of each sequence in 𝐸 is nonempty and compact. Then, 𝑇 has a fixed point.

We give some examples of mappings satisfying condition (). The first example is of course the mapping described in Theorem 3.12.

Condition 3.2 (𝐶𝜆). García-Falset et al. [9] introduced the following mappings.
Let E be a nonempty subset of a Banach space 𝑋. For 𝜆(0,1), we say that a mapping 𝑇𝐸𝑋 satisfies Condition (𝐶𝜆) on 𝐸 if, for each 𝑥,𝑦𝐸, 𝜆𝑥𝑇𝑥𝑥𝑦implies𝑇𝑥𝑇𝑦𝑥𝑦.(3.34) It is natural to define a multivalued version of Condition (𝐶𝜆) (see [18]).
Let 𝐸 be a nonempty subset of a Banach space 𝑋, and let 𝑇𝐸𝐶𝐵(𝑋) be a multivalued mapping. Then 𝑇 is said to satisfy condition (𝐶𝜆) for some 𝜆(0,1) if, for each 𝑥,𝑦𝐸, 𝜆dist(𝑥,𝑇𝑥)𝑥𝑦implies𝐻(𝑇𝑥,𝑇𝑦)𝑥𝑦.(3.35) Clearly, 𝑇 satisfies (2) of condition ().

Proposition 3.20. Let 𝐸 be a nonempty bounded closed and convex subset of a Banach space 𝑋. If 𝑇𝐸𝐶𝐵(𝐸) satisfies condition (𝐶𝜆) for some 𝜆(0,1), then 𝑇 satisfies condition ().

Proof. We only show that 𝐸 contains an afps for 𝑇. But this follows from [37, Lemma  2.8].

Generalized Nonexpansive Mappings
Let 𝐸 be a nonempty subset of a Banach space 𝑋. Following [19], a mapping 𝑇𝐸𝑋 is a generalized nonexpansive mapping if for some nonnegative constants 𝛼1,,𝛼5 with 5𝑖=1𝛼𝑖=1, 𝑇𝑥𝑇𝑦𝛼1𝑥𝑦+𝛼2𝑥𝑇𝑥+𝛼3𝑦𝑇𝑦+𝛼4𝑥𝑇𝑦+𝛼5𝑦𝑇𝑥,(3.36) for each 𝑥,𝑦𝐸.

We will use the following equivalent condition.

For some nonnegative constants 𝛼,𝛽,𝛾 with 𝛼+2𝛽+2𝛾1,𝑇𝑥𝑇𝑦𝛼𝑥𝑦+𝛽(𝑥𝑇𝑥+𝑦𝑇𝑦)+𝛾(𝑥𝑇𝑦+𝑦𝑇𝑥),(3.37)

for all 𝑥,𝑦𝐸.

We introduce a multivalued version of these mappings.

Let 𝑇𝐸𝐶𝐵(𝑋) be a multivalued mapping. 𝑇 is called a generalized nonexpansive mapping if there exist nonnegative constants 𝛼,𝛽,𝛾 with 𝛼+2𝛽+2𝛾1 such that, for each 𝑥,𝑦𝐸, there holds𝐻(𝑇𝑥,𝑇𝑦)𝛼𝑥𝑦+𝛽(dist(𝑥,𝑇𝑥)+dist(𝑦,𝑇𝑦))+𝛾(dist(𝑥,𝑇𝑦)+dist(𝑦,𝑇𝑥)).(3.38)

Proposition 3.21. Let 𝐸 be a nonempty subset of a Banach space 𝑋. If 𝑇𝐸𝐶𝐵(𝑋) is a generalized nonexpansive mapping, then 𝑇 satisfies (2) of condition ().

