Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

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

S. Dhompongsa, N. Nanan, "Fixed Point Theorems by Ways of Ultra-Asymptotic Centers", Abstract and Applied Analysis, vol. 2011, Article ID 826851, 21 pages, 2011. 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

DomΓ­nguez Benavides and Lorenzo RamΓ­rez [1–3] 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 DomΓ­nguez-Lorenzo condition ((DL) condition, in short), if there exists πœ†βˆˆ[0,1) such that for every weakly compact convex subset 𝐸 of 𝑋 and for every bounded sequence {π‘₯𝑛} in 𝐸 which is regular relative to 𝐸, π‘ŸπΈξ€·π΄ξ€·ξ€½π‘₯𝐸,𝑛π‘₯ξ€Ύξ€Έξ€Έβ‰€πœ†π‘ŸπΈ,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 π‘‡βˆΆπΈβ†’πΎπΆ(𝑋) is a nonexpansive and 1βˆ’πœ’-contractive mapping such that 𝑇(E) is a bounded set and which satisfies the inwardness condition:𝑇π‘₯βŠ‚πΌπΈ(π‘₯)forallπ‘₯∈𝐸, then 𝑇 has a fixed point.

WiΕ›nicki and WoΕ›ko [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 β„•. The coefficient DL𝒰(𝑋) of a Banach space 𝑋 is defined as DLπ’°ξƒ―πœ’(𝑋)=sup𝐸𝐴𝒰π‘₯𝐸,π‘›ξ€Ύξ€Έξ€Έπœ’πΈπ‘₯𝑛,(1.2) where the supremum is taken over all nonempty weakly compact convex subsets 𝐸 of 𝑋 and all weakly, not norm-convergent sequences {π‘₯𝑛} 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 π‘‡βˆΆπΈβ†’πΎπΆ(𝑋) is a nonexpansive and 1βˆ’πœ’-contractive mapping such that 𝑇π‘₯βŠ‚πΌπΈ(π‘₯) for all π‘₯∈𝐸. Then 𝑇 has a fixed point.

Observe that, unlike Theorem 1.2, it does not assume the β€œseparability” condition on 𝐸 in Theorem 1.4. However, it should be mentioned that the idea of the proof came from the original one of DomΓ­nguez Benavides and Lorenzo RamΓ­rez [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 πœ†βˆˆ[0,1) such that for every 𝜏-compact convex subset 𝐸 of 𝑋 and for every bounded ultranet {π‘₯𝛼} in πΈπ‘ŸπΈξ€·π΄ξ€·ξ€½π‘₯𝐸,𝛼π‘₯ξ€Ύξ€Έξ€Έβ‰€πœ†π‘ŸπΈ,𝛼.(1.3) When 𝜏 is the weak topology πœ” we write (DL)𝛼 condition instead of πœ”(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 π‘‡βˆΆπΈβ†’πΎπΆ(𝑋) is a nonexpansive and 1βˆ’πœ’-contractive mapping such that 𝑇π‘₯βŠ‚πΌπΈ(π‘₯) for π‘₯∈𝐸, 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 πœ†βˆˆ[0,1) such that for any nonempty weakly compact convex subset 𝐸 of 𝑋, any sequence {π‘₯𝑛}βŠ‚πΈ which is regular and asymptotically uniform relative to 𝐸, and any sequence {𝑦𝑛}βŠ‚π΄(𝐸,{π‘₯𝑛}) which is regular and asymptotically uniform relative to 𝐸 one has π‘Ÿξ€·ξ€½π‘¦πΈ,𝑛π‘₯ξ€Ύξ€Έβ‰€πœ†π‘ŸπΈ,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 π‘‡βˆΆπΈβ†’πΎπΆ(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 π‘‡βˆΆπΈβ†’πΈ be a mapping on a subset 𝐸 of a Banach space 𝑋. Then 𝑇 is said to satisfy condition (βˆ—) 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 π‘Š of 𝐸, 𝐴(π‘Š,{π‘₯𝑛}) is 𝑇-invariant for each afps {π‘₯𝑛} in 𝐾.

The main fixed point theorem concerning condition (βˆ—) 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 π‘‡βˆΆπΈβ†’πΈ satisfy conditon (βˆ—). If 𝑇 is continuous, then 𝑇 has a fixed point.

In fact, we can replace β€œcontinuity” by a weaker condition, namely β€œπΌβˆ’π‘‡ is strongly demiclosed at 0”: for every sequence {π‘₯𝑛} in 𝐸 strongly converges to π‘§βˆˆπΈ and such that π‘₯π‘›βˆ’π‘‡π‘₯𝑛→0 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 β„•. The coefficient 𝐷𝒰(𝑋) of a Banach space 𝑋 is defined as π·π’°ξƒ―πœ’(𝑋)=supπΈπ‘¦ξ€·ξ€½π‘›ξ€Ύξ€Έπœ’πΈπ‘₯𝑛,(1.5) where the supremum is taken over all nonempty weakly compact convex subsets 𝐸 of 𝑋, all sequences {π‘₯𝑛} in 𝐸 which are weakly, not norm-convergent and are regular relative to 𝐸 and all weakly, not norm-convergent sequences {𝑦𝑛}βŠ‚π΄π’°(𝐸,{π‘₯𝑛}) 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 πœ†βˆˆ[0,1) such that for any nonempty weakly compact convex subset 𝐸 of 𝑋, any sequence {π‘₯𝑛}βŠ‚πΈ which is regular relative to 𝐸, and any sequence {𝑦𝑛}βŠ‚π΄(𝐸,{π‘₯𝑛}) which is regular relative to 𝐸 one has π‘Ÿξ€·ξ€½π‘¦πΈ,𝑛π‘₯ξ€Ύξ€Έβ‰€πœ†π‘ŸπΈ,𝑛.(1.6)

Obviously, DL𝒰(𝑋)<1β‡’(DL)β‡’(π·ξ…ž)β‡’(𝐷). Several well-known spaces have been proved to satisfy the (DL) condition and to have property (𝐷′) (see, e.g., [1, 4, 7, 11–16]). 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 π‘‡βˆΆπΈβ†’πΎπΆ(𝑋) is a nonexpansive and 1βˆ’πœ’-contractive mapping such that 𝑇π‘₯βŠ‚πΌπΈ(π‘₯) for every π‘₯∈𝐸. 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 β€œseparability” 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 β„•. 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 [8–10, 18–25] 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, 𝐢𝐡(𝑋) the family of all nonempty bounded and closed subsets of 𝑋 and denote by 𝐾𝐢(𝑋) the family of all nonempty compact convex subsets of 𝑋. For a given mapping π‘‡βˆΆπΈβ†’πΆπ΅(𝑋) the set of all fixed points of 𝑇 will be denoted by 𝐹(𝑇), that is, 𝐹(𝑇)∢={π‘₯∈𝐸∢π‘₯βˆˆπ‘‡π‘₯}. Let 𝐻(β‹…,β‹…) be the Hausdorff distance defined on 𝐢𝐡(𝑋), that is,𝐻(𝐴,𝐡)∢=maxsupπ‘Žβˆˆπ΄dist(a,B),supπ‘βˆˆπ΅ξ‚Όdist(𝑏,𝐴),𝐴,𝐡∈𝐢𝐡(𝑋),(2.1)

where dist(π‘Ž,𝐡)∢=inf{β€–π‘Žβˆ’π‘β€–βˆΆπ‘βˆˆπ΅} is the distance from a point π‘Ž to a subset 𝐡. A multivalued mapping π‘‡βˆΆπΈβ†’πΆπ΅(𝑋) is said to be nonexpansive if𝐻(𝑇π‘₯,𝑇𝑦)≀‖π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΈ,(2.2)

and 𝑇 is said to be a contraction if there exists a constant π‘˜<1 such that𝐻(𝑇π‘₯,𝑇𝑦)β‰€π‘˜β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΈ.(2.3) A multivalued mapping π‘‡βˆΆπΈβ†’2𝑋 is called πœ™-condensing (resp., 1βˆ’πœ™-contractive), where πœ™ is a measure of noncompactness, if for each bounded subset 𝐡 of 𝐸 with πœ™(𝐡)>0, there holds the inequalityπœ™(𝑇(𝐡))<πœ™(𝐡)(resp.πœ™(𝑇(𝐡))β‰€πœ™(𝐡)),(2.4)

where ⋃𝑇(𝐡)=π‘₯βˆˆπ΅π‘‡π‘₯.

