# Fixed Point Theorems by Ways of Ultra-Asymptotic Centers

Authors: S. Dhompongsa; N. Nanan
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 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 spaceX 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,
(1.1)rE(A(E,{xn}))≤λr(E,{xn}).Theorem 1.2 (see [4, Theorem  3.3]).
LetX 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⊂IE(x)forallx∈E, 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]).
Let𝒰 be a free ultrafilter defined on the set of natural numbers ℕ. The coefficient DL𝒰(X) of a Banach space X is defined as
(1.2)DLU(X)=sup⁡⁡{χE(AU(E,{xn}))χE({xn})},
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]).
LetE be a nonempty weakly compact convex subset of a Banach space X with DL𝒰(X)<1. Assume that T:E→KC(X) is a nonexpansive and 1-χ-contractive mapping such that Tx⊂IE(x) for all x∈E. Then T has a fixed point.Observe that, unlike Theorem1.2, 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 spaceX 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 E(1.3)rE(A(E,{xα}))≤λr(E,{xα}).
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]).
LetX 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⊂IE(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 spaceX 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
(1.4)r(E,{yn})≤λr(E,{xn}).Theorem 1.8 (see [7, Theorem  3.6]).
LetE 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]).
LetT:E→E be a mapping on a subset E of a Banach space X. Then T is said to satisfy condition (*) if (1)
for eachT-invariant subset K of E, T has an afps in K,(2)
for each pair ofT-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]).
LetX 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 {xn} in E strongly converges to z∈E and such that xn-Txn→0 we have z=Tz (cf. [9]).Following the concept ofDL𝒰(X), Dhompongsa and Inthakon [10] introduced the following coefficient.Definition 1.11 (see [10, Definition  3.2]).
Let𝒰 be a free ultrafilter defined on ℕ. The coefficient D𝒰(X) of a Banach space X is defined as
(1.5)DU(X)=sup⁡⁡{χE({yn})χE({xn})},
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 coefficientD𝒰(X) is the following property.Definition 1.12 (see [10, Definition  3.1]).
A Banach spaceX 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
(1.6)r(E,{yn})≤λr(E,{xn}).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, 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]).
LetE 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⊂IE(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]).
LetE be a weakly compact convex subset of a Banach space X and 𝒰 a free ultrafilter defined on ℕ. Then D𝒰(X)<1 if and only if X has property (D′).Thus, unlike the coefficientDL𝒰(X), we have, for any two free ultrafilters 𝒰 and 𝒱, D𝒰(X)<1 if and only if D𝒱(X)<1.In Section3.1, 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

LetE 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 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,(2.1)H(A,B)∶=max⁡⁡{sup⁡a∈A⁡dist(a,B),sup⁡b∈B⁡dist(b,A)},A,B∈CB(X),wheredist(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 if(2.2)H(Tx,Ty)≤‖x-y‖,∀x,y∈E,andT is said to be a contraction if there exists a constant k<1 such that(2.3)H(Tx,Ty)≤k‖x-y‖,∀x,y∈E.
A multivalued mapping T:E→2X 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 inequality(2.4)ϕ(T(B))<ϕ(B)(resp.ϕ(T(B))≤ϕ(B)),whereT(B)=⋃x∈BTx.Recall that the inward set ofE at x∈E is defined by(2.5)IE(x)={x+α(y-x):α≥1,y∈E}.
A sequence {xn} in E for which limn→∞∥xn-Txn∥=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 {xn} in E of a Banach space X for which limn→∞dist(xn,Txn)=0 is called an approximate fixed point sequence (afps for short) for T.We denote byxn→x to indicate that the sequence {xn} in X converges to x∈X.LetE be a nonempty closed and convex subset of a Banach space X and {xn} a bounded sequence in X. For x∈X, define the asymptotic radius of {xn} at x as the number(2.6)r(x,{xn})=limsup⁡n→∞⁡‖xn-x‖.
Let(2.7)r(E,{xn})∶=inf⁡⁡{r(x,{xn}):x∈E},A(E,{xn})∶={x∈E:r(x,{xn})=r(E,{xn})}.
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. Let𝒰 be a free ultrafilter on ℕ. Recall ([26, 28–30]) that the ultrapower (X)𝒰 of a Banach space X is the quotient space of(2.8)l∞(X)={{xn}:xn∈X∀n∈N,‖{xn}‖=sup⁡n⁡‖xn‖<∞}
by(2.9)ker⁡N={{xn}∈l∞(X):lim⁡n→U⁡‖xn‖=0}.
