Fixed Point Theorems by Ways of Ultra-Asymptotic Centers

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 [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 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 {𝑥_𝑛} in 𝐸 which is regular relative to 𝐸, 𝑟_𝐸𝐴𝐸,𝑥_𝑛≤𝜆𝑟𝐸,𝑥_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 𝑇∶𝐸→𝐾𝐶(𝑋) is a nonexpansive and 1−𝜒-contractive mapping such that 𝑇(E) is a bounded set and which satisfies the inwardness condition:𝑇𝑥⊂𝐼_𝐸(𝑥)forall𝑥∈𝐸, then 𝑇 has a fixed point.

Wiśnicki and Wośko [5] introduced an ultrafilter coefficient DL_𝒰(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 ℕ. The coefficient DL_𝒰(𝑋) of a Banach space 𝑋 is defined as DL_𝒰(𝑋)=sup𝜒^𝐸𝐴_𝒰𝐸,𝑥_𝑛𝜒_𝐸𝑥_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 {𝑥_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 𝐷𝐿_𝒰(𝑋)<1. Assume that 𝑇∶𝐸→𝐾𝐶(𝑋) is a nonexpansive and 1−𝜒-contractive mapping such that 𝑇𝑥⊂𝐼_𝐸(𝑥) for all 𝑥∈𝐸. 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 {𝑥_𝛼} in 𝐸𝑟_𝐸𝐴𝐸,𝑥_𝛼≤𝜆𝑟𝐸,𝑥_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 𝑇∶𝐸→𝐾𝐶(𝑋) is a nonexpansive and 1−𝜒-contractive mapping such that 𝑇𝑥⊂𝐼_𝐸(𝑥) for 𝑥∈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 𝜆∈[0,1) such that for any nonempty weakly compact convex subset 𝐸 of 𝑋, any sequence {𝑥_𝑛}⊂𝐸 which is regular and asymptotically uniform relative to 𝐸, and any sequence {𝑦_𝑛}⊂𝐴(𝐸,{𝑥_𝑛}) which is regular and asymptotically uniform relative to 𝐸 one has 𝑟𝐸,𝑦_𝑛≤𝜆𝑟𝐸,𝑥_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 𝑇∶𝐸→𝐾𝐶(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 𝑇∶𝐸→𝐸 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,{𝑥_𝑛}) is 𝑇-invariant for each afps {𝑥_𝑛} 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 𝑇∶𝐸→𝐸 satisfy conditon (∗). If 𝑇 is continuous, then 𝑇 has a fixed point.

In fact, we can replace “continuity” by a weaker condition, namely “𝐼−𝑇 is strongly demiclosed at 0”: for every sequence {𝑥_𝑛} in 𝐸 strongly converges to 𝑧∈𝐸 and such that 𝑥_𝑛−𝑇𝑥_𝑛→0 we have 𝑧=𝑇𝑧 (cf. [9]).

Following the concept of 𝐷𝐿_𝒰(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 ℕ. The coefficient 𝐷_𝒰(𝑋) of a Banach space 𝑋 is defined as 𝐷_𝒰(𝑋)=sup𝜒^𝐸𝑦_𝑛𝜒_𝐸𝑥_𝑛,(1.5) where the supremum is taken over all nonempty weakly compact convex subsets 𝐸 of 𝑋, all sequences {𝑥_𝑛} in 𝐸 which are weakly, not norm-convergent and are regular relative to 𝐸 and all weakly, not norm-convergent sequences {𝑦_𝑛}⊂𝐴_𝒰(𝐸,{𝑥_𝑛}) which are regular relative to 𝐸.

A concept corresponding to the coefficient 𝐷_𝒰(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 {𝑥_𝑛}⊂𝐸 which is regular relative to 𝐸, and any sequence {𝑦_𝑛}⊂𝐴(𝐸,{𝑥_𝑛}) which is regular relative to 𝐸 one has 𝑟𝐸,𝑦_𝑛≤𝜆𝑟𝐸,𝑥_𝑛.(1.6)

Obviously, DL_𝒰(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, 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 𝑇∶𝐸→𝐾𝐶(𝑋) is a nonexpansive and 1−𝜒-contractive mapping such that 𝑇𝑥⊂𝐼_𝐸(𝑥) for every 𝑥∈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 “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 ℕ. Then 𝐷_𝒰(𝑋)<1 if and only if 𝑋 has property (𝐷′).

Thus, unlike the coefficient DL_𝒰(𝑋), we have, for any two free ultrafilters 𝒰 and 𝒱, 𝐷_𝒰(𝑋)<1 if and only if 𝐷_𝒱(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 2^𝑋 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 𝑇∶𝐸→𝐶𝐵(𝑋) the set of all fixed points of 𝑇 will be denoted by 𝐹(𝑇), that is, 𝐹(𝑇)∶={𝑥∈𝐸∶𝑥∈𝑇𝑥}. Let 𝐻(⋅,⋅) be the Hausdorff distance defined on 𝐶𝐵(𝑋), that is,𝐻(𝐴,𝐵)∶=maxsup_𝑎∈𝐴dist(a,B),sup_𝑏∈𝐵dist(𝑏,𝐴),𝐴,𝐵∈𝐶𝐵(𝑋),(2.1)

where dist(𝑎,𝐵)∶=inf{‖𝑎−𝑏‖∶𝑏∈𝐵} is the distance from a point 𝑎 to a subset 𝐵. A multivalued mapping 𝑇∶𝐸→𝐶𝐵(𝑋) is said to be nonexpansive if𝐻(𝑇𝑥,𝑇𝑦)≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸,(2.2)

and 𝑇 is said to be a contraction if there exists a constant 𝑘<1 such that𝐻(𝑇𝑥,𝑇𝑦)≤𝑘‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸.(2.3) A multivalued mapping 𝑇∶𝐸→2^𝑋 is called 𝜙-condensing (resp., 1−𝜙-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)=⋃_𝑥∈𝐵𝑇𝑥.

Recall that the inward set of 𝐸 at 𝑥∈𝐸 is defined by𝐼_𝐸(𝑥)={𝑥+𝛼(𝑦−𝑥)∶𝛼≥1,𝑦∈𝐸}.(2.5) A sequence {𝑥_𝑛} in 𝐸 for which lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0 for a mapping 𝑇∶𝐸→𝐸 is called an approximate fixed point sequence (afps for short) for 𝑇. Analogously for a multivalued mapping 𝑇∶𝐸→𝐶𝐵(𝑋), a sequence {𝑥_𝑛} in 𝐸 of a Banach space 𝑋 for which lim_𝑛→∞dist(𝑥_𝑛,𝑇𝑥_𝑛)=0 is called an approximate fixed point sequence (afps for short) for 𝑇.