Recall that the inward set of 𝐸 at π‘₯∈𝐸 is defined by𝐼𝐸(π‘₯)={π‘₯+𝛼(π‘¦βˆ’π‘₯)βˆΆπ›Όβ‰₯1,π‘¦βˆˆπΈ}.(2.5) A sequence {π‘₯𝑛} in 𝐸 for which limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖=0 for a mapping π‘‡βˆΆπΈβ†’πΈ is called an approximate fixed point sequence (afps for short) for 𝑇. Analogously for a multivalued mapping π‘‡βˆΆπΈβ†’πΆπ΅(𝑋), a sequence {π‘₯𝑛} in 𝐸 of a Banach space 𝑋 for which limπ‘›β†’βˆždist(π‘₯𝑛,𝑇π‘₯𝑛)=0 is called an approximate fixed point sequence (afps for short) for 𝑇.

We denote by π‘₯𝑛→π‘₯ to indicate that the sequence {π‘₯𝑛} in 𝑋 converges to π‘₯βˆˆπ‘‹.

Let 𝐸 be a nonempty closed and convex subset of a Banach space 𝑋 and {π‘₯𝑛} a bounded sequence in 𝑋. For π‘₯βˆˆπ‘‹, define the asymptotic radius of {π‘₯𝑛} at π‘₯ as the numberπ‘Ÿξ€·ξ€½π‘₯π‘₯,𝑛=limsupπ‘›β†’βˆžβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘₯.(2.6) Letπ‘Ÿξ€·ξ€½π‘₯𝐸,π‘›ξ€½π‘Ÿξ€·ξ€½π‘₯ξ€Ύξ€ΈβˆΆ=infπ‘₯,𝑛,𝐴π‘₯ξ€Ύξ€ΈβˆΆπ‘₯∈𝐸𝐸,𝑛π‘₯ξ€Ύξ€ΈβˆΆ=π‘₯βˆˆπΈβˆΆπ‘Ÿπ‘₯,𝑛π‘₯ξ€Ύξ€Έ=π‘ŸπΈ,𝑛.ξ€Ύξ€Έξ€Ύ(2.7) The number π‘Ÿ(𝐸,{π‘₯𝑛}) and the set 𝐴(𝐸,{π‘₯𝑛}) are, respectively, called the asymptotic radius and asymptotic center of {π‘₯𝑛} relative to 𝐸. The sequence {π‘₯𝑛} is called regular relative to 𝐸 if π‘Ÿ(𝐸,{π‘₯𝑛})=π‘Ÿ(𝐸,{π‘₯π‘›ξ…ž}) for each subsequence {π‘₯π‘›ξ…ž} of {π‘₯𝑛}. It was noted in [26] that if 𝐸 is nonempty and weakly compact, then 𝐴(𝐸,{π‘₯𝑛}) is nonempty and weakly compact, and if 𝐸 is convex, then 𝐴(𝐸,{π‘₯𝑛}) is convex.

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

We now present the formulation of an ultrapower of Banach spaces. Let 𝒰 be a free ultrafilter on β„•. Recall ([26, 28–30]) that the ultrapower (𝑋)𝒰 of a Banach space 𝑋 is the quotient space ofπ‘™βˆžξ‚»ξ€½π‘₯(𝑋)=π‘›ξ€ΎβˆΆπ‘₯𝑛‖‖π‘₯βˆˆπ‘‹βˆ€π‘›βˆˆβ„•,𝑛‖‖=sup𝑛‖‖π‘₯𝑛‖‖<∞(2.8) byξ‚»ξ€½π‘₯ker𝒩=π‘›ξ€Ύβˆˆπ‘™βˆž(𝑋)∢lim𝑛→𝒰‖‖π‘₯𝑛‖‖=0.(2.9) One can proof that 𝑋=(𝑋)𝒰 is a Banach space with the quotient norm given by β€–{π‘₯𝑛}𝒰‖=lim𝑛→𝒰‖π‘₯𝑛‖, where {π‘₯𝑛}𝒰 is the equivalence class of {π‘₯𝑛}. It is also clear that 𝑋 is isometric to a subspace of 𝑋 by the canonical embedding π‘₯↦{π‘₯,π‘₯,…}𝒰. If πΈβŠ‚π‘‹, we will use the symbols ̇𝐸 and Μ‡π‘₯ to denote the image of 𝐸 and π‘₯ in 𝑋 under this isometry, respectively, and denoteπ‘₯𝐸=Μƒπ‘₯βˆˆπ‘‹βˆΆβˆƒπ‘›ξ€Ύξ€½π‘₯suchthatΜƒπ‘₯=𝑛𝒰,π‘₯π‘›ξ‚‡βˆˆπΈβˆ€π‘›βˆˆβ„•.(2.10)

Thus Μ‡π‘₯={π‘₯,π‘₯,…}𝒰 and ̇𝐸={Μ‡π‘₯βˆˆπ‘‹βˆΆπ‘₯∈𝐸}.

If π‘‡βˆΆπΈβ†’πΆπ΅(𝑋) is a multivalued mapping, we define a corresponding multivalued mapping ξ‚ξ‚ξ‚π‘‡βˆΆπΈβ†’πΆB(𝑋) by𝑒𝑇(Μƒπ‘₯)∢=Μƒπ‘’βˆˆπ‘‹βˆΆβˆƒπ‘›ξ€Ύξ€½π‘’suchthat̃𝑒=𝑛𝒰,π‘’π‘›βˆˆπ‘‡π‘₯π‘›ξ‚‡βˆ€π‘›βˆˆβ„•,(2.11)

where Μƒπ‘₯={π‘₯𝑛}π’°βˆˆξ‚πΈ. Moreover the set 𝑇(Μƒπ‘₯) is bounded and closed (see [28, 29]). The Hausdorff metric on 𝐢𝐡(𝑋) will be denoted by 𝐻.

Proposition 2.2 (see [5, Proposition  3.1]). For every {π‘₯𝑛}𝒰 and {𝑦𝑛}𝒰 in 𝐸, 𝐻𝑇π‘₯𝑛𝒰,𝑇𝑦𝑛𝒰=lim𝑛→𝒰𝐻𝑇π‘₯𝑛,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 π‘‡βˆΆπΈβ†’πΆπ΅(𝑋). (i)If 𝑇 is convex-valued, then 𝑇 is convex-valued. (ii)If 𝑇 is compact-valued, then 𝑇 is compact-valued and ̇𝑇̇π‘₯=(𝑇π‘₯) for every π‘₯∈𝐸. (iii)If 𝑇 is nonexpansive, then 𝑇 is nonexpansive.

Let 𝒰 denote a free ultrafilter defined on β„•. WiΕ›nicki and WoΕ›ko [5] defined the ultra-asymptotic radius π‘Ÿπ’°(𝐸,{π‘₯𝑛}) and the ultra-asymptotic center 𝐴𝒰(𝐸,{π‘₯𝑛}) of {π‘₯𝑛} relative to 𝐸 by π‘Ÿπ’°ξ€·ξ€½π‘₯𝐸,𝑛=inflim𝑛→𝒰‖‖π‘₯𝑛‖‖,π΄βˆ’π‘₯∢π‘₯βˆˆπΈπ’°ξ€·ξ€½π‘₯𝐸,𝑛=ξ‚»ξ€Ύξ€Έπ‘₯∈𝐸∢lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘₯=π‘Ÿπ’°ξ€·ξ€½π‘₯𝐸,𝑛.ξ€Ύξ€Έ(2.13) It is not difficult to see that 𝐴𝒰(𝐸,{π‘₯𝑛}) is a nonempty weakly compact convex set if 𝐸 is. Notice that the above notions have a natural interpretation in the ultrapower 𝑋 [5]:π‘Ÿπ’°ξ€·ξ€½π‘₯𝐸,𝑛=infπ‘₯βˆˆπΈβ€–β€–ξ€½π‘₯π‘›ξ€Ύπ’°β€–β€–βˆ’Μ‡π‘₯(2.14)