One can proof that X̃=(X)𝒰 is a Banach space with the quotient norm given by ∥{xn}𝒰∥=limn→𝒰∥xn∥, where {xn}𝒰 is the equivalence class of {xn}. It is also clear that X is isometric to a subspace of X̃ 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 X̃ under this isometry, respectively, and denote(2.10)Ẽ={x̃∈X̃:∃{xn}suchthatx̃={xn}U,xn∈E∀n∈N}.Thusẋ={x,x,…}𝒰 and Ė={ẋ∈X̃:x∈E}.IfT:E→CB(X) is a multivalued mapping, we define a corresponding multivalued mapping T̃:Ẽ→CB(X̃) by(2.11)T̃(x̃)∶={ũ∈X̃:∃{un}suchthatũ={un}U,un∈Txn∀n∈N},wherex̃={xn}𝒰∈Ẽ. Moreover the set T̃(x̃) is bounded and closed (see [28, 29]). The Hausdorff metric on CB(X̃) will be denoted by H̃.Proposition 2.2 (see [5, Proposition  3.1]).
For every{xn}𝒰 and {yn}𝒰 in Ẽ,
(2.12)H̃(T̃{xn}U,T̃{yn}U)=lim⁡n→U⁡H(Txn,Tyn).Proposition 2.3 (see [31, Page 37], [5, Proposition  3.2]).
LetE be a nonempty subset of a Banach space X and T:E→CB(X). (i)
IfT is convex-valued, then T̃ is convex-valued.(ii)
IfT is compact-valued, then T̃ is compact-valued and T̃ẋ=(Ṫx) for every x∈E.(iii)
IfT is nonexpansive, then T̃ is nonexpansive.Let𝒰 denote a free ultrafilter defined on ℕ. 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 (2.13)rU(E,{xn})=inf⁡⁡{lim⁡n→U⁡‖xn-x‖:x∈E},AU(E,{xn})={x∈E:lim⁡n→U⁡‖xn-x‖=rU(E,{xn})}.
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 X̃ [5]:(2.14)rU(E,{xn})=inf⁡x∈E⁡‖{xn}U-ẋ‖is the relative Chebyshev radius of{xn}𝒰, and(2.15)(AU(Ė,{xn}))=Ė∩BX̃({xn}U,r)is the relative Chebyshev center of{xn}𝒰 relative to Ė in the ultrapower X̃. (Here BX̃({xn}𝒰,r) denotes the ball in X̃ 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] as(2.16)χE(A)=inf⁡⁡{ε>0:AcanbecoveredbyfinitelymanyballsinEofradii≤ε}.Proposition 2.4 (see [5, Proposition  4.5]).
If{xn} is a bounded sequence which is regular relative to E, then
(2.17)r(E,{xn})=rU(E,{xn})=χE({xn}).From Proposition2.4, we have, for w∈A(E,{xn}),(2.18)lim⁡n→U⁡‖xn-w‖≤limsup⁡n→∞⁡‖xn-w‖=r(E,{xn})=rU(E,{xn}).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]).
LetE 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 Lemma2.5 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.
LetE be a nonempty subset of a Banach space X and T:E→CB(X). Then (1)
ifT is uniformly continuous, then T̃ is uniformly continuous;(2)
ifT is continuous at z∈E, then T̃ is continuous at ż.Proof.
(1) Letε>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 {xn}𝒰,{yn}𝒰∈Ẽ, and ∥{xn}𝒰-{yn}𝒰∥<δ. Let A={n:∥xn-yn∥<δ} and B={n:H(Txn,Tyn)<ε}. Since A∈𝒰 and A⊂B, B∈𝒰. Thus, by Proposition 2.2H̃(T̃{xn}𝒰,T̃{yn}𝒰)≤ε.
(2) Letε>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 {xn}𝒰∈Ẽ such that ∥{xn}𝒰-ż∥<δ, then, letting A={n:∥xn-z∥<δ} and B={n:H(Txn,Tz)<ε}, we see that A∈𝒰 and B∈𝒰. Thus, by Proposition 2.2H̃(T̃{xn}𝒰,T̃ż)≤ε.We now introduce condition(*) for multivalued mappings.Definition 3.2.