We denote by 𝑥_𝑛→𝑥 to indicate that the sequence {𝑥_𝑛} in 𝑋 converges to 𝑥∈𝑋.

Let 𝐸 be a nonempty closed and convex subset of a Banach space 𝑋 and {𝑥_𝑛} a bounded sequence in 𝑋. For 𝑥∈𝑋, define the asymptotic radius of {𝑥_𝑛} at 𝑥 as the number𝑟𝑥,𝑥_𝑛=limsup_𝑛→∞‖‖𝑥_𝑛−𝑥‖‖.(2.6) Let𝐴𝐸,𝑟𝑥𝐸_𝑛,𝑥_𝑛∶=inf𝑟𝑥,𝑥_𝑛∶𝑥∈𝐸,∶=𝑥∈𝐸∶𝑟𝑥,𝑥_𝑛=𝑟𝐸,𝑥_𝑛.(2.7) The number 𝑟(𝐸,{𝑥_𝑛}) and the set 𝐴(𝐸,{𝑥_𝑛}) are, respectively, called the asymptotic radius and asymptotic center of {𝑥_𝑛} relative to 𝐸. The sequence {𝑥_𝑛} is called regular relative to 𝐸 if 𝑟(𝐸,{𝑥_𝑛})=𝑟(𝐸,{𝑥_𝑛}) for each subsequence {𝑥_𝑛} of {𝑥_𝑛}. It was noted in [26] that if 𝐸 is nonempty and weakly compact, then 𝐴(𝐸,{𝑥_𝑛}) is nonempty and weakly compact, and if 𝐸 is convex, then 𝐴(𝐸,{𝑥_𝑛}) is convex.

Proposition 2.1 (see [27, Theorem  1]). Let {𝑥_𝑛} and 𝐸 be as above. Then there exists a subsequence of {𝑥_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 ℕ. Recall ([26, 28–30]) that the ultrapower (𝑋)_𝒰 of a Banach space 𝑋 is the quotient space of𝑙_∞(𝑋)=𝑥_𝑛∶𝑥_𝑛∈𝑋∀𝑛∈ℕ,‖‖𝑥_𝑛‖‖=sup_𝑛‖‖𝑥_𝑛‖‖<∞(2.8) byker𝒩=𝑥_𝑛∈𝑙_∞(𝑋)∶lim_𝑛→𝒰‖‖𝑥_𝑛‖‖=0.(2.9) One can proof that 𝑋=(𝑋)_𝒰 is a Banach space with the quotient norm given by ‖{𝑥_𝑛}_𝒰‖=lim_𝑛→𝒰‖𝑥_𝑛‖, where {𝑥_𝑛}_𝒰 is the equivalence class of {𝑥_𝑛}. It is also clear that 𝑋 is isometric to a subspace of 𝑋 by the canonical embedding 𝑥↦{𝑥,𝑥,…}_𝒰. If 𝐸⊂𝑋, we will use the symbols 𝐸̇ and 𝑥̇ to denote the image of 𝐸 and 𝑥 in 𝑋 under this isometry, respectively, and denote𝐸=𝑥̃∈𝑋∶∃𝑥_𝑛suchthat𝑥̃=𝑥_𝑛_𝒰,𝑥_𝑛∈𝐸∀𝑛∈ℕ.(2.10)

Thus 𝑥̇={𝑥,𝑥,…}_𝒰 and 𝐸̇={𝑥̇∈𝑋∶𝑥∈𝐸}.

If 𝑇∶𝐸→𝐶𝐵(𝑋) is a multivalued mapping, we define a corresponding multivalued mapping 𝑇∶𝐸→𝐶B(𝑋) by𝑇(𝑥̃)∶=𝑢̃∈𝑋∶∃𝑢_𝑛suchthat𝑢̃=𝑢_𝑛_𝒰,𝑢_𝑛∈𝑇𝑥_𝑛∀𝑛∈ℕ,(2.11)

where 𝑥̃={𝑥_𝑛}_𝒰∈𝐸. Moreover the set 𝑇(𝑥̃) 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 {𝑥_𝑛}_𝒰 and {𝑦_𝑛}_𝒰 in 𝐸, 𝐻𝑇𝑥_𝑛_𝒰,𝑇𝑦_𝑛_𝒰=lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑦_𝑛.(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 𝑇∶𝐸→𝐶𝐵(𝑋). (i)If 𝑇 is convex-valued, then 𝑇 is convex-valued. (ii)If 𝑇 is compact-valued, then 𝑇 is compact-valued and 𝑇𝑥̇=(𝑇̇𝑥) for every 𝑥∈𝐸. (iii)If 𝑇 is nonexpansive, then 𝑇 is nonexpansive.

Let 𝒰 denote a free ultrafilter defined on ℕ. Wiśnicki and Wośko [5] defined the ultra-asymptotic radius 𝑟_𝒰(𝐸,{𝑥_𝑛}) and the ultra-asymptotic center 𝐴_𝒰(𝐸,{𝑥_𝑛}) of {𝑥_𝑛} relative to 𝐸 by 𝐴𝑟_𝒰𝐸,𝑥_𝑛=inflim_𝑛→𝒰‖‖𝑥_𝑛−𝑥‖‖∶𝑥∈𝐸_𝒰,𝐸,𝑥_𝑛=𝑥∈𝐸∶lim_𝑛→𝒰‖‖𝑥_𝑛−𝑥‖‖=𝑟_𝒰𝐸,𝑥_𝑛.(2.13) It is not difficult to see that 𝐴_𝒰(𝐸,{𝑥_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]:𝑟_𝒰𝐸,𝑥_𝑛=inf_𝑥∈𝐸‖‖𝑥_𝑛_𝒰−𝑥̇‖‖(2.14)

is the relative Chebyshev radius of {𝑥_𝑛}_𝒰, and𝐴_𝒰𝐸̇,𝑥_𝑛=𝐸̇∩𝐵^𝑋𝑥_𝑛_𝒰,𝑟(2.15)

is the relative Chebyshev center of {𝑥_𝑛}_𝒰 relative to 𝐸̇ in the ultrapower 𝑋. (Here 𝐵_𝑋({𝑥_𝑛}_𝒰,𝑟) denotes the ball in 𝑋 centered at {𝑥_𝑛}_𝒰 and of radius 𝑟=𝑟_𝒰(𝐸,{𝑥_𝑛}).) It should be noted that, in general, 𝐴(𝐸,{𝑥_𝑛}) and 𝐴_𝒰(𝐸,{𝑥_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 Domínguez Benavides and Lorenzo Ramírez [1] as𝜒_𝐸(𝐴)=inf{𝜀>0∶𝐴canbecoveredbyfinitelymanyballsin𝐸ofradii≤𝜀}.(2.16)