is the relative Chebyshev radius of {π‘₯𝑛}𝒰, and𝐴𝒰̇π‘₯𝐸,𝑛=Μ‡ξ€Ύξ€Έξ€ΈπΈβˆ©π΅ξ‚π‘‹π‘₯𝑛𝒰,π‘Ÿ(2.15)

is the relative Chebyshev center of {π‘₯𝑛}𝒰 relative to ̇𝐸 in the ultrapower 𝑋. (Here 𝐡𝑋({π‘₯𝑛}𝒰,π‘Ÿ) denotes the ball in 𝑋 centered at {π‘₯𝑛}𝒰 and of radius π‘Ÿ=π‘Ÿπ’°(𝐸,{π‘₯𝑛}).) It should be noted that, in general, 𝐴(𝐸,{π‘₯𝑛}) and 𝐴𝒰(𝐸,{π‘₯𝑛}) may be different. The notion of the asymptotic radius is closely related to the notion of the relative Hausdorff measure of noncompactness defined by DomΓ­nguez Benavides and Lorenzo RamΓ­rez [1] asπœ’πΈ(𝐴)=inf{πœ€>0∢𝐴canbecoveredbyfinitelymanyballsin𝐸ofradiiβ‰€πœ€}.(2.16)

Proposition 2.4 (see [5, Proposition  4.5]). If {π‘₯𝑛} is a bounded sequence which is regular relative to 𝐸, then π‘Ÿξ€·ξ€½π‘₯𝐸,𝑛=π‘Ÿπ’°ξ€·ξ€½π‘₯𝐸,𝑛=πœ’πΈπ‘₯𝑛.(2.17)

From Proposition 2.4, we have, for π‘€βˆˆπ΄(𝐸,{π‘₯𝑛}),lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘€β‰€limsupπ‘›β†’βˆžβ€–β€–π‘₯𝑛‖‖π‘₯βˆ’π‘€=π‘ŸπΈ,𝑛=π‘Ÿπ’°ξ€·ξ€½π‘₯𝐸,𝑛.(2.18)

Therefore, 𝐴(𝐸,{π‘₯𝑛})βŠ‚π΄π’°(𝐸,{π‘₯𝑛}).

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 {π‘₯𝑛} a bounded sequence in 𝑋 which is regular relative to 𝐸. For each {𝑦𝑛}βŠ‚π΄π’°(𝐸,{π‘₯𝑛}), there exists a subsequence {π‘₯π‘›ξ…ž} of {π‘₯𝑛} such that {𝑦𝑛}βŠ‚π΄(𝐸,{π‘₯π‘›ξ…ž}).

A direct consequence of Lemma 2.5 is as follows. If every center 𝐴(𝐸,{π‘₯𝑛}) is compact for every bounded sequence {π‘₯𝑛} in E which is regular relative to 𝐸, then 𝐴𝒰(𝐸,{π‘₯𝑛}) is also compact for every bounded sequence {π‘₯𝑛} 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 π‘‡βˆΆπΈβ†’πΆπ΅(𝑋). Then (1)if 𝑇 is uniformly continuous, then 𝑇 is uniformly continuous; (2)if 𝑇 is continuous at π‘§βˆˆπΈ, then 𝑇 is continuous at ΕΌ.

Proof. (1) Let πœ€>0. Since 𝑇 is uniformly continuous, there exists 𝛿>0 such that 𝐻(𝑇π‘₯,𝑇𝑦)<πœ€ for each π‘₯,π‘¦βˆˆπΈ with β€–π‘₯βˆ’π‘¦β€–<𝛿. Suppose {π‘₯𝑛}𝒰,{𝑦𝑛}π’°βˆˆξ‚πΈ, and β€–{π‘₯𝑛}π’°βˆ’{𝑦𝑛}𝒰‖<𝛿. Let 𝐴={π‘›βˆΆβ€–π‘₯π‘›βˆ’π‘¦π‘›β€–<𝛿} and 𝐡={π‘›βˆΆπ»(𝑇π‘₯𝑛,𝑇𝑦𝑛)<πœ€}. Since π΄βˆˆπ’° and π΄βŠ‚π΅, π΅βˆˆπ’°. Thus, by Proposition 2.2𝐻(𝑇{π‘₯𝑛}𝒰,𝑇{𝑦𝑛}𝒰)β‰€πœ€.
(2) Let πœ€>0. Since 𝑇 is continuous at 𝑧, there exists 𝛿>0 such that 𝐻(𝑇π‘₯,𝑇𝑧)<πœ€ for each π‘₯∈𝐸 with β€–π‘₯βˆ’π‘§β€–<𝛿. If {π‘₯𝑛}π’°βˆˆξ‚πΈ such that β€–{π‘₯𝑛}π’°βˆ’Μ‡π‘§β€–<𝛿, then, letting 𝐴={π‘›βˆΆβ€–π‘₯π‘›βˆ’π‘§β€–<𝛿} and 𝐡={π‘›βˆΆπ»(𝑇π‘₯𝑛,𝑇𝑧)<πœ€}, we see that π΄βˆˆπ’° and π΅βˆˆπ’°. Thus, by Proposition 2.2𝐻(𝑇{π‘₯𝑛}𝒰,𝑇̇𝑧)β‰€πœ€.

We now introduce condition (βˆ—) for multivalued mappings.