LetE 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,{xn′}) for some subsequence {xn′} of any given afps {xn} for T in E.Theorem 3.3.
LetX 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 {xn0} be an afps for T in E. We can assume by Proposition 2.1 that {xn0} is regular relative to E. Condition (2) of Definition 3.2 gives us a subsequence {xn00} of {xn0} so that the center A(E,{xn00}) contains an afps for T. Denote A0=A(E,{xn00}), and let {xn1} be an afps in A0. Assume that {xn1} is regular relative to E. As before, T has an afps in A(E,{xn11}) for some subsequence {xn11} of {xn1}. Since X has property (D′), put λ=D𝒰(X)<1. Then, by Proposition 2.4 and Definition 1.11,
(3.1)r(E,{xn11})=χE({xn11})≤λχE({xn00})=λr(E,{xn00}).
Continue the procedure to obtain, for each m≥0, a regular sequence {xnmm} relative to E in Am-1∶=A(E,{xnm-1m-1}) such that
(3.2)lim⁡n→∞⁡dist⁡(xnmm,Txnmm)=0,
and for all m≥1,
(3.3)χE({xnmm})≤λr(E,{xnm-1m-1}).
Consequently,
(3.4)r(E,{xnmm})≤λr(E,{xnm-1m-1})≤⋯≤λmr(E,{xn00}).
We show that {xnmm}𝒰 is a Cauchy sequence in X̃. Indeed, for each m≥1, take an element ẏm∈Ȧm-1. Then
(3.5)‖ẋnmm-ẏm{xnm-1m-1}U‖≤‖ẋnmm-{xnm-1m-1}U‖+‖{xnm-1m-1}U-ẏm‖≤2r(E,{xnm-1m-1})
for all m≥1, and hence
(3.6)‖{xnmm}U-{xnm-1m-1}U‖≤‖{xnmm}U-ẏm‖+‖ẏm-{xnm-1m-1}U‖≤3r(E,{xnm-1m-1}).
Thus
(3.7)‖{xnmm}U-{xnm-1m-1}U‖≤3λm-1r(E,{xn00}),
implying that {xnmm}𝒰 is a Cauchy sequence and hence converges to some {zn}𝒰 in Ẽ as m→∞. Next, we will show that {zn}𝒰∈Ė. For each m≥0,
(3.8)dist⁡({zn}U,Ė)≤‖{zn}U-{xnmm}U‖+dist⁡({xnmm}U,Ė)≤‖{zn}U-{xnmm}U‖+‖{xnmm}U-ẋ1m+1m+1‖=‖{zn}U-{xnmm}U‖+r(E,{xnmm})≤‖{zn}U-{xnmm}U‖+λmr(E,{xn00}).
Taking m→∞ we see that
(3.9)dist({zn}U,Ė)=0.
Thus, it follows that there exists z∈E such that {zn}𝒰=ż. By Lemma 3.1, T̃ is continuous at ż, and thus H̃(T̃{xnmm}𝒰,T̃ż)→0 as m→∞. For every m≥0,
(3.10)dist(ż,T̃ż)≤‖ż-{xnmm}U‖+dist({xnmm}U,T̃{xnmm}U)+H̃(T̃{xnmm}U,T̃ż).
Taking m→∞ we then obtain ż∈T̃ż. By Proposition 2.3, T̃ż=(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 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)
Theorem3.3 (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)
Theorem3.3 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 Remark3.15(ii) below, we show the following implication:(3.11)(**)+(A)⟹(*).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 Remark3.24(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 T:E→KC(X) is nonexpansive and 1-χ-contractive such that Tx⊂IE(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]).
LetE 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)∩IE(x)¯≠∅ for all x∈E, then F has a fixed point.
Proposition 3.6.
LetE 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⊂IE(x) for every x∈E. Then T satisfies condition (*).
Proof.
First, we will show thatT has an afps in E. Let y0∈E, and consider, for each n≥1, the contraction Tn:E→KC(X) defined by
(3.12)Tn(x)=1ny0+(1-1n)Tx,x∈E.
It is not difficult to see that Tn(x)⊂IE(x) for every x∈E. Since T is 1-χ-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 thatT 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
(3.13)Tx∩IAU(x)≠∅for everyx∈AU.