Proposition 2.4 (see [5, Proposition  4.5]). If {𝑥_𝑛} is a bounded sequence which is regular relative to 𝐸, then 𝑟𝐸,𝑥_𝑛=𝑟_𝒰𝐸,𝑥_𝑛=𝜒_𝐸𝑥_𝑛.(2.17)

From Proposition 2.4, we have, for 𝑤∈𝐴(𝐸,{𝑥_𝑛}),lim_𝑛→𝒰‖‖𝑥_𝑛−𝑤‖‖≤limsup_𝑛→∞‖‖𝑥_𝑛−𝑤‖‖=𝑟𝐸,𝑥_𝑛=𝑟_𝒰𝐸,𝑥_𝑛.(2.18)

Therefore, 𝐴(𝐸,{𝑥_𝑛})⊂𝐴_𝒰(𝐸,{𝑥_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 {𝑥_𝑛} a bounded sequence in 𝑋 which is regular relative to 𝐸. For each {𝑦_𝑛}⊂𝐴_𝒰(𝐸,{𝑥_𝑛}), there exists a subsequence {𝑥_𝑛} of {𝑥_𝑛} such that {𝑦_𝑛}⊂𝐴(𝐸,{𝑥_𝑛}).

A direct consequence of Lemma 2.5 is as follows. If every center 𝐴(𝐸,{𝑥_𝑛}) is compact for every bounded sequence {𝑥_𝑛} in E which is regular relative to 𝐸, then 𝐴_𝒰(𝐸,{𝑥_𝑛}) is also compact for every bounded sequence {𝑥_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 𝑇∶𝐸→𝐶𝐵(𝑋). Then (1)if 𝑇 is uniformly continuous, then 𝑇 is uniformly continuous; (2)if 𝑇 is continuous at 𝑧∈𝐸, 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,𝑦∈𝐸 with ‖𝑥−𝑦‖<𝛿. Suppose {𝑥_𝑛}_𝒰,{𝑦_𝑛}_𝒰∈𝐸, and ‖{𝑥_𝑛}_𝒰−{𝑦_𝑛}_𝒰‖<𝛿. Let 𝐴={𝑛∶‖𝑥_𝑛−𝑦_𝑛‖<𝛿} and 𝐵={𝑛∶𝐻(𝑇𝑥_𝑛,𝑇𝑦_𝑛)<𝜀}. Since 𝐴∈𝒰 and 𝐴⊂𝐵, 𝐵∈𝒰. Thus, by Proposition 2.2𝐻(𝑇{𝑥_𝑛}_𝒰,𝑇{𝑦_𝑛}_𝒰)≤𝜀.
(2) Let 𝜀>0. Since 𝑇 is continuous at 𝑧, there exists 𝛿>0 such that 𝐻(𝑇𝑥,𝑇𝑧)<𝜀 for each 𝑥∈𝐸 with ‖𝑥−𝑧‖<𝛿. If {𝑥_𝑛}_𝒰∈𝐸 such that ‖{𝑥_𝑛}_𝒰−𝑧‖̇<𝛿, then, letting 𝐴={𝑛∶‖𝑥_𝑛−𝑧‖<𝛿} and 𝐵={𝑛∶𝐻(𝑇𝑥_𝑛,𝑇𝑧)<𝜀}, we see that 𝐴∈𝒰 and 𝐵∈𝒰. Thus, by Proposition 2.2𝐻(𝑇{𝑥_𝑛}_𝒰,𝑇𝑧)̇≤𝜀.

We now introduce condition (∗) for multivalued mappings.