Definition 3.2. Let 𝐸 be a nonempty subset of a Banach space 𝑋. A mapping π‘‡βˆΆπΈβ†’πΆπ΅(𝑋) is said to satisfy condition (βˆ—) if (1)𝑇 has an afps in 𝐸, (2)𝑇 has an afps in 𝐴(𝐸,{π‘₯π‘›ξ…ž}) for some subsequence {π‘₯π‘›ξ…ž} of any given afps {π‘₯𝑛} 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 π‘‡βˆΆπΈβ†’πΎπΆ(𝑋) is a multivalued mapping satisfying condition (βˆ—). 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 {π‘₯0𝑛} be an afps for 𝑇 in 𝐸. We can assume by Proposition 2.1 that {π‘₯0𝑛} is regular relative to 𝐸. Condition (2) of Definition 3.2 gives us a subsequence {π‘₯0𝑛0} of {π‘₯0𝑛} so that the center 𝐴(𝐸,{π‘₯0𝑛0}) contains an afps for 𝑇. Denote 𝐴0=𝐴(𝐸,{π‘₯0𝑛0}), and let {π‘₯1𝑛} be an afps in 𝐴0. Assume that {π‘₯1𝑛} is regular relative to 𝐸. As before, 𝑇 has an afps in 𝐴(𝐸,{π‘₯1𝑛1}) for some subsequence {π‘₯1𝑛1} of {π‘₯1𝑛}. Since 𝑋 has property (𝐷′), put πœ†=𝐷𝒰(𝑋)<1. Then, by Proposition 2.4 and Definition 1.11, π‘Ÿξ€·ξ€½π‘₯𝐸,1𝑛1ξ€Ύξ€Έ=πœ’Eπ‘₯ξ€·ξ€½1𝑛1ξ€Ύξ€Έβ‰€πœ†πœ’πΈπ‘₯ξ€·ξ€½0𝑛0ξ€·ξ€½π‘₯ξ€Ύξ€Έ=πœ†π‘ŸπΈ,0𝑛0.ξ€Ύξ€Έ(3.1) Continue the procedure to obtain, for each π‘šβ‰₯0, a regular sequence {π‘₯π‘šπ‘›π‘š} relative to 𝐸 in π΄π‘šβˆ’1∢=𝐴(𝐸,{π‘₯π‘›π‘šβˆ’1π‘šβˆ’1}) such that limπ‘›β†’βˆžξ€·π‘₯distπ‘šπ‘›π‘š,𝑇π‘₯π‘šπ‘›π‘šξ€Έ=0,(3.2) and for all π‘šβ‰₯1, πœ’πΈπ‘₯ξ€·ξ€½π‘šπ‘›π‘šξ€·ξ€½π‘₯ξ€Ύξ€Έβ‰€πœ†π‘ŸπΈ,π‘›π‘šβˆ’1π‘šβˆ’1ξ€Ύξ€Έ.(3.3) Consequently, π‘Ÿξ€·ξ€½π‘₯𝐸,π‘šπ‘›π‘šξ€·ξ€½π‘₯ξ€Ύξ€Έβ‰€πœ†π‘ŸπΈ,π‘›π‘šβˆ’1π‘šβˆ’1ξ€Ύξ€Έβ‰€β‹―β‰€πœ†π‘šπ‘Ÿξ€·ξ€½π‘₯𝐸,0𝑛0.ξ€Ύξ€Έ(3.4) We show that {π‘₯π‘šπ‘›π‘š}𝒰 is a Cauchy sequence in 𝑋. Indeed, for each π‘šβ‰₯1, take an element Μ‡π‘¦π‘šβˆˆΜ‡π΄π‘šβˆ’1. Then β€–β€–Μ‡π‘₯π‘šπ‘›π‘šβˆ’Μ‡π‘¦π‘šβ€–β€–β‰€β€–β€–Μ‡π‘₯π‘šπ‘›π‘šβˆ’ξ€½π‘₯π‘›π‘šβˆ’1π‘šβˆ’1𝒰‖‖+β€–β€–ξ€½π‘₯π‘›π‘šβˆ’1π‘šβˆ’1ξ€Ύπ’°βˆ’Μ‡π‘¦π‘šβ€–β€–ξ€·ξ€½π‘₯≀2π‘ŸπΈ,π‘›π‘šβˆ’1π‘šβˆ’1ξ€Ύξ€Έ(3.5) for all π‘šβ‰₯1, and hence β€–β€–ξ€½π‘₯π‘šπ‘›π‘šξ€Ύπ’°βˆ’ξ€½π‘₯π‘›π‘šβˆ’1π‘šβˆ’1𝒰‖‖≀‖‖π‘₯π‘šπ‘›π‘šξ€Ύπ’°βˆ’Μ‡π‘¦π‘šβ€–β€–+β€–β€–Μ‡π‘¦π‘šβˆ’ξ€½π‘₯π‘›π‘šβˆ’1π‘šβˆ’1𝒰‖‖π‘₯≀3π‘ŸπΈ,π‘›π‘šβˆ’1π‘šβˆ’1ξ€Ύξ€Έ.(3.6) Thus β€–β€–ξ€½π‘₯π‘šπ‘›π‘šξ€Ύπ’°βˆ’ξ€½π‘₯π‘›π‘šβˆ’1π‘šβˆ’1𝒰‖‖≀3πœ†π‘šβˆ’1π‘Ÿξ€·ξ€½π‘₯𝐸,0𝑛0ξ€Ύξ€Έ,(3.7) implying that {π‘₯π‘šπ‘›π‘š}𝒰 is a Cauchy sequence and hence converges to some {𝑧𝑛}𝒰 in 𝐸 as π‘šβ†’βˆž. Next, we will show that {𝑧𝑛}π’°βˆˆΜ‡πΈ. For each π‘šβ‰₯0, 𝑧dist𝑛𝒰,Μ‡πΈξ€Έβ‰€β€–β€–ξ€½π‘§π‘›ξ€Ύπ’°βˆ’ξ€½π‘₯π‘šπ‘›π‘šξ€Ύπ’°β€–β€–ξ‚€ξ€½π‘₯+𝑑𝑖𝑠tπ‘šπ‘›π‘šξ€Ύπ’°,̇𝐸≀‖‖{𝑧𝑛}π’°βˆ’ξ€½π‘₯π‘šπ‘›π‘šξ€Ύπ’°β€–β€–+β€–β€–ξ€½π‘₯π‘šπ‘›π‘šξ€Ύπ’°βˆ’Μ‡π‘₯1π‘š+1π‘š+1β€–β€–=β€–β€–ξ€½π‘§π‘›ξ€Ύπ’°βˆ’ξ€½π‘₯π‘šπ‘›π‘šξ€Ύπ’°β€–β€–ξ€·ξ€½π‘₯+π‘ŸπΈ,π‘šπ‘›π‘šβ‰€β€–β€–ξ€½π‘§ξ€Ύξ€Έπ‘›ξ€Ύπ’°βˆ’ξ€½π‘₯π‘šπ‘›π‘šξ€Ύπ’°β€–β€–+πœ†π‘šπ‘Ÿξ€·ξ€½π‘₯𝐸,0𝑛0.ξ€Ύξ€Έ(3.8) Taking π‘šβ†’βˆž we see that 𝑧dist𝑛𝒰,̇𝐸=0.(3.9) Thus, it follows that there exists π‘§βˆˆπΈ such that {𝑧𝑛}𝒰=̇𝑧. By Lemma 3.1, 𝑇 is continuous at ΕΌ, and thus 𝐻(𝑇{π‘₯π‘šπ‘›π‘š}𝒰,𝑇̇𝑧)β†’0 as π‘šβ†’βˆž. For every π‘šβ‰₯0, ≀‖‖π‘₯disṫ𝑧,π‘‡Μ‡π‘§Μ‡π‘§βˆ’π‘šπ‘›π‘šξ€Ύπ’°β€–β€–ξ‚€ξ€½π‘₯+distπ‘šπ‘›π‘šξ€Ύπ’°,𝑇π‘₯π‘šπ‘›π‘šξ€Ύπ’°ξ‚+𝐻𝑇π‘₯π‘šπ‘›π‘šξ€Ύπ’°,𝑇̇𝑧.(3.10) Taking π‘šβ†’βˆž we then obtain ξ‚Μ‡π‘§βˆˆπ‘‡Μ‡π‘§. By Proposition 2.3, ̇𝑇̇𝑧=(𝑇𝑧), and therefore, π‘§βˆˆπ‘‡π‘§.

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 WiΕ›nicki and WoΕ›ko. 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 (βˆ—) 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:(βˆ—βˆ—)+(𝐴)⟹(βˆ—).(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 (βˆ—). We will see that the ultracenter 𝐴𝒰(𝐸,{π‘₯𝑛}) plays a significant role in verifying condition (2) of condition (βˆ—) for a given mapping.