Let x∈A𝒰. Observe first that {xn}𝒰∈T̃{xn}𝒰. By Proposition 2.3, T̃ẋ=(Ṫx) is compact, and hence there exists u∈Tx such that
(3.14)‖{xn}U-u̇‖=H̃(T̃{xn}U,T̃ẋ)≤‖{xn}U-ẋ‖=rU(E,{xn}).
Since u∈Tx⊂IE(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∈A𝒰. If α>1 then y=(1/α)u+(1-1/α)x, and therefore, we have
(3.15)‖{xn}U-ẏ‖≤1α‖{xn}U-u̇‖+(1-1α)‖{xn}U-ẋ‖≤rU(E,{xn}).
Hence y∈A𝒰 and consequently u∈IA𝒰(x). Thus (3.13) is justified.
Fixedy0∈A𝒰, and consider for each n≥1, the contraction Tn:A𝒰→KC(X) defined by
(3.16)Tn(x)=1ny0+(1-1n)Tx,x∈AU.
As before, Tn is (1-1/n)-χ-contractive, and by Theorem 3.5, there exists a fixed point zn∈A𝒰 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 mappingT 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]).
LetE 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]).
LetE 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 Theorem3.9 partially.
Proposition 3.10.
LetE 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 thatT has an afps in E. Let y0∈E, and consider, for each n≥1, the contraction Tn:E→KC(X) defined by
(3.17)Tn(x)=1ny0+(1-1n)Tx,x∈E.
For x∈E, let a∈Tx∩E. Thus (1/n)y0+(1-1/n)a∈Tnx∩E, and therefore, Tnx∩E≠∅ for every x∈E. We show that δ(TnK)<δ(K)forallclosedsetsKwithδ(K)>0. Let K be a closed subset of E with δ(K)>0. For x,y∈TnK, there exist x′,y′∈TK such that
(3.18)x=1ny0+(1-1n)x′,y=1ny0+(1-1n)y′,
and this entails ∥x-y∥=(1-1/n)∥x′-y′∥≤(1-1/n)δ(TK). Hence δ(TnK)≤(1-1/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 thatT 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
(3.19)AU∩Tx≠∅,for everyx∈AU.
Let x∈A𝒰, and for each n≥1, we see that H(Txn,Tx)≤δ(T{xn,x})≤δ({xn,x})=∥xn-x∥. Take yn∈Txn so that
(3.20)‖xn-yn‖=dist(xn,Txn),
and select zn∈Tx for each n such that
(3.21)‖zn-yn‖=dist⁡(yn,Tx).
Let limn→𝒰zn=z∈Tx. Note that
(3.22)‖xn-z‖≤‖xn-yn‖+‖yn-zn‖+‖zn-z‖.
We obtain
(3.23)lim⁡n→U⁡‖xn-z‖≤lim⁡n→U⁡‖yn-zn‖=lim⁡n→U⁡dist⁡(yn,Tx)≤lim⁡n→U⁡H(Txn,Tx)≤lim⁡n→U⁡‖xn-x‖=rU(E,{xn})
proving that z∈A𝒰. Thus (3.19) is satisfied. Fix y0∈A𝒰, and consider, for each n≥1, the contraction Tn:A𝒰→KC(X) defined by
(3.24)Tn(x)=1ny0+(1-1n)Tx,x∈AU.
For x∈E, let a∈A𝒰∩Tx. Thus (1/n)y0+(1-1/n)a∈A𝒰∩Tnx. Therefore, A𝒰∩Tnx≠∅ for every x∈A𝒰. Let K be a closed subset of E with δ(K)>0. As before, δ(TnK)≤(1-1/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 {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).
LetE 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]).
LetX 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 Section3.1.Definition 3.14.
Let𝒰 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{xn} is an apfs for T in E and x∈E, then limn→𝒰H(Txn,Tx)≤limn→𝒰∥xn-x∥.Remark 3.15.
(i)  LetE 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)  everyT-invariant, closed, and convex subset possesses an afps,
thenT 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 x∈A𝒰. The compactness of Txn implies that for each n we can take yn∈Txn so that
(3.25)‖xn-yn‖=dist⁡(xn,Txn).
Since Tx is compact, select zn∈Tx for each n such that
(3.26)‖zn-yn‖=dist⁡(yn,Tx).
Let limn→𝒰zn=z∈Tx. Note that
(3.27)‖xn-z‖≤‖xn-yn‖+‖yn-zn‖+‖zn-z‖.