Definition 3.2. Let 𝐸 be a nonempty subset of a Banach space 𝑋. A mapping 𝑇∶𝐸→𝐶𝐵(𝑋) is said to satisfy condition (∗) if (1)𝑇 has an afps in 𝐸, (2)𝑇 has an afps in 𝐴(𝐸,{𝑥_𝑛}) for some subsequence {𝑥_𝑛} of any given afps {𝑥_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 𝑇∶𝐸→𝐾𝐶(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 {𝑥^0𝑛} be an afps for 𝑇 in 𝐸. We can assume by Proposition 2.1 that {𝑥^0𝑛} is regular relative to 𝐸. Condition (2) of Definition 3.2 gives us a subsequence {𝑥^0𝑛₀} of {𝑥^0𝑛} so that the center 𝐴(𝐸,{𝑥^0𝑛₀}) contains an afps for 𝑇. Denote 𝐴⁰=𝐴(𝐸,{𝑥^0𝑛₀}), and let {𝑥^1𝑛} be an afps in 𝐴⁰. Assume that {𝑥^1𝑛} is regular relative to 𝐸. As before, 𝑇 has an afps in 𝐴(𝐸,{𝑥^1𝑛₁}) for some subsequence {𝑥^1𝑛₁} of {𝑥^1𝑛}. Since 𝑋 has property (𝐷′), put 𝜆=𝐷_𝒰(𝑋)<1. Then, by Proposition 2.4 and Definition 1.11, 𝑟𝐸,𝑥^1𝑛₁=𝜒_E𝑥^1𝑛₁≤𝜆𝜒_𝐸𝑥^0𝑛₀=𝜆𝑟𝐸,𝑥^0𝑛₀.(3.1) Continue the procedure to obtain, for each 𝑚≥0, a regular sequence {𝑥^𝑚𝑛_𝑚} relative to 𝐸 in 𝐴^𝑚−1∶=𝐴(𝐸,{𝑥^𝑛𝑚−1_𝑚−1}) such that lim_𝑛→∞dist𝑥^𝑚𝑛_𝑚,𝑇𝑥^𝑚𝑛_𝑚=0,(3.2) and for all 𝑚≥1, 𝜒_𝐸𝑥^𝑚𝑛_𝑚≤𝜆𝑟𝐸,𝑥^𝑛𝑚−1_𝑚−1.(3.3) Consequently, 𝑟𝐸,𝑥^𝑚𝑛_𝑚≤𝜆𝑟𝐸,𝑥^𝑛𝑚−1_𝑚−1≤⋯≤𝜆^𝑚𝑟𝐸,𝑥^0𝑛₀.(3.4) We show that {𝑥^𝑚𝑛_𝑚}_𝒰 is a Cauchy sequence in 𝑋. Indeed, for each 𝑚≥1, take an element 𝑦̇_𝑚∈𝐴̇^𝑚−1. Then ‖‖𝑥̇^𝑚𝑛_𝑚−𝑦̇_𝑚‖‖≤‖‖𝑥̇^𝑚𝑛_𝑚−𝑥^𝑛𝑚−1_𝑚−1_𝒰‖‖+‖‖𝑥^𝑛𝑚−1_𝑚−1_𝒰−𝑦̇_𝑚‖‖≤2𝑟𝐸,𝑥^𝑛𝑚−1_𝑚−1(3.5) for all 𝑚≥1, and hence ‖‖𝑥^𝑚𝑛_𝑚_𝒰−𝑥^𝑛𝑚−1_𝑚−1_𝒰‖‖≤‖‖𝑥^𝑚𝑛_𝑚_𝒰−𝑦̇_𝑚‖‖+‖‖𝑦̇_𝑚−𝑥^𝑛𝑚−1_𝑚−1_𝒰‖‖≤3𝑟𝐸,𝑥^𝑛𝑚−1_𝑚−1.(3.6) Thus ‖‖𝑥^𝑚𝑛_𝑚_𝒰−𝑥^𝑛𝑚−1_𝑚−1_𝒰‖‖≤3𝜆^𝑚−1𝑟𝐸,𝑥^0𝑛₀,(3.7) implying that {𝑥^𝑚𝑛_𝑚}_𝒰 is a Cauchy sequence and hence converges to some {𝑧_𝑛}_𝒰 in 𝐸 as 𝑚→∞. Next, we will show that {𝑧_𝑛}_𝒰∈𝐸̇. For each 𝑚≥0, dist𝑧_𝑛_𝒰,𝐸̇≤≤‖‖‖‖{𝑧𝑧_𝑛𝑛_𝒰−𝑥^𝑚𝑛_𝑚_𝒰‖‖+𝑑𝑖𝑠t𝑥^𝑚𝑛_𝑚_𝒰,𝐸̇=‖‖𝑧_𝑛}_𝒰−𝑥^𝑚𝑛_𝑚_𝒰‖‖+‖‖𝑥^𝑚𝑛_𝑚_𝒰−𝑥̇^1𝑚+1_𝑚+1‖‖≤‖‖𝑧_𝒰−𝑥^𝑚𝑛_𝑚_𝒰‖‖+𝑟𝐸,𝑥^𝑚𝑛_𝑚_𝑛_𝒰−𝑥^𝑚𝑛_𝑚_𝒰‖‖+𝜆^𝑚𝑟𝐸,𝑥^0𝑛₀.(3.8) Taking 𝑚→∞ we see that dist𝑧_𝑛_𝒰,𝐸̇=0.(3.9) Thus, it follows that there exists 𝑧∈𝐸 such that {𝑧_𝑛}_𝒰=𝑧̇. By Lemma 3.1, 𝑇 is continuous at ż, and thus 𝐻(𝑇{𝑥^𝑚𝑛_𝑚}_𝒰,𝑇𝑧)̇→0 as 𝑚→∞. For every 𝑚≥0, dist𝑧,̇𝑇𝑧̇≤‖‖𝑧̇−𝑥^𝑚𝑛_𝑚_𝒰‖‖+dist𝑥^𝑚𝑛_𝑚_𝒰,𝑇𝑥^𝑚𝑛_𝑚_𝒰+𝐻𝑇𝑥^𝑚𝑛_𝑚_𝒰,𝑇𝑧̇.(3.10) Taking 𝑚→∞ we then obtain 𝑧̇∈𝑇𝑧̇. By Proposition 2.3, 𝑇𝑧̇=(𝑇̇𝑧), and therefore, 𝑧∈𝑇𝑧.