Nonexpansive Mappings
We will show by following the proof of Theorem  5.3 in [5] that if π‘‡βˆΆπΈβ†’πΎπΆ(𝑋) is nonexpansive and 1βˆ’πœ’-contractive such that 𝑇π‘₯βŠ‚πΌπΈ(π‘₯) for every π‘₯∈𝐸. Then 𝑇 satisfies condition (βˆ—). 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 πΉβˆΆπΈβ†’πΎπΆ(𝑋) an upper semicontinuous and πœ’-condensing mapping. If 𝐹(π‘₯)∩𝐼𝐸(π‘₯)β‰ βˆ… for all π‘₯∈𝐸, 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 π‘‡βˆΆπΈβ†’πΎπΆ(𝑋) is nonexpansive and 1βˆ’πœ’-contractive such that 𝑇π‘₯βŠ‚πΌπΈ(π‘₯) for every π‘₯∈𝐸. Then 𝑇 satisfies condition (βˆ—).
Proof. First, we will show that 𝑇 has an afps in 𝐸. Let 𝑦0∈𝐸, and consider, for each 𝑛β‰₯1, the contraction π‘‡π‘›βˆΆπΈβ†’πΎπΆ(𝑋) defined by 𝑇𝑛1(π‘₯)=𝑛𝑦0+ξ‚€11βˆ’π‘›ξ‚π‘‡π‘₯,π‘₯∈𝐸.(3.12) It is not difficult to see that 𝑇𝑛(π‘₯)βŠ‚πΌπΈ(π‘₯) for every π‘₯∈𝐸. Since 𝑇 is 1βˆ’πœ’-contractive, 𝑇𝑛 is (1βˆ’(1/𝑛))βˆ’πœ’-contractive, and by Theorem 3.5, there exists a fixed point π‘₯𝑛 of 𝑇𝑛. Clearly, {π‘₯𝑛} is an afps for 𝑇 in 𝐸.
Next, let us see that 𝑇 has an afps in 𝐴(𝐸,{π‘₯𝑛′}) for some subsequence {π‘₯𝑛′} of an afps {π‘₯𝑛} for 𝑇 in 𝐸. Let {π‘₯𝑛} be an afps in 𝐸. By Proposition 2.1, we can assume that {π‘₯𝑛} is weakly convergent and regular relative to 𝐸. Let π΄π’°βˆΆ=𝐴𝒰(𝐸,{π‘₯𝑛}). We show that 𝑇π‘₯βˆ©πΌπ΄π’°(π‘₯)β‰ βˆ…foreveryπ‘₯βˆˆπ΄π’°.(3.13) Let π‘₯βˆˆπ΄π’°. Observe first that {π‘₯𝑛}π’°βˆˆξ‚π‘‡{π‘₯𝑛}𝒰. By Proposition 2.3, ̇𝑇̇π‘₯=(𝑇π‘₯) is compact, and hence there exists π‘’βˆˆπ‘‡π‘₯ such that β€–β€–ξ€½π‘₯𝑛𝒰‖‖=𝐻𝑇π‘₯βˆ’Μ‡π‘’π‘›ξ€Ύπ’°,≀‖‖π‘₯𝑇̇π‘₯π‘›ξ€Ύπ’°β€–β€–βˆ’Μ‡π‘₯=π‘Ÿπ’°ξ€·ξ€½π‘₯𝐸,𝑛.(3.14) Since π‘’βˆˆπ‘‡π‘₯βŠ‚πΌπΈ(π‘₯), there exists 𝛼β‰₯1 and π‘¦βˆˆπΈ such that 𝑒=π‘₯+𝛼(π‘¦βˆ’π‘₯). If 𝛼=1 then 𝑒=π‘¦βˆˆπΈ, and it follows from (3.14) that π‘’βˆˆπ΄π’°. If 𝛼>1 then 𝑦=(1/𝛼)𝑒+(1βˆ’1/𝛼)π‘₯, and therefore, we have β€–β€–ξ€½π‘₯𝑛𝒰‖‖≀1βˆ’Μ‡π‘¦π›Όβ€–β€–ξ€½π‘₯𝑛𝒰‖‖+ξ‚€1βˆ’Μ‡π‘’1βˆ’π›Όξ‚β€–β€–ξ€½π‘₯π‘›ξ€Ύπ’°β€–β€–βˆ’Μ‡π‘₯β‰€π‘Ÿπ’°ξ€·ξ€½π‘₯𝐸,𝑛.(3.15) Hence π‘¦βˆˆπ΄π’° and consequently π‘’βˆˆπΌπ΄π’°(π‘₯). Thus (3.13) is justified.
Fixed 𝑦0βˆˆπ΄π’°, and consider for each 𝑛β‰₯1, the contraction π‘‡π‘›βˆΆπ΄π’°β†’πΎπΆ(𝑋) defined by 𝑇𝑛1(π‘₯)=𝑛𝑦0+ξ‚€11βˆ’π‘›ξ‚π‘‡π‘₯,π‘₯βˆˆπ΄π’°.(3.16) As before, 𝑇𝑛 is (1βˆ’1/𝑛)βˆ’πœ’-contractive, and by Theorem 3.5, there exists a fixed point π‘§π‘›βˆˆπ΄π’° of 𝑇𝑛. Again, as above, {𝑧𝑛} is an afps for 𝑇 in 𝐴𝒰. By Lemma 2.5, there exists a subsequence {π‘₯𝑛′} of {π‘₯𝑛} such that {𝑧𝑛}βŠ‚π΄(𝐸,{π‘₯𝑛′}).

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(π‘₯,𝑦)∢π‘₯,π‘¦βˆˆπΎ} denotes the diameter of πΎβŠ‚π‘‹.)
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 π‘‡βˆΆπΈβ†’πΈ 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=π‘˜βˆˆπΎπ‘‡π‘˜ and 𝐸 is said to be 𝑇-invariant if 𝑇π‘₯βˆ©πΈβ‰ βˆ… for all π‘₯∈𝐸.
Theorem 3.9 (see [35, Theorem  2.2]). Let 𝐸 be a weakly compact subset of a Banach space 𝑋, and let π‘‡βˆΆπΈβ†’πΎπΆ(𝑋) be a multivalued mapping such that 𝛿(π‘‡πΎβˆ©πΎ)<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 π‘‡βˆΆπΈβ†’πΎπΆ(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 (βˆ—).
Proof. First, we will see that 𝑇 has an afps in 𝐸. Let 𝑦0∈𝐸, and consider, for each 𝑛β‰₯1, the contraction π‘‡π‘›βˆΆπΈβ†’πΎπΆ(𝑋) defined by 𝑇𝑛1(π‘₯)=𝑛𝑦0+ξ‚€11βˆ’π‘›ξ‚π‘‡π‘₯,π‘₯∈𝐸.(3.17) For π‘₯∈𝐸, let π‘Žβˆˆπ‘‡π‘₯∩𝐸. Thus (1/𝑛)𝑦0+(1βˆ’1/𝑛)π‘Žβˆˆπ‘‡π‘›π‘₯∩𝐸, and therefore, 𝑇𝑛π‘₯βˆ©πΈβ‰ βˆ… for every π‘₯∈𝐸. We show that 𝛿(𝑇𝑛𝐾)<𝛿(𝐾)forallclosedsets𝐾with𝛿(𝐾)>0. Let 𝐾 be a closed subset of 𝐸 with 𝛿(𝐾)>0. For π‘₯,π‘¦βˆˆπ‘‡π‘›πΎ, there exist π‘₯β€²,π‘¦β€²βˆˆπ‘‡πΎ such that 1π‘₯=𝑛𝑦0+ξ‚€11βˆ’π‘›ξ‚1π‘₯β€²,𝑦=𝑛𝑦0+ξ‚€11βˆ’π‘›ξ‚π‘¦ξ…ž,(3.18) and this entails β€–π‘₯βˆ’π‘¦β€–=(1βˆ’1/𝑛)β€–π‘₯β€²βˆ’π‘¦β€²β€–β‰€(1βˆ’1/𝑛)𝛿(𝑇𝐾). Hence 𝛿(𝑇𝑛𝐾)≀(1βˆ’1/𝑛)𝛿(𝑇𝐾)<𝛿(𝐾). By Theorem 3.9, there exists a fixed point π‘₯𝑛 of 𝑇𝑛, and thus the sequence {π‘₯𝑛} forms an afps for 𝑇 in 𝐸.
Next, let us see that 𝑇 has an afps in 𝐴(𝐸,{π‘₯𝑛′}) for some subsequence {π‘₯𝑛′} of an afps {π‘₯𝑛} for 𝑇 in 𝐸. Let {π‘₯𝑛} be an afps in 𝐸. We can assume that {π‘₯𝑛} is weakly convergent and regular relative to 𝐸. Let 𝐴𝒰=𝐴𝒰(𝐸,{π‘₯𝑛}). First, we show that π΄π’°βˆ©π‘‡π‘₯β‰ βˆ…,foreveryπ‘₯βˆˆπ΄π’°.(3.19) Let π‘₯βˆˆπ΄π’°, and for each 𝑛β‰₯1, we see that 𝐻(𝑇π‘₯𝑛,𝑇π‘₯)≀𝛿(𝑇{π‘₯𝑛,π‘₯})≀𝛿({π‘₯𝑛,π‘₯})=β€–π‘₯π‘›βˆ’π‘₯β€–. Take π‘¦π‘›βˆˆπ‘‡π‘₯𝑛 so that β€–β€–π‘₯π‘›βˆ’π‘¦π‘›β€–β€–ξ€·π‘₯=dist𝑛,𝑇π‘₯𝑛,(3.20) and select π‘§π‘›βˆˆπ‘‡π‘₯ for each 𝑛 such that β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–ξ€·π‘¦=dist𝑛,𝑇π‘₯.(3.21) Let lim𝑛→𝒰𝑧𝑛=π‘§βˆˆπ‘‡π‘₯. Note that β€–β€–π‘₯𝑛‖‖≀‖‖π‘₯βˆ’π‘§π‘›βˆ’π‘¦π‘›β€–β€–+β€–β€–π‘¦π‘›βˆ’π‘§π‘›β€–β€–+β€–β€–π‘§π‘›β€–β€–βˆ’π‘§.(3.22) We obtain lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘§β‰€limπ‘›β†’π’°β€–β€–π‘¦π‘›βˆ’π‘§π‘›β€–β€–=lim𝑛→𝒰𝑦dist𝑛,𝑇π‘₯≀lim𝑛→𝒰𝐻𝑇π‘₯𝑛,𝑇π‘₯≀lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘₯=π‘Ÿπ’°ξ€·ξ€½π‘₯𝐸,𝑛(3.23) proving that π‘§βˆˆπ΄π’°. Thus (3.19) is satisfied. Fix 𝑦0βˆˆπ΄π’°, and consider, for each 𝑛β‰₯1, the contraction π‘‡π‘›βˆΆπ΄π’°β†’πΎπΆ(𝑋) defined by 𝑇𝑛1(π‘₯)=𝑛𝑦0+ξ‚€11βˆ’π‘›ξ‚π‘‡π‘₯,π‘₯βˆˆπ΄π’°.(3.24) For π‘₯∈𝐸, let π‘Žβˆˆπ΄π’°βˆ©π‘‡π‘₯. Thus (1/𝑛)𝑦0+(1βˆ’1/𝑛)π‘Žβˆˆπ΄π’°βˆ©π‘‡π‘›π‘₯. Therefore, π΄π’°βˆ©π‘‡π‘›π‘₯β‰ βˆ… for every π‘₯βˆˆπ΄π’°. Let 𝐾 be a closed subset of 𝐸 with 𝛿(𝐾)>0. As before, 𝛿(𝑇𝑛𝐾)≀(1βˆ’1/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 {π‘₯𝑛′} of {π‘₯𝑛} such that {𝑧𝑛}βŠ‚π΄(𝐸,{π‘₯𝑛′}).