We obtain
(3.28)lim⁡n→U⁡‖xn-z‖≤lim⁡n→U⁡‖yn-zn‖=lim⁡n→U⁡dist⁡(yn,Tx)≤lim⁡n→U⁡H(Txn,Tx)≤lim⁡n→U⁡‖xn-x‖=rU(E,{xn})
proving that z∈A𝒰 and hence A𝒰∩Tx≠∅ for all x∈A𝒰, 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 T:E→E satisfies condition (L) on E provided that it fulfills the following two conditions. (1)
If a setD⊂E 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 x∈E,(3.29)limsup⁡n→∞∥xn-Tx∥≤limsup⁡n→∞∥xn-x∥.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.
LetX 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{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 yn∈Txn such that
(3.30)‖xn-yn‖=dist⁡(xn,Txn).
If x∈A𝒰, since Tx is compact, select zn∈Tx for each n such that
(3.31)‖zn-yn‖=dist⁡(yn,Tx).
Let limn→𝒰zn=z∈Tx. Note that
(3.32)‖xn-z‖≤‖xn-yn‖+‖yn-zn‖+‖zn-z‖.
Thus
(3.33)lim⁡n→U⁡‖xn-z‖≤lim⁡n→U⁡‖yn-zn‖=lim⁡n→U⁡dist⁡(yn,Tx)≤lim⁡n→U⁡H(Txn,Tx)≤lim⁡n→U⁡‖xn-x‖=rU(E,{xn})
proving that z∈A𝒰 and hence A𝒰∩Tx≠∅ for all x∈A𝒰. By assumption, A(E,{xn}) is nonempty and compact which implies that A𝒰 is also nonempty and compact. Now define a mapping F:A𝒰→KC(A𝒰) by Fx∶=A𝒰∩Tx for all x∈A𝒰. Thus F is upper semicontinuous. Indeed, let {un}⊂A be such that limn→∞un=u, and let vn∈Fun be such that limn→∞vn=v. Since T is upper semicontinuous and A𝒰 is compact, we have v∈Tu and v∈A𝒰, that is v∈Fu. 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 Theorem3.16 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]).
LetE 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]).
LetE 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 (Cλ).
García-Falset et al. [9] introduced the following mappings.
Let E be a nonempty subset of a Banach spaceX. For λ∈(0,1), we say that a mapping T:E→X satisfies Condition (Cλ) on E if, for each x,y∈E,
(3.34)λ‖x-Tx‖≤‖x-y‖implies‖Tx-Ty‖≤‖x-y‖.
It is natural to define a multivalued version of Condition (Cλ) (see [18]).
LetE 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 (Cλ) for some λ∈(0,1) if, for each x,y∈E,
(3.35)λdist⁡(x,Tx)≤‖x-y‖impliesH(Tx,Ty)≤‖x-y‖.
Clearly, T satisfies (2) of condition (**).Proposition 3.20.
LetE be a nonempty bounded closed and convex subset of a Banach space X. If T:E→CB(E) satisfies condition (Cλ) for some λ∈(0,1), then T satisfies condition (**).Proof.
We only show thatE contains an afps for T. But this follows from [37, Lemma  2.8].Generalized Nonexpansive Mappings
LetE 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 ∑i=15αi=1,
(3.36)‖Tx-Ty‖≤α1‖x-y‖+α2‖x-Tx‖+α3‖y-Ty‖+α4‖x-Ty‖+α5‖y-Tx‖,
for each x,y∈E.We will use the following equivalent condition.For some nonnegative constantsα,β,γ with α+2β+2γ≤1,(3.37)‖Tx-Ty‖≤α‖x-y‖+β(‖x-Tx‖+‖y-Ty‖)+γ(‖x-Ty‖+‖y-Tx‖),for allx,y∈E.We introduce a multivalued version of these mappings.LetT: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 holds(3.38)H(Tx,Ty)≤α‖x-y‖+β(dist⁡(x,Tx)+dist⁡(y,Ty))+γ(dist⁡(x,Ty)+dist⁡(y,Tx)).Proposition 3.21.
LetE 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{xn} be an afps for T in E and x∈E. By assumption we obtain
(3.39)H(Txn,Tx)≤α‖xn-x‖+β(dist⁡(xn,Txn)+dist⁡(x,Tx))+γ(dist⁡(xn,Tx)+dist⁡(x,Txn)).