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 𝐴_𝒰 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 𝐴_𝒰(𝐸,{𝑥_𝑛}) 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 𝑇∶𝐸→𝐾𝐶(𝑋) is nonexpansive and 1−𝜒-contractive such that 𝑇𝑥⊂𝐼_𝐸(𝑥) for every 𝑥∈𝐸. 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 𝐹∶𝐸→𝐾𝐶(𝑋) an upper semicontinuous and 𝜒-condensing mapping. If 𝐹(𝑥)∩𝐼_𝐸(𝑥)≠∅ for all 𝑥∈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 𝑇∶𝐸→𝐾𝐶(𝑋) is nonexpansive and 1−𝜒-contractive such that 𝑇𝑥⊂𝐼_𝐸(𝑥) for every 𝑥∈𝐸. Then 𝑇 satisfies condition (∗).
Proof. First, we will show that 𝑇 has an afps in 𝐸. Let 𝑦₀∈𝐸, and consider, for each 𝑛≥1, the contraction 𝑇_𝑛∶𝐸→𝐾𝐶(𝑋) defined by 𝑇_𝑛(𝑥)=1𝑛𝑦⁰+1−1𝑛𝑇𝑥,𝑥∈𝐸.(3.12) It is not difficult to see that 𝑇_𝑛(𝑥)⊂𝐼_𝐸(𝑥) for every 𝑥∈𝐸. Since 𝑇 is 1−𝜒-contractive, 𝑇_𝑛 is (1−(1/𝑛))−𝜒-contractive, and by Theorem 3.5, there exists a fixed point 𝑥_𝑛 of 𝑇_𝑛. Clearly, {𝑥_𝑛} is an afps for 𝑇 in 𝐸.
Next, let us see that 𝑇 has an afps in 𝐴(𝐸,{𝑥_𝑛′}) for some subsequence {𝑥_𝑛′} of an afps {𝑥_𝑛} for 𝑇 in 𝐸. Let {𝑥_𝑛} be an afps in 𝐸. By Proposition 2.1, we can assume that {𝑥_𝑛} is weakly convergent and regular relative to 𝐸. Let 𝐴_𝒰∶=𝐴_𝒰(𝐸,{𝑥_𝑛}). We show that 𝑇𝑥∩𝐼_𝐴𝒰(𝑥)≠∅forevery𝑥∈𝐴_𝒰.(3.13) Let 𝑥∈𝐴_𝒰. Observe first that {𝑥_𝑛}_𝒰∈𝑇{𝑥_𝑛}_𝒰. By Proposition 2.3, 𝑇𝑥̇=(𝑇̇𝑥) is compact, and hence there exists 𝑢∈𝑇𝑥 such that ‖‖𝑥_𝑛_𝒰−𝑢‖‖̇=𝐻𝑇𝑥_𝑛_𝒰,𝑇𝑥̇≤‖‖𝑥_𝑛_𝒰−𝑥̇‖‖=𝑟_𝒰𝐸,𝑥_𝑛.(3.14) Since 𝑢∈𝑇𝑥⊂𝐼_𝐸(𝑥), there exists 𝛼≥1 and 𝑦∈𝐸 such that 𝑢=𝑥+𝛼(𝑦−𝑥). If 𝛼=1 then 𝑢=𝑦∈𝐸, and it follows from (3.14) that 𝑢∈𝐴_𝒰. If 𝛼>1 then 𝑦=(1/𝛼)𝑢+(1−1/𝛼)𝑥, and therefore, we have ‖‖𝑥_𝑛_𝒰−𝑦‖‖̇≤1𝛼‖‖𝑥^𝑛_𝒰−𝑢‖‖̇+1−1𝛼‖‖𝑥^𝑛_𝒰−𝑥̇‖‖≤𝑟_𝒰𝐸,𝑥_𝑛.(3.15) Hence 𝑦∈𝐴_𝒰 and consequently 𝑢∈𝐼_𝐴𝒰(𝑥). Thus (3.13) is justified.
Fixed 𝑦₀∈𝐴_𝒰, and consider for each 𝑛≥1, the contraction 𝑇_𝑛∶𝐴_𝒰→𝐾𝐶(𝑋) defined by 𝑇_𝑛(𝑥)=1𝑛𝑦⁰+1−1𝑛𝑇𝑥,𝑥∈𝐴^𝒰.(3.16) As before, 𝑇_𝑛 is (1−1/𝑛)−𝜒-contractive, and by Theorem 3.5, there exists a fixed point 𝑧_𝑛∈𝐴_𝒰 of 𝑇_𝑛. Again, as above, {𝑧_𝑛} is an afps for 𝑇 in 𝐴_𝒰. By Lemma 2.5, there exists a subsequence {𝑥_𝑛′} of {𝑥_𝑛} such that {𝑧_𝑛}⊂𝐴(𝐸,{𝑥_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(𝑥,𝑦)∶𝑥,𝑦∈𝐾} denotes the diameter of 𝐾⊂𝑋.)
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 𝑇∶𝐸→𝐸 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=⋃_𝑘∈𝐾𝑇𝑘 and 𝐸 is said to be 𝑇-invariant if 𝑇𝑥∩𝐸≠∅ for all 𝑥∈𝐸.
Theorem 3.9 (see [35, Theorem  2.2]). Let 𝐸 be a weakly compact subset of a Banach space 𝑋, and let 𝑇∶𝐸→𝐾𝐶(𝑋) be a multivalued mapping such that 𝛿(𝑇𝐾∩𝐾)<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 𝑇∶𝐸→𝐾𝐶(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 𝑦₀∈𝐸, and consider, for each 𝑛≥1, the contraction 𝑇_𝑛∶𝐸→𝐾𝐶(𝑋) defined by 𝑇_𝑛(𝑥)=1𝑛𝑦⁰+1−1𝑛𝑇𝑥,𝑥∈𝐸.(3.17) For 𝑥∈𝐸, let 𝑎∈𝑇𝑥∩𝐸. Thus (1/𝑛)𝑦₀+(1−1/𝑛)𝑎∈𝑇_𝑛𝑥∩𝐸, and therefore, 𝑇_𝑛𝑥∩𝐸≠∅ for every 𝑥∈𝐸. We show that 𝛿(𝑇_𝑛𝐾)<𝛿(𝐾)forallclosedsets𝐾with𝛿(𝐾)>0. Let 𝐾 be a closed subset of 𝐸 with 𝛿(𝐾)>0. For 𝑥,𝑦∈𝑇_𝑛𝐾, there exist 𝑥′,𝑦′∈𝑇𝐾 such that 𝑥=1𝑛𝑦⁰+1−1𝑦=1𝑛𝑥′,𝑛𝑦⁰+1−1𝑛𝑦^,(3.18) and this entails ‖𝑥−𝑦‖=(1−1/𝑛)‖𝑥′−𝑦′‖≤(1−1/𝑛)𝛿(𝑇𝐾). Hence 𝛿(𝑇_𝑛𝐾)≤(1−1/𝑛)𝛿(𝑇𝐾)<𝛿(𝐾). By Theorem 3.9, there exists a fixed point 𝑥_𝑛 of 𝑇_𝑛, and thus the sequence {𝑥_𝑛} forms an afps for 𝑇 in 𝐸.
Next, let us see that 𝑇 has an afps in 𝐴(𝐸,{𝑥_𝑛′}) for some subsequence {𝑥_𝑛′} of an afps {𝑥_𝑛} for 𝑇 in 𝐸. Let {𝑥_𝑛} be an afps in 𝐸. We can assume that {𝑥_𝑛} is weakly convergent and regular relative to 𝐸. Let 𝐴_𝒰=𝐴_𝒰(𝐸,{𝑥_𝑛}). First, we show that 𝐴_𝒰∩𝑇𝑥≠∅,forevery𝑥∈𝐴_𝒰.(3.19) Let 𝑥∈𝐴_𝒰, and for each 𝑛≥1, we see that 𝐻(𝑇𝑥_𝑛,𝑇𝑥)≤𝛿(𝑇{𝑥_𝑛,𝑥})≤𝛿({𝑥_𝑛,𝑥})=‖𝑥_𝑛−𝑥‖. Take 𝑦_𝑛∈𝑇𝑥_𝑛 so that ‖‖𝑥_𝑛−𝑦_𝑛‖‖=dist𝑥_𝑛,𝑇𝑥_𝑛,(3.20) and select 𝑧_𝑛∈𝑇𝑥 for each 𝑛 such that ‖‖𝑧_𝑛−𝑦_𝑛‖‖=dist𝑦_𝑛,𝑇𝑥.(3.21) Let lim_𝑛→𝒰𝑧_𝑛=𝑧∈𝑇𝑥. Note that ‖‖𝑥_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑧_𝑛‖‖+‖‖𝑧_𝑛−𝑧‖‖.(3.22) We obtain lim_𝑛→𝒰‖‖𝑥_𝑛−𝑧‖‖≤≤lliimm_{𝑛→𝒰𝑛→𝒰}‖‖𝑦_𝑛−𝑧_𝑛‖‖=lim_𝑛→𝒰dist𝑦_𝑛,𝑇𝑥≤lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥‖‖𝑥_𝑛−𝑥‖‖=𝑟_𝒰𝐸,𝑥_𝑛(3.23) proving that 𝑧∈𝐴_𝒰. Thus (3.19) is satisfied. Fix 𝑦₀∈𝐴_𝒰, and consider, for each 𝑛≥1, the contraction 𝑇_𝑛∶𝐴_𝒰→𝐾𝐶(𝑋) defined by 𝑇_𝑛(𝑥)=1𝑛𝑦⁰+1−1𝑛𝑇𝑥,𝑥∈𝐴^𝒰.(3.24) For 𝑥∈𝐸, let 𝑎∈𝐴_𝒰∩𝑇𝑥. Thus (1/𝑛)𝑦₀+(1−1/𝑛)𝑎∈𝐴_𝒰∩𝑇_𝑛𝑥. Therefore, 𝐴_𝒰∩𝑇_𝑛𝑥≠∅ for every 𝑥∈𝐴_𝒰. Let 𝐾 be a closed subset of 𝐸 with 𝛿(𝐾)>0. As before, 𝛿(𝑇_𝑛𝐾)≤(1−1/𝑛)𝛿(𝑇𝐾)<𝛿(𝐾). By Theorem 3.9 (or we can apply Theorem 3.5), there exists a fixed point 𝑧_𝑛 of 𝑇_𝑛. Again, as above, {𝑧_𝑛} is an afps for 𝑇 in 𝐴_𝒰. Finally, by Lemma 2.5, there exists a subsequence {𝑥_𝑛′} of {𝑥_𝑛} such that {𝑧_𝑛}⊂𝐴(𝐸,{𝑥_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 𝑇∶𝐸→𝐾𝐶(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 𝑇∶𝐸→𝐾𝐶(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 ℕ. Let 𝐸 be a bounded closed and convex subset of a Banach space 𝑋. A mapping 𝑇∶𝐸→𝐶𝐵(𝑋) is said to satisfy condition (∗∗) if it fulfills the following conditions. (1)𝑇 has an afps in 𝐸; (2)if {𝑥_𝑛} is an apfs for 𝑇 in 𝐸 and 𝑥∈𝐸, then lim_𝑛→𝒰𝐻(𝑇𝑥_𝑛,𝑇𝑥)≤lim_𝑛→𝒰‖𝑥_𝑛−𝑥‖.