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 π‘‡βˆΆπΈβ†’πΎπΆ(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 π‘‡βˆΆπΈβ†’πΎπΆ(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 (π·ξ…ž). 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 β„•. Let 𝐸 be a bounded closed and convex subset of a Banach space 𝑋. A mapping π‘‡βˆΆπΈβ†’πΆπ΅(𝑋) is said to satisfy condition (βˆ—βˆ—) if it fulfills the following conditions. (1)𝑇 has an afps in 𝐸; (2)if {π‘₯𝑛} is an apfs for 𝑇 in 𝐸 and π‘₯∈𝐸, then lim𝑛→𝒰𝐻(𝑇π‘₯𝑛,𝑇π‘₯)≀lim𝑛→𝒰‖π‘₯π‘›βˆ’π‘₯β€–.

Remark 3.15. (i)  Let 𝐸 be a bounded closed and convex subset of a Banach space 𝑋, and let a mapping π‘‡βˆΆπΈβ†’πΎπΆ(𝑋) satisfy condition (βˆ—βˆ—). If in addition, 𝑇 satisfies the following:
     (A)  every 𝑇-invariant, closed, and convex subset possesses an afps,
then 𝑇 satisfies condition (βˆ—).

Proof. By (1) of condition (βˆ—βˆ—), let {π‘₯𝑛} be an apfs for 𝑇 in 𝐸. From Proposition 2.1 by passing through a subsequence, we may assume that {π‘₯𝑛} is regular relative to 𝐸. Let 𝐴𝒰=𝐴𝒰(𝐸,{π‘₯𝑛}) and π‘₯βˆˆπ΄π’°. The compactness of 𝑇π‘₯𝑛 implies that for each 𝑛 we can take π‘¦π‘›βˆˆπ‘‡π‘₯𝑛 so that β€–β€–π‘₯π‘›βˆ’π‘¦π‘›β€–β€–ξ€·π‘₯=dist𝑛,𝑇π‘₯𝑛.(3.25) Since 𝑇π‘₯ is compact, select π‘§π‘›βˆˆπ‘‡π‘₯ for each 𝑛 such that β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–ξ€·π‘¦=dist𝑛,𝑇π‘₯.(3.26) Let lim𝑛→𝒰𝑧𝑛=π‘§βˆˆπ‘‡π‘₯. Note that β€–β€–π‘₯𝑛‖‖≀‖‖π‘₯βˆ’π‘§π‘›βˆ’π‘¦π‘›β€–β€–+β€–β€–π‘¦π‘›βˆ’π‘§π‘›β€–β€–+β€–β€–π‘§π‘›β€–β€–βˆ’π‘§.(3.27) We obtain lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘§β‰€limπ‘›β†’π’°β€–β€–π‘¦π‘›βˆ’π‘§π‘›β€–β€–=lim𝑛→𝒰𝑦dist𝑛,𝑇π‘₯≀lim𝑛→𝒰𝐻𝑇π‘₯𝑛,𝑇π‘₯≀lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘₯=π‘Ÿπ’°ξ€·ξ€½π‘₯𝐸,𝑛(3.28) proving that π‘§βˆˆπ΄π’° and hence π΄π’°βˆ©π‘‡π‘₯β‰ βˆ… for all π‘₯βˆˆπ΄π’°, that is, 𝐴𝒰 is 𝑇-invariant. By assumption, there exists an afps in 𝐴𝒰. By Lemma 2.5, there exists a subsequence {π‘₯𝑛′} of {π‘₯𝑛} such that {𝑧𝑛}βŠ‚π΄(𝐸,{π‘₯𝑛′}). Thus, 𝑇 satisfies condition (βˆ—).

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

(ii)  In [23, Definition  3.1] the following concept of mappings is defined: a mapping π‘‡βˆΆπΈβ†’πΈ satisfies condition (𝐿) on 𝐸 provided that it fulfills the following two conditions. (1)If a set π·βŠ‚πΈ is nonempty, closed, convex, and 𝑇-invariant, then there exists an afps for 𝑇 in 𝐷. (2)For any afps {π‘₯𝑛} of 𝑇 in 𝐸 and each π‘₯∈𝐸,limsupπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘‡π‘₯‖≀limsupπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘₯β€–.(3.29)

Therefore, (i) shows that the class of mappings satisfying condition (βˆ—) 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 π‘‡βˆΆπΈβ†’πΎπΆ(𝑋) be a multivalued mapping satisfying condition (βˆ—βˆ—). If 𝑇 is an upper semicontinuous mapping, then 𝑇 has a fixed point.

Proof. Let {π‘₯𝑛} be an afps for 𝑇 in 𝐸. From Proposition 2.1 by passing through a subsequence, we may assume that {π‘₯𝑛} is regular relative to 𝐸. Let 𝐴𝒰=𝐴𝒰(𝐸,{π‘₯𝑛}). The compactness of 𝑇π‘₯𝑛 implies that for each 𝑛 we can take π‘¦π‘›βˆˆπ‘‡π‘₯𝑛 such that β€–β€–π‘₯π‘›βˆ’π‘¦π‘›β€–β€–ξ€·π‘₯=dist𝑛,𝑇π‘₯𝑛.(3.30) If π‘₯βˆˆπ΄π’°, since 𝑇π‘₯ is compact, select π‘§π‘›βˆˆπ‘‡π‘₯ for each 𝑛 such that β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–ξ€·π‘¦=dist𝑛,𝑇π‘₯.(3.31) Let lim𝑛→𝒰𝑧𝑛=π‘§βˆˆπ‘‡π‘₯. Note that β€–β€–π‘₯𝑛‖‖≀‖‖π‘₯βˆ’π‘§π‘›βˆ’π‘¦π‘›β€–β€–+β€–β€–π‘¦π‘›βˆ’π‘§π‘›β€–β€–+β€–β€–π‘§π‘›β€–β€–βˆ’π‘§.(3.32) Thus lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘§β‰€limπ‘›β†’π’°β€–β€–π‘¦π‘›βˆ’π‘§π‘›β€–β€–=lim𝑛→𝒰𝑦dist𝑛,𝑇π‘₯≀lim𝑛→𝒰𝐻𝑇π‘₯𝑛,𝑇π‘₯≀lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘₯=π‘Ÿπ’°ξ€·ξ€½π‘₯𝐸,𝑛(3.33) proving that π‘§βˆˆπ΄π’° and hence π΄π’°βˆ©π‘‡π‘₯β‰ βˆ… for all π‘₯βˆˆπ΄π’°. By assumption, 𝐴(𝐸,{π‘₯𝑛}) is nonempty and compact which implies that 𝐴𝒰 is also nonempty and compact. Now define a mapping πΉβˆΆπ΄π’°β†’πΎπΆ(𝐴𝒰) by 𝐹π‘₯∢=π΄π’°βˆ©π‘‡π‘₯ for all π‘₯βˆˆπ΄π’°. Thus 𝐹 is upper semicontinuous. Indeed, let {𝑒𝑛}βŠ‚π΄ be such that limπ‘›β†’βˆžπ‘’π‘›=𝑒, and let π‘£π‘›βˆˆπΉπ‘’π‘› be such that limπ‘›β†’βˆžπ‘£π‘›=𝑣. Since 𝑇 is upper semicontinuous and 𝐴𝒰 is compact, we have π‘£βˆˆπ‘‡π‘’ and π‘£βˆˆπ΄π’°, that is π‘£βˆˆπΉπ‘’. 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 β€œupper 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 π‘‡βˆΆπΈβ†’πΈ 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 π‘‡βˆΆπΈβ†’πΈ 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 (βˆ—βˆ—). The first example is of course the mapping described in Theorem 3.12.