Since dist(x,Tx)≤∥x-xn∥+dist(xn,Txn)+H(Txn,Tx),dist(xn,Tx)≤dist(xn,Txn)+H(Txn,Tx), dist(x,Txn)≤∥x-xn∥+dist(xn,Txn),
(3.40)dist⁡(x,Tx)≤lim⁡n→U⁡‖x-xn‖+lim⁡n→U⁡H(Txn,Tx),(3.41)lim⁡n→U⁡dist⁡(xn,Tx)≤lim⁡n→U⁡H(Txn,Tx),(3.42)lim⁡n→Udist⁡(x,Txn)≤lim⁡n→U∥x-xn∥.
By (3.39),
(3.43)lim⁡n→U⁡H(Txn,Tx)≤αlim⁡n→U⁡‖xn-x‖+βlim⁡n→U⁡‖x-xn‖+βlim⁡n→U⁡H(Txn,Tx)+γlim⁡n→U⁡H(Txn,Tx)+γlim⁡n→U⁡‖x-xn‖.
Thus
(3.44)(1-β-γ)lim⁡n→U⁡H(Txn,Tx)≤(α+β+γ)lim⁡n→U⁡‖x-xn‖,
and therefore,
(3.45)lim⁡n→U⁡H(Txn,Tx)≤lim⁡n→U⁡‖xn-x‖.Takahashi Generalized Nonexpansive Mappings
Definition 3.22.
LetE 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
(3.46)‖Tx-Ty‖2≤α‖x-y‖2+β(‖y-Tx‖2+‖x-Ty‖2)forx,y∈E.
The following are examples of Takahashi generalized nonexpansive mappings:(i)
nonexpansive mappingsT:∥Tx-Ty∥≤∥x-y∥;(ii)
nonspreading mappingsT [38]: 2∥Tx-Ty∥2≤∥y-Tx∥2+∥x-Ty∥2;(iii)
hybrid mappingsT [39]: 3∥Tx-Ty∥2≤∥y-Tx∥2+∥x-Ty∥2;(iv)
mappingsT [39]: 2∥Tx-Ty∥2≤∥x-y∥2+∥y-Tx∥2;(v)
mappingsT: 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.
LetE 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
(3.47)H2(Tx,Ty)≤α‖x-y‖2+β(dist⁡2(x,Ty)+dist⁡2(y,Tx)),
then T satisfies (2) of condition (**).
Proof.
Let{xn} be an afps for T in E and x∈E. By (3.41) and (3.42),
(3.48)lim⁡n→U⁡H2(Txn,Tx)≤αlim⁡n→U⁡‖xn-x‖2+βlim⁡n→U⁡dist⁡2(x,Txn)+βlim⁡n→U⁡dist⁡2(xn,Tx)≤αlim⁡n→U⁡‖xn-x‖2+βlim⁡n→U⁡‖xn-x‖2+βlim⁡n→U⁡H2(Txn,Tx).
Thus
(3.49)(1-β)lim⁡n→U⁡H2(Txn,Tx)≤(α+β)lim⁡n→U⁡‖x-xn‖2.
Therefore,
(3.50)lim⁡n→U⁡H(Txn,Tx)≤lim⁡n→U⁡‖xn-x‖.
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,x3]. Since 0 is a fixed point of T, the sequence {xn} given by xn≡0 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 xn≡0, if x∈(0,1/2], then
(3.51)lim⁡n→U⁡H(Txn,Tx)=x>x=lim⁡n→U⁡‖xn-x‖.
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 Theorem3.16 is still valid when “limn→u” 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 x∈A𝒰, then (3.33) becomes
(3.52)lim⁡n→U⁡‖xn-z‖≤lim⁡n→U⁡‖yn-zn‖=lim⁡n→U⁡dist(yn,Tx)≤lim⁡n→U⁡H(Txn,Tx)≤limsup⁡n→∞⁡H(Txn,Tx)≤limsup⁡n→∞⁡‖xn-x‖=r(E,{xn})=rU(E,{xn}).
Hence A𝒰∩Tx≠∅, and the rest of the proof follows.
It is observed that ifA𝒰=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)  IfT: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 Theorem3.16, 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.


## 3.1. Property(D′)

Lemma 3.1.