Remark 3.15. (i)  Let 𝐸 be a bounded closed and convex subset of a Banach space 𝑋, and let a mapping 𝑇∶𝐸→𝐾𝐶(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 {𝑥_𝑛} be an apfs for 𝑇 in 𝐸. From Proposition 2.1 by passing through a subsequence, we may assume that {𝑥_𝑛} is regular relative to 𝐸. Let 𝐴_𝒰=𝐴_𝒰(𝐸,{𝑥_𝑛}) and 𝑥∈𝐴_𝒰. The compactness of 𝑇𝑥_𝑛 implies that for each 𝑛 we can take 𝑦_𝑛∈𝑇𝑥_𝑛 so that ‖‖𝑥_𝑛−𝑦_𝑛‖‖=dist𝑥_𝑛,𝑇𝑥_𝑛.(3.25) Since 𝑇𝑥 is compact, select 𝑧_𝑛∈𝑇𝑥 for each 𝑛 such that ‖‖𝑧_𝑛−𝑦_𝑛‖‖=dist𝑦_𝑛,𝑇𝑥.(3.26) Let lim_𝑛→𝒰𝑧_𝑛=𝑧∈𝑇𝑥. Note that ‖‖𝑥_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑧_𝑛‖‖+‖‖𝑧_𝑛−𝑧‖‖.(3.27) We obtain lim_𝑛→𝒰‖‖𝑥_𝑛−𝑧‖‖≤≤lliimm_{𝑛→𝒰𝑛→𝒰}‖‖𝑦_𝑛−𝑧_𝑛‖‖=lim_𝑛→𝒰dist𝑦_𝑛,𝑇𝑥≤lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥‖‖𝑥_𝑛−𝑥‖‖=𝑟_𝒰𝐸,𝑥_𝑛(3.28) proving that 𝑧∈𝐴_𝒰 and hence 𝐴_𝒰∩𝑇𝑥≠∅ for all 𝑥∈𝐴_𝒰, that is, 𝐴_𝒰 is 𝑇-invariant. By assumption, there exists an afps in 𝐴_𝒰. By Lemma 2.5, there exists a subsequence {𝑥_𝑛′} of {𝑥_𝑛} such that {𝑧_𝑛}⊂𝐴(𝐸,{𝑥_𝑛′}). 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 𝑇∶𝐸→𝐸 satisfies condition (𝐿) on 𝐸 provided that it fulfills the following two conditions. (1)If a set 𝐷⊂𝐸 is nonempty, closed, convex, and 𝑇-invariant, then there exists an afps for 𝑇 in 𝐷. (2)For any afps {𝑥_𝑛} of 𝑇 in 𝐸 and each 𝑥∈𝐸,limsup_𝑛→∞‖𝑥_𝑛−𝑇𝑥‖≤limsup_𝑛→∞‖𝑥_𝑛−𝑥‖.(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 𝑇∶𝐸→𝐾𝐶(𝑋) be a multivalued mapping satisfying condition (∗∗). If 𝑇 is an upper semicontinuous mapping, then 𝑇 has a fixed point.