Condition 3.2 (πΆπœ†). GarcΓ­a-Falset et al. [9] introduced the following mappings.
Let E be a nonempty subset of a Banach space 𝑋. For πœ†βˆˆ(0,1), we say that a mapping π‘‡βˆΆπΈβ†’π‘‹ satisfies Condition (πΆπœ†) on 𝐸 if, for each π‘₯,π‘¦βˆˆπΈ, πœ†β€–π‘₯βˆ’π‘‡π‘₯‖≀‖π‘₯βˆ’π‘¦β€–implies‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–.(3.34) It is natural to define a multivalued version of Condition (πΆπœ†) (see [18]).
Let 𝐸 be a nonempty subset of a Banach space 𝑋, and let π‘‡βˆΆπΈβ†’πΆπ΅(𝑋) be a multivalued mapping. Then 𝑇 is said to satisfy condition (πΆπœ†) for some πœ†βˆˆ(0,1) if, for each π‘₯,π‘¦βˆˆπΈ, πœ†dist(π‘₯,𝑇π‘₯)≀‖π‘₯βˆ’π‘¦β€–implies𝐻(𝑇π‘₯,𝑇𝑦)≀‖π‘₯βˆ’π‘¦β€–.(3.35) Clearly, 𝑇 satisfies (2) of condition (βˆ—βˆ—).

Proposition 3.20. Let 𝐸 be a nonempty bounded closed and convex subset of a Banach space 𝑋. If π‘‡βˆΆπΈβ†’πΆπ΅(𝐸) satisfies condition (πΆπœ†) for some πœ†βˆˆ(0,1), then 𝑇 satisfies condition (βˆ—βˆ—).

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 π‘‡βˆΆπΈβ†’π‘‹ is a generalized nonexpansive mapping if for some nonnegative constants 𝛼1,…,𝛼5 with βˆ‘5𝑖=1𝛼𝑖=1, ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€π›Ό1β€–π‘₯βˆ’π‘¦β€–+𝛼2β€–π‘₯βˆ’π‘‡π‘₯β€–+𝛼3β€–π‘¦βˆ’π‘‡π‘¦β€–+𝛼4β€–π‘₯βˆ’π‘‡π‘¦β€–+𝛼5β€–π‘¦βˆ’π‘‡π‘₯β€–,(3.36) for each π‘₯,π‘¦βˆˆπΈ.

We will use the following equivalent condition.

For some nonnegative constants 𝛼,𝛽,𝛾 with 𝛼+2𝛽+2𝛾≀1,‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€π›Όβ€–π‘₯βˆ’π‘¦β€–+𝛽(β€–π‘₯βˆ’π‘‡π‘₯β€–+β€–π‘¦βˆ’π‘‡π‘¦β€–)+𝛾(β€–π‘₯βˆ’π‘‡π‘¦β€–+β€–π‘¦βˆ’π‘‡π‘₯β€–),(3.37)

for all π‘₯,π‘¦βˆˆπΈ.

We introduce a multivalued version of these mappings.

Let π‘‡βˆΆπΈβ†’πΆπ΅(𝑋) be a multivalued mapping. 𝑇 is called a generalized nonexpansive mapping if there exist nonnegative constants 𝛼,𝛽,𝛾 with 𝛼+2𝛽+2𝛾≀1 such that, for each π‘₯,π‘¦βˆˆπΈ, there holds𝐻(𝑇π‘₯,𝑇𝑦)≀𝛼‖π‘₯βˆ’π‘¦β€–+𝛽(dist(π‘₯,𝑇π‘₯)+dist(𝑦,𝑇𝑦))+𝛾(dist(π‘₯,𝑇𝑦)+dist(𝑦,𝑇π‘₯)).(3.38)

Proposition 3.21. Let 𝐸 be a nonempty subset of a Banach space 𝑋. If π‘‡βˆΆπΈβ†’πΆπ΅(𝑋) is a generalized nonexpansive mapping, then 𝑇 satisfies (2) of condition (βˆ—βˆ—).

Proof. Let {π‘₯𝑛} be an afps for 𝑇 in 𝐸 and π‘₯∈𝐸. By assumption we obtain 𝐻𝑇π‘₯𝑛‖‖π‘₯,𝑇π‘₯≀𝛼𝑛‖‖π‘₯βˆ’π‘₯+𝛽dist𝑛,𝑇π‘₯𝑛π‘₯+dist(π‘₯,𝑇π‘₯)+𝛾dist𝑛,𝑇π‘₯+distπ‘₯,𝑇π‘₯𝑛.ξ€Έξ€Έ(3.39) Since dist(π‘₯,𝑇π‘₯)≀‖π‘₯βˆ’π‘₯𝑛‖+dist(π‘₯𝑛,𝑇π‘₯𝑛)+𝐻(𝑇π‘₯𝑛,𝑇π‘₯),dist(π‘₯𝑛,𝑇π‘₯)≀dist(π‘₯𝑛,𝑇π‘₯𝑛)+𝐻(𝑇π‘₯𝑛,𝑇π‘₯), dist(π‘₯,𝑇π‘₯𝑛)≀‖π‘₯βˆ’π‘₯𝑛‖+dist(π‘₯𝑛,𝑇π‘₯𝑛), dist(π‘₯,𝑇π‘₯)≀lim𝑛→𝒰‖‖π‘₯βˆ’π‘₯𝑛‖‖+lim𝑛→𝒰𝐻𝑇π‘₯𝑛,,𝑇π‘₯(3.40)lim𝑛→𝒰π‘₯dist𝑛,𝑇π‘₯≀lim𝑛→𝒰𝐻𝑇π‘₯𝑛,𝑇π‘₯,(3.41)lim𝑛→𝒰distπ‘₯,𝑇π‘₯𝑛≀lim𝑛→𝒰‖π‘₯βˆ’π‘₯𝑛‖.(3.42) By (3.39), lim𝑛→𝒰𝐻𝑇π‘₯𝑛,𝑇π‘₯≀𝛼lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘₯+𝛽lim𝑛→𝒰‖‖π‘₯βˆ’π‘₯𝑛‖‖+𝛽lim𝑛→𝒰𝐻𝑇π‘₯𝑛,𝑇π‘₯+𝛾lim𝑛→𝒰𝐻𝑇π‘₯𝑛,𝑇π‘₯+𝛾lim𝑛→𝒰‖‖π‘₯βˆ’π‘₯𝑛‖‖.(3.43) Thus (1βˆ’π›½βˆ’π›Ύ)lim𝑛→𝒰𝐻𝑇π‘₯𝑛≀,𝑇π‘₯(𝛼+𝛽+𝛾)lim𝑛→𝒰‖‖π‘₯βˆ’π‘₯𝑛‖‖,(3.44) and therefore, lim𝑛→𝒰𝐻𝑇π‘₯𝑛,𝑇π‘₯≀lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘₯.(3.45)