Proof. Let {𝑥𝑛} be an afps for 𝑇 in 𝐸 and 𝑥𝐸. By assumption we obtain 𝐻𝑇𝑥𝑛𝑥,𝑇𝑥𝛼𝑛𝑥𝑥+𝛽dist𝑛,𝑇𝑥𝑛𝑥+dist(𝑥,𝑇𝑥)+𝛾dist𝑛,𝑇𝑥+dist𝑥,𝑇𝑥𝑛.(3.39) Since dist(𝑥,𝑇𝑥)𝑥𝑥𝑛+dist(𝑥𝑛,𝑇𝑥𝑛)+𝐻(𝑇𝑥𝑛,𝑇𝑥),dist(𝑥𝑛,𝑇𝑥)dist(𝑥𝑛,𝑇𝑥𝑛)+𝐻(𝑇𝑥𝑛,𝑇𝑥), dist(𝑥,𝑇𝑥𝑛)𝑥𝑥𝑛+dist(𝑥𝑛,𝑇𝑥𝑛), dist(𝑥,𝑇𝑥)lim𝑛𝒰𝑥𝑥𝑛+lim𝑛𝒰𝐻𝑇𝑥𝑛,,𝑇𝑥(3.40)lim𝑛𝒰𝑥dist𝑛,𝑇𝑥lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑥,(3.41)lim𝑛𝒰dist𝑥,𝑇𝑥𝑛lim𝑛𝒰𝑥𝑥𝑛.(3.42) By (3.39), lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑥𝛼lim𝑛𝒰𝑥𝑛𝑥+𝛽lim𝑛𝒰𝑥𝑥𝑛+𝛽lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑥+𝛾lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑥+𝛾lim𝑛𝒰𝑥𝑥𝑛.(3.43) Thus (1𝛽𝛾)lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑥(𝛼+𝛽+𝛾)lim𝑛𝒰𝑥𝑥𝑛,(3.44) and therefore, lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑥lim𝑛𝒰𝑥𝑛𝑥.(3.45)