Proof. Let {𝑥_𝑛} be an afps for 𝑇 in 𝐸. From Proposition 2.1 by passing through a subsequence, we may assume that {𝑥_𝑛} is regular relative to 𝐸. Let 𝐴_𝒰=𝐴_𝒰(𝐸,{𝑥_𝑛}). The compactness of 𝑇𝑥_𝑛 implies that for each 𝑛 we can take 𝑦_𝑛∈𝑇𝑥_𝑛 such that ‖‖𝑥_𝑛−𝑦_𝑛‖‖=dist𝑥_𝑛,𝑇𝑥_𝑛.(3.30) If 𝑥∈𝐴_𝒰, since 𝑇𝑥 is compact, select 𝑧_𝑛∈𝑇𝑥 for each 𝑛 such that ‖‖𝑧_𝑛−𝑦_𝑛‖‖=dist𝑦_𝑛,𝑇𝑥.(3.31) Let lim_𝑛→𝒰𝑧_𝑛=𝑧∈𝑇𝑥. Note that ‖‖𝑥_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑧_𝑛‖‖+‖‖𝑧_𝑛−𝑧‖‖.(3.32) Thus lim_𝑛→𝒰‖‖𝑥_𝑛−𝑧‖‖≤≤lliimm_{𝑛→𝒰𝑛→𝒰}‖‖𝑦_𝑛−𝑧_𝑛‖‖=lim_𝑛→𝒰dist𝑦_𝑛,𝑇𝑥≤lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥‖‖𝑥_𝑛−𝑥‖‖=𝑟_𝒰𝐸,𝑥_𝑛(3.33) proving that 𝑧∈𝐴_𝒰 and hence 𝐴_𝒰∩𝑇𝑥≠∅ for all 𝑥∈𝐴_𝒰. By assumption, 𝐴(𝐸,{𝑥_𝑛}) is nonempty and compact which implies that 𝐴_𝒰 is also nonempty and compact. Now define a mapping 𝐹∶𝐴_𝒰→𝐾𝐶(𝐴_𝒰) by 𝐹𝑥∶=𝐴_𝒰∩𝑇𝑥 for all 𝑥∈𝐴_𝒰. Thus 𝐹 is upper semicontinuous. Indeed, let {𝑢_𝑛}⊂𝐴 be such that lim_𝑛→∞𝑢_𝑛=𝑢, and let 𝑣_𝑛∈𝐹𝑢_𝑛 be such that lim_𝑛→∞𝑣_𝑛=𝑣. Since 𝑇 is upper semicontinuous and 𝐴_𝒰 is compact, we have 𝑣∈𝑇𝑢 and 𝑣∈𝐴_𝒰, that is 𝑣∈𝐹𝑢. By the Bohnenblust-Karlin fixed point theorem [36], 𝐹 and hence 𝑇, have a fixed point in 𝐴_𝒰.

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 𝑇∶𝐸→𝐸 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 𝑇∶𝐸→𝐸 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 (𝐶_𝜆). 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 𝑇∶𝐸→𝑋 satisfies Condition (𝐶_𝜆) on 𝐸 if, for each 𝑥,𝑦∈𝐸, 𝜆‖𝑥−𝑇𝑥‖≤‖𝑥−𝑦‖implies‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖.(3.34) It is natural to define a multivalued version of Condition (𝐶_𝜆) (see [18]).
Let 𝐸 be a nonempty subset of a Banach space 𝑋, and let 𝑇∶𝐸→𝐶𝐵(𝑋) be a multivalued mapping. Then 𝑇 is said to satisfy condition (𝐶_𝜆) for some 𝜆∈(0,1) if, for each 𝑥,𝑦∈𝐸, 𝜆dist(𝑥,𝑇𝑥)≤‖𝑥−𝑦‖implies𝐻(𝑇𝑥,𝑇𝑦)≤‖𝑥−𝑦‖.(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 𝑇∶𝐸→𝐶𝐵(𝐸) satisfies condition (𝐶_𝜆) 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 𝑇∶𝐸→𝑋 is a generalized nonexpansive mapping if for some nonnegative constants 𝛼₁,…,𝛼₅ with ∑^5𝑖=1𝛼_𝑖=1, ‖𝑇𝑥−𝑇𝑦‖≤𝛼₁‖𝑥−𝑦‖+𝛼₂‖𝑥−𝑇𝑥‖+𝛼₃‖𝑦−𝑇𝑦‖+𝛼₄‖𝑥−𝑇𝑦‖+𝛼₅‖𝑦−𝑇𝑥‖,(3.36) for each 𝑥,𝑦∈𝐸.

We will use the following equivalent condition.

For some nonnegative constants 𝛼,𝛽,𝛾 with 𝛼+2𝛽+2𝛾≤1,‖𝑇𝑥−𝑇𝑦‖≤𝛼‖𝑥−𝑦‖+𝛽(‖𝑥−𝑇𝑥‖+‖𝑦−𝑇𝑦‖)+𝛾(‖𝑥−𝑇𝑦‖+‖𝑦−𝑇𝑥‖),(3.37)

for all 𝑥,𝑦∈𝐸.

We introduce a multivalued version of these mappings.

Let 𝑇∶𝐸→𝐶𝐵(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 𝑥,𝑦∈𝐸, there holds𝐻(𝑇𝑥,𝑇𝑦)≤𝛼‖𝑥−𝑦‖+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 𝑇∶𝐸→𝐶𝐵(𝑋) is a generalized nonexpansive mapping, then 𝑇 satisfies (2) of condition (∗∗).

Proof. Let {𝑥_𝑛} be an afps for 𝑇 in 𝐸 and 𝑥∈𝐸. By assumption we obtain 𝐻𝑇𝑥_𝑛,𝑇𝑥≤+𝛼𝛾‖‖𝑥_𝑛di−st𝑥‖‖𝑥_𝑛+𝛽dist𝑥_𝑛,𝑇𝑥_𝑛+dist(𝑥,𝑇𝑥),𝑇𝑥+dist𝑥,𝑇𝑥_𝑛.(3.39) Since dist(𝑥,𝑇𝑥)≤‖𝑥−𝑥_𝑛‖+dist(𝑥_𝑛,𝑇𝑥_𝑛)+𝐻(𝑇𝑥_𝑛,𝑇𝑥),dist(𝑥_𝑛,𝑇𝑥)≤dist(𝑥_𝑛,𝑇𝑥_𝑛)+𝐻(𝑇𝑥_𝑛,𝑇𝑥), dist(𝑥,𝑇𝑥_𝑛)≤‖𝑥−𝑥_𝑛‖+dist(𝑥_𝑛,𝑇𝑥_𝑛), dist(𝑥li,m𝑇_𝑛→𝒰𝑥)≤lim_𝑛→𝒰‖‖𝑥−𝑥_𝑛‖‖+lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥,(3.40)limd_𝑛→𝒰ist𝑥_𝑛,𝑇𝑥≤lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥,(3.41)dist𝑥,𝑇𝑥_𝑛≤lim_𝑛→𝒰‖𝑥−𝑥_𝑛‖.(3.42) By (3.39), lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥≤+𝛼𝛾limli_𝑛→𝒰m_𝑛→𝒰‖‖𝑥_𝑛−𝑥‖‖+𝛽lim_𝑛→𝒰‖‖𝑥−𝑥_𝑛‖‖+𝛽lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥𝐻𝑇𝑥_𝑛,𝑇𝑥+𝛾lim_𝑛→𝒰‖‖𝑥−𝑥_𝑛‖‖.(3.43) Thus (1−𝛽−𝛾)lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥≤(𝛼+𝛽+𝛾)lim_𝑛→𝒰‖‖𝑥−𝑥_𝑛‖‖,(3.44) and therefore, lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥≤lim_𝑛→𝒰‖‖𝑥_𝑛−𝑥‖‖.(3.45)