Takahashi Generalized Nonexpansive Mappings
Definition 3.22. Let 𝐸 be a nonempty subset of a Banach space 𝑋. A mapping π‘‡βˆΆπΈβ†’π‘‹ is said to be a Takahashi generalized nonexpansive mapping if, for some 𝛼,π›½βˆˆ[0,1] with 𝛼+2𝛽≀1, there holds ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–2≀𝛼‖π‘₯βˆ’π‘¦β€–2ξ€·+π›½β€–π‘¦βˆ’π‘‡π‘₯β€–2+β€–π‘₯βˆ’π‘‡π‘¦β€–2ξ€Έforπ‘₯,π‘¦βˆˆπΈ.(3.46)
The following are examples of Takahashi generalized nonexpansive mappings: (i)nonexpansive mappings π‘‡βˆΆβ€–π‘‡π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–; (ii)nonspreading mappings 𝑇 [38]: 2‖𝑇π‘₯βˆ’π‘‡π‘¦β€–2β‰€β€–π‘¦βˆ’π‘‡π‘₯β€–2+β€–π‘₯βˆ’π‘‡π‘¦β€–2; (iii)hybrid mappings 𝑇 [39]: 3‖𝑇π‘₯βˆ’π‘‡π‘¦β€–2β‰€β€–π‘¦βˆ’π‘‡π‘₯β€–2+β€–π‘₯βˆ’π‘‡π‘¦β€–2; (iv)mappings 𝑇 [39]: 2‖𝑇π‘₯βˆ’π‘‡π‘¦β€–2≀‖π‘₯βˆ’π‘¦β€–2+β€–π‘¦βˆ’π‘‡π‘₯β€–2; (v)mappings 𝑇: 3‖𝑇π‘₯βˆ’π‘‡π‘¦β€–2≀2β€–π‘¦βˆ’π‘‡π‘₯β€–2+β€–π‘₯βˆ’π‘‡π‘¦β€–2.
We define a multivalued version of Takahashi generalized nonexpansive mappings and prove that these mappings satisfy (2) of condition (βˆ—βˆ—).
Proposition 3.23. Let 𝐸 be a nonempty subset of a Banach space 𝑋. For nonnegative constants 𝛼,𝛽 with 𝛼+2𝛽≀1, if π‘‡βˆΆπΈβ†’πΎπΆ(𝑋) is a multivalued mapping such that 𝐻2(𝑇π‘₯,𝑇𝑦)≀𝛼‖π‘₯βˆ’π‘¦β€–2ξ€·+𝛽dist2(π‘₯,𝑇𝑦)+dist2ξ€Έ(𝑦,𝑇π‘₯),(3.47) then 𝑇 satisfies (2) of condition (βˆ—βˆ—).
Proof. Let {π‘₯𝑛} be an afps for 𝑇 in 𝐸 and π‘₯∈𝐸. By (3.41) and (3.42), lim𝑛→𝒰𝐻2𝑇π‘₯𝑛,𝑇π‘₯≀𝛼lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘₯2+𝛽lim𝑛→𝒰dist2ξ€·π‘₯,𝑇π‘₯𝑛+𝛽lim𝑛→𝒰dist2ξ€·π‘₯𝑛,𝑇π‘₯≀𝛼lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘₯2+𝛽lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘₯2+𝛽lim𝑛→𝒰𝐻2𝑇π‘₯𝑛.,𝑇π‘₯(3.48) Thus (1βˆ’π›½)lim𝑛→𝒰𝐻2𝑇π‘₯𝑛,𝑇π‘₯≀(𝛼+𝛽)lim𝑛→𝒰‖‖π‘₯βˆ’π‘₯𝑛‖‖2.(3.49) Therefore, lim𝑛→𝒰𝐻𝑇π‘₯𝑛,𝑇π‘₯≀lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘₯.(3.50)
Remark 3.24. (i)  A mapping that satisfies condition (βˆ—) need not satisfy condition (βˆ—βˆ—). Consider a mapping π‘‡βˆΆ[0,1/2]β†’2[0,1/2] defined by βˆšπ‘‡(π‘₯)=[π‘₯,3√π‘₯]. Since 0 is a fixed point of 𝑇, the sequence {π‘₯𝑛} given by π‘₯𝑛≑0 for all 𝑛 forms an afps for 𝑇. Thus, 𝑇 fulfills condition (1) of Definition 3.2. If {π‘₯𝑛} is an apfs for 𝑇, then {π‘₯𝑛} converges to 0 and 𝐴(𝐸,{π‘₯𝑛})={0}. This implies that 𝐴(𝐸,{π‘₯𝑛}) has an apfs for 𝑇, and 𝑇 satisfies condition (βˆ—). On the other hand, for the afps {π‘₯𝑛} given by π‘₯𝑛≑0, if π‘₯∈(0,1/2], then lim𝑛→𝒰𝐻𝑇π‘₯𝑛=√,𝑇π‘₯π‘₯>π‘₯=lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘₯.(3.51) Thus, 𝑇 fails to satisfy condition (βˆ—βˆ—).
As mentioned earlier, it is unclear if a mapping, satisfies condition (βˆ—βˆ—) also satisfies condition (βˆ—).
(ii)  We do not know if Theorem 3.16 is still valid when β€œlim𝑛→𝑒” in Definition 3.14 is replaced by β€œlimsupπ‘›β†’βˆž.” 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 {π‘₯𝑛} for 𝑇 to be regular and asymptotically uniform relative to 𝐸. Thus, 𝐴(𝐸,{π‘₯𝑛′})=𝐴(𝐸,{π‘₯𝑛}) for all subsequences {π‘₯𝑛′} of {π‘₯𝑛}. Therefore, by Lemma 2.5, it is easy to see that 𝐴𝒰=𝐴(𝐸,{π‘₯𝑛}). If π‘₯βˆˆπ΄π’°, then (3.33) becomes lim𝑛→𝒰‖‖π‘₯π‘›β€–β€–βˆ’π‘§β‰€limπ‘›β†’π’°β€–β€–π‘¦π‘›βˆ’π‘§π‘›β€–β€–=lim𝑛→𝒰𝑦dist𝑛,𝑇π‘₯≀lim𝑛→𝒰𝐻𝑇π‘₯𝑛,𝑇π‘₯≀limsupπ‘›β†’βˆžπ»ξ€·π‘‡π‘₯𝑛,𝑇π‘₯≀limsupπ‘›β†’βˆžβ€–β€–π‘₯𝑛‖‖π‘₯βˆ’π‘₯=π‘ŸπΈ,𝑛=π‘Ÿπ’°ξ€·ξ€½π‘₯𝐸,𝑛.ξ€Ύξ€Έ(3.52) Hence π΄π’°βˆ©π‘‡π‘₯β‰ βˆ…, and the rest of the proof follows.
It is observed that if 𝐴𝒰=𝐴(𝐸,{π‘₯𝑛}) and 𝐴(𝐸,{π‘₯𝑛})={π‘₯} is a singleton, then π‘₯ 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 π‘‡βˆΆπΈβ†’πΈ is a generalized nonexpansive mapping with any of the following conditions holding, then 𝑇 satisfies condition (βˆ—βˆ—):(1)𝛼+2𝛽+2𝛾<1 (see [43, Theorem  4]);(2)𝛼+2𝛽+2𝛾=1 and 𝛽>0,𝛾>0,𝛼β‰₯0 (see [19, Theorem  1]);(3)𝛼+2𝛽+2𝛾=1 and 𝛽>0,𝛾=0,𝛼>0 (see [44, Theorem  1.1]);(4)𝛼+2𝛽+2𝛾=1 and 𝛽=0,𝛾>0,𝛼β‰₯0 (see [45, Lemma  2.1]).
(iv)  Regarding the proof of Theorem 3.16, the fixed point result also holds for weak*-nonexpansive mappings (see [46, Definition  1.3]). Thereby [46, Theorem  1.7] is extended to another circumstance.