LetE be a nonempty subset of a Banach space X and T:E→CB(X). Then (1)
ifT is uniformly continuous, then T̃ is uniformly continuous;(2)
ifT is continuous at z∈E, then T̃ is continuous at ż.Proof.
(1) Letε>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 {xn}𝒰,{yn}𝒰∈Ẽ, and ∥{xn}𝒰-{yn}𝒰∥<δ. Let A={n:∥xn-yn∥<δ} and B={n:H(Txn,Tyn)<ε}. Since A∈𝒰 and A⊂B, B∈𝒰. Thus, by Proposition 2.2H̃(T̃{xn}𝒰,T̃{yn}𝒰)≤ε.
(2) Letε>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 {xn}𝒰∈Ẽ such that ∥{xn}𝒰-ż∥<δ, then, letting A={n:∥xn-z∥<δ} and B={n:H(Txn,Tz)<ε}, we see that A∈𝒰 and B∈𝒰. Thus, by Proposition 2.2H̃(T̃{xn}𝒰,T̃ż)≤ε.We now introduce condition(*) for multivalued mappings.Definition 3.2.
LetE 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,{xn′}) for some subsequence {xn′} of any given afps {xn} for T in E.Theorem 3.3.
LetX 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 {xn0} be an afps for T in E. We can assume by Proposition 2.1 that {xn0} is regular relative to E. Condition (2) of Definition 3.2 gives us a subsequence {xn00} of {xn0} so that the center A(E,{xn00}) contains an afps for T. Denote A0=A(E,{xn00}), and let {xn1} be an afps in A0. Assume that {xn1} is regular relative to E. As before, T has an afps in A(E,{xn11}) for some subsequence {xn11} of {xn1}. Since X has property (D′), put λ=D𝒰(X)<1. Then, by Proposition 2.4 and Definition 1.11,
(3.1)r(E,{xn11})=χE({xn11})≤λχE({xn00})=λr(E,{xn00}).
Continue the procedure to obtain, for each m≥0, a regular sequence {xnmm} relative to E in Am-1∶=A(E,{xnm-1m-1}) such that
(3.2)lim⁡n→∞⁡dist⁡(xnmm,Txnmm)=0,
and for all m≥1,
(3.3)χE({xnmm})≤λr(E,{xnm-1m-1}).
Consequently,
(3.4)r(E,{xnmm})≤λr(E,{xnm-1m-1})≤⋯≤λmr(E,{xn00}).
We show that {xnmm}𝒰 is a Cauchy sequence in X̃. Indeed, for each m≥1, take an element ẏm∈Ȧm-1. Then
(3.5)‖ẋnmm-ẏm{xnm-1m-1}U‖≤‖ẋnmm-{xnm-1m-1}U‖+‖{xnm-1m-1}U-ẏm‖≤2r(E,{xnm-1m-1})
for all m≥1, and hence
(3.6)‖{xnmm}U-{xnm-1m-1}U‖≤‖{xnmm}U-ẏm‖+‖ẏm-{xnm-1m-1}U‖≤3r(E,{xnm-1m-1}).
Thus
(3.7)‖{xnmm}U-{xnm-1m-1}U‖≤3λm-1r(E,{xn00}),
implying that {xnmm}𝒰 is a Cauchy sequence and hence converges to some {zn}𝒰 in Ẽ as m→∞. Next, we will show that {zn}𝒰∈Ė. For each m≥0,
(3.8)dist⁡({zn}U,Ė)≤‖{zn}U-{xnmm}U‖+dist⁡({xnmm}U,Ė)≤‖{zn}U-{xnmm}U‖+‖{xnmm}U-ẋ1m+1m+1‖=‖{zn}U-{xnmm}U‖+r(E,{xnmm})≤‖{zn}U-{xnmm}U‖+λmr(E,{xn00}).
Taking m→∞ we see that
(3.9)dist({zn}U,Ė)=0.
Thus, it follows that there exists z∈E such that {zn}𝒰=ż. By Lemma 3.1, T̃ is continuous at ż, and thus H̃(T̃{xnmm}𝒰,T̃ż)→0 as m→∞. For every m≥0,
(3.10)dist(ż,T̃ż)≤‖ż-{xnmm}U‖+dist({xnmm}U,T̃{xnmm}U)+H̃(T̃{xnmm}U,T̃ż).