Takahashi Generalized Nonexpansive Mappings
Definition 3.22. Let 𝐸 be a nonempty subset of a Banach space 𝑋. A mapping 𝑇∶𝐸→𝑋 is said to be a Takahashi generalized nonexpansive mapping if, for some 𝛼,𝛽∈[0,1] with 𝛼+2𝛽≤1, there holds ‖𝑇𝑥−𝑇𝑦‖²≤𝛼‖𝑥−𝑦‖²+𝛽‖𝑦−𝑇𝑥‖²+‖𝑥−𝑇𝑦‖²for𝑥,𝑦∈𝐸.(3.46)
The following are examples of Takahashi generalized nonexpansive mappings: (i)nonexpansive mappings 𝑇∶‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖; (ii)nonspreading mappings 𝑇 [38]: 2‖𝑇𝑥−𝑇𝑦‖²≤‖𝑦−𝑇𝑥‖²+‖𝑥−𝑇𝑦‖²; (iii)hybrid mappings 𝑇 [39]: 3‖𝑇𝑥−𝑇𝑦‖²≤‖𝑦−𝑇𝑥‖²+‖𝑥−𝑇𝑦‖²; (iv)mappings 𝑇 [39]: 2‖𝑇𝑥−𝑇𝑦‖²≤‖𝑥−𝑦‖²+‖𝑦−𝑇𝑥‖²; (v)mappings 𝑇: 3‖𝑇𝑥−𝑇𝑦‖²≤2‖𝑦−𝑇𝑥‖²+‖𝑥−𝑇𝑦‖².
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+2𝛽≤1, if 𝑇∶𝐸→𝐾𝐶(𝑋) is a multivalued mapping such that 𝐻²(𝑇𝑥,𝑇𝑦)≤𝛼‖𝑥−𝑦‖²+𝛽dist²(𝑥,𝑇𝑦)+dist²(𝑦,𝑇𝑥),(3.47) then 𝑇 satisfies (2) of condition (∗∗).
Proof. Let {𝑥_𝑛} be an afps for 𝑇 in 𝐸 and 𝑥∈𝐸. By (3.41) and (3.42), lim_𝑛→𝒰𝐻²𝑇𝑥_𝑛,𝑇𝑥≤≤𝛼𝛼lliimm_{𝑛→𝒰𝑛→𝒰}‖‖𝑥_𝑛−𝑥‖‖²+𝛽lim_𝑛→𝒰dist²𝑥,𝑇𝑥_𝑛+𝛽lim_𝑛→𝒰dist²𝑥_𝑛,𝑇𝑥‖‖𝑥_𝑛−𝑥‖‖²+𝛽lim_𝑛→𝒰‖‖𝑥_𝑛−𝑥‖‖²+𝛽lim_𝑛→𝒰𝐻²𝑇𝑥_𝑛,𝑇𝑥.(3.48) Thus (1−𝛽)lim_𝑛→𝒰𝐻²𝑇𝑥_𝑛,𝑇𝑥≤(𝛼+𝛽)lim_𝑛→𝒰‖‖𝑥−𝑥_𝑛‖‖².(3.49) Therefore, lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥≤lim_𝑛→𝒰‖‖𝑥_𝑛−𝑥‖‖.(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 {𝑥_𝑛} given by 𝑥_𝑛≡0 for all 𝑛 forms an afps for 𝑇. Thus, 𝑇 fulfills condition (1) of Definition 3.2. If {𝑥_𝑛} is an apfs for 𝑇, then {𝑥_𝑛} converges to 0 and 𝐴(𝐸,{𝑥_𝑛})={0}. This implies that 𝐴(𝐸,{𝑥_𝑛}) has an apfs for 𝑇, and 𝑇 satisfies condition (∗). On the other hand, for the afps {𝑥_𝑛} given by 𝑥_𝑛≡0, if 𝑥∈(0,1/2], then lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥=√𝑥>𝑥=lim_𝑛→𝒰‖‖𝑥_𝑛−𝑥‖‖.(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 “lim_𝑛→𝑢” in Definition 3.14 is replaced by “limsup_𝑛→∞.” 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 {𝑥_𝑛} for 𝑇 to be regular and asymptotically uniform relative to 𝐸. Thus, 𝐴(𝐸,{𝑥_𝑛′})=𝐴(𝐸,{𝑥_𝑛}) for all subsequences {𝑥_𝑛′} of {𝑥_𝑛}. Therefore, by Lemma 2.5, it is easy to see that 𝐴_𝒰=𝐴(𝐸,{𝑥_𝑛}). If 𝑥∈𝐴_𝒰, then (3.33) becomes lim_𝑛→𝒰‖‖𝑥_𝑛−𝑧‖‖≤≤lliimm_𝑛→𝒰su‖‖p_𝑛→∞𝑦_𝑛−𝑧_𝑛‖‖=lim_𝑛→𝒰dist𝑦_𝑛,𝑇𝑥≤lim_𝑛→𝒰𝐻𝑇𝑥_𝑛,𝑇𝑥≤limsup_𝑛→∞𝐻𝑇𝑥_𝑛,𝑇𝑥‖‖𝑥_𝑛−𝑥‖‖=𝑟𝐸,𝑥_𝑛=𝑟_𝒰𝐸,𝑥_𝑛.(3.52) Hence 𝐴_𝒰∩𝑇𝑥≠∅, and the rest of the proof follows.
It is observed that if 𝐴_𝒰=𝐴(𝐸,{𝑥_𝑛}) and 𝐴(𝐸,{𝑥_𝑛})={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 𝑇∶𝐸→𝐸 is a generalized nonexpansive mapping with any of the following conditions holding, then 𝑇 satisfies condition (∗∗):(1)𝛼+2𝛽+2𝛾<1 (see [43, Theorem  4]);(2)𝛼+2𝛽+2𝛾=1 and 𝛽>0,𝛾>0,𝛼≥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,𝛼≥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.

Acknowledgment

The authors wish to thank the National Research University Project under Thailand's Office of the Higher Education Commission for financial support.