Takahashi Generalized Nonexpansive Mappings
Definition 3.22. Let 𝐸 be a nonempty subset of a Banach space 𝑋. A mapping 𝑇𝐸𝑋 is said to be a Takahashi generalized nonexpansive mapping if, for some 𝛼,𝛽[0,1] with 𝛼+2𝛽1, there holds 𝑇𝑥𝑇𝑦2𝛼𝑥𝑦2+𝛽𝑦𝑇𝑥2+𝑥𝑇𝑦2for𝑥,𝑦𝐸.(3.46)
The following are examples of Takahashi generalized nonexpansive mappings: (i)nonexpansive mappings 𝑇𝑇𝑥𝑇𝑦𝑥𝑦; (ii)nonspreading mappings 𝑇 [38]: 2𝑇𝑥𝑇𝑦2𝑦𝑇𝑥2+𝑥𝑇𝑦2; (iii)hybrid mappings 𝑇 [39]: 3𝑇𝑥𝑇𝑦2𝑦𝑇𝑥2+𝑥𝑇𝑦2; (iv)mappings 𝑇 [39]: 2𝑇𝑥𝑇𝑦2𝑥𝑦2+𝑦𝑇𝑥2; (v)mappings 𝑇: 3𝑇𝑥𝑇𝑦22𝑦𝑇𝑥2+𝑥𝑇𝑦2.
We define a multivalued version of Takahashi generalized nonexpansive mappings and prove that these mappings satisfy (2) of condition ().
Proposition 3.23. Let 𝐸 be a nonempty subset of a Banach space 𝑋. For nonnegative constants 𝛼,𝛽 with 𝛼+2𝛽1, if 𝑇𝐸𝐾𝐶(𝑋) is a multivalued mapping such that 𝐻2(𝑇𝑥,𝑇𝑦)𝛼𝑥𝑦2+𝛽dist2(𝑥,𝑇𝑦)+dist2(𝑦,𝑇𝑥),(3.47) then 𝑇 satisfies (2) of condition ().
Proof. Let {𝑥𝑛} be an afps for 𝑇 in 𝐸 and 𝑥𝐸. By (3.41) and (3.42), lim𝑛𝒰𝐻2𝑇𝑥𝑛,𝑇𝑥𝛼lim𝑛𝒰𝑥𝑛𝑥2+𝛽lim𝑛𝒰dist2𝑥,𝑇𝑥𝑛+𝛽lim𝑛𝒰dist2𝑥𝑛,𝑇𝑥𝛼lim𝑛𝒰𝑥𝑛𝑥2+𝛽lim𝑛𝒰𝑥𝑛𝑥2+𝛽lim𝑛𝒰𝐻2𝑇𝑥𝑛.,𝑇𝑥(3.48) Thus (1𝛽)lim𝑛𝒰𝐻2𝑇𝑥𝑛,𝑇𝑥(𝛼+𝛽)lim𝑛𝒰𝑥𝑥𝑛2.(3.49) Therefore, lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑥lim𝑛𝒰𝑥𝑛𝑥.(3.50)
Remark 3.24. (i)  A mapping that satisfies condition () need not satisfy condition (). Consider a mapping 𝑇[0,1/2]2[0,1/2] defined by 𝑇(𝑥)=[𝑥,3𝑥]. Since 0 is a fixed point of 𝑇, the sequence {𝑥𝑛} given by 𝑥𝑛0 for all 𝑛 forms an afps for 𝑇. Thus, 𝑇 fulfills condition (1) of Definition 3.2. If {𝑥𝑛} is an apfs for 𝑇, then {𝑥𝑛} converges to 0 and 𝐴(𝐸,{𝑥𝑛})={0}. This implies that 𝐴(𝐸,{𝑥𝑛}) has an apfs for 𝑇, and 𝑇 satisfies condition (). On the other hand, for the afps {𝑥𝑛} given by 𝑥𝑛0, if 𝑥(0,1/2], then lim𝑛𝒰𝐻𝑇𝑥𝑛=,𝑇𝑥𝑥>𝑥=lim𝑛𝒰𝑥𝑛𝑥.(3.51) Thus, 𝑇 fails to satisfy condition ().
As mentioned earlier, it is unclear if a mapping, satisfies condition () also satisfies condition ().
(ii)  We do not know if Theorem 3.16 is still valid when “lim𝑛𝑢” in Definition 3.14 is replaced by “limsup𝑛.” It is possible that the theorem holds true when the domain 𝐸 is separable. Indeed, by [27, 40] and Kirk [41], we assume the afps {𝑥𝑛} for 𝑇 to be regular and asymptotically uniform relative to 𝐸. Thus, 𝐴(𝐸,{𝑥𝑛})=𝐴(𝐸,{𝑥𝑛}) for all subsequences {𝑥𝑛} of {𝑥𝑛}. Therefore, by Lemma 2.5, it is easy to see that 𝐴𝒰=𝐴(𝐸,{𝑥𝑛}). If 𝑥𝐴𝒰, then (3.33) becomes lim𝑛𝒰𝑥𝑛𝑧lim𝑛𝒰𝑦𝑛𝑧𝑛=lim𝑛𝒰𝑦dist𝑛,𝑇𝑥lim𝑛𝒰𝐻𝑇𝑥𝑛,𝑇𝑥limsup𝑛𝐻𝑇𝑥𝑛,𝑇𝑥limsup𝑛𝑥𝑛𝑥𝑥=𝑟𝐸,𝑛=𝑟𝒰𝑥𝐸,𝑛.(3.52) Hence 𝐴𝒰𝑇𝑥, and the rest of the proof follows.
It is observed that if 𝐴𝒰=𝐴(𝐸,{𝑥𝑛}) and 𝐴(𝐸,{𝑥𝑛})={𝑥} is a singleton, then 𝑥 is automatically a fixed point of 𝑇. Thus our method provides another proof of Lim [42, Theorem  8], where 𝑋 is a uniformly convex Banach space, 𝑇 assumes only compact values and 𝐸 need not be separable.
(iii)  If 𝑇𝐸𝐸 is a generalized nonexpansive mapping with any of the following conditions holding, then 𝑇 satisfies condition ():(1)𝛼+2𝛽+2𝛾<1 (see [43<