Taking m→∞ we then obtain ż∈T̃ż. By Proposition 2.3, T̃ż=(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 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)
Theorem3.3 (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)
Theorem3.3 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 Remark3.15(ii) below, we show the following implication:(3.11)(**)+(A)⟹(*).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 Remark3.24(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 T:E→KC(X) is nonexpansive and 1-χ-contractive such that Tx⊂IE(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]).
LetE 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)∩IE(x)¯≠∅ for all x∈E, then F has a fixed point.
Proposition 3.6.
LetE 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⊂IE(x) for every x∈E. Then T satisfies condition (*).
Proof.
First, we will show thatT has an afps in E. Let y0∈E, and consider, for each n≥1, the contraction Tn:E→KC(X) defined by
(3.12)Tn(x)=1ny0+(1-1n)Tx,x∈E.
It is not difficult to see that Tn(x)⊂IE(x) for every x∈E. Since T is 1-χ-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 thatT 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
(3.13)Tx∩IAU(x)≠∅for everyx∈AU.
Let x∈A𝒰. Observe first that {xn}𝒰∈T̃{xn}𝒰. By Proposition 2.3, T̃ẋ=(Ṫx) is compact, and hence there exists u∈Tx such that
(3.14)‖{xn}U-u̇‖=H̃(T̃{xn}U,T̃ẋ)≤‖{xn}U-ẋ‖=rU(E,{xn}).
Since u∈Tx⊂IE(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∈A𝒰. If α>1 then y=(1/α)u+(1-1/α)x, and therefore, we have
(3.15)‖{xn}U-ẏ‖≤1α‖{xn}U-u̇‖+(1-1α)‖{xn}U-ẋ‖≤rU(E,{xn}).
Hence y∈A𝒰 and consequently u∈IA𝒰(x). Thus (3.13) is justified.
Fixedy0∈A𝒰, and consider for each n≥1, the contraction Tn:A𝒰→KC(X) defined by
(3.16)Tn(x)=1ny0+(1-1n)Tx,x∈AU.
As before, Tn is (1-1/n)-χ-contractive, and by Theorem 3.5, there exists a fixed point zn∈A𝒰 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 mappingT 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]).
LetE 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]).
LetE 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 Theorem3.9 partially.
Proposition 3.10.
LetE 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 thatT has an afps in E. Let y0∈E, and consider, for each n≥1, the contraction Tn:E→KC(X) defined by
(3.17)Tn(x)=1ny0+(1-1n)Tx,x∈E.
For x∈E, let a∈Tx∩E. Thus (1/n)y0+(1-1/n)a∈Tnx∩E, and therefore, Tnx∩E≠∅ for every x∈E. We show that δ(TnK)<δ(K)forallclosedsetsKwithδ(K)>0. Let K be a closed subset of E with δ(K)>0. For x,y∈TnK, there exist x′,y′∈TK such that
(3.18)x=1ny0+(1-1n)x′,y=1ny0+(1-1n)y′,
and this entails ∥x-y∥=(1-1/n)∥x′-y′∥≤(1-1/n)δ(TK). Hence δ(TnK)≤(1-1/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 thatT 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
(3.19)AU∩Tx≠∅,for everyx∈AU.
Let x∈A𝒰, and for each n≥1, we see that H(Txn,Tx)≤δ(T{xn,x})≤δ({xn,x})=∥xn-x∥. Take yn∈Txn so that
(3.20)‖xn-yn‖=dist(xn,Txn),
and select zn∈Tx for each n such that
(3.21)‖zn-yn‖=dist⁡(yn,Tx).
Let limn→𝒰zn=z∈Tx. Note that
(3.22)‖xn-z‖≤‖xn-yn‖+‖yn-zn‖+‖zn-z‖.
We obtain
(3.23)lim⁡n→U⁡‖xn-z‖≤lim⁡n→U⁡‖yn-zn‖=lim⁡n→U⁡dist⁡(yn,Tx)≤lim⁡n→U⁡H(Txn,Tx)≤lim⁡n→U⁡‖xn-x‖=rU(E,{xn})
proving that z∈A𝒰. Thus (3.19) is satisfied. Fix y0∈A𝒰, and consider, for each n≥1, the contraction Tn:A𝒰→KC(X) defined by
(3.24)Tn(x)=1ny0+(1-1n)Tx,x∈AU.
For x∈E, let a∈A𝒰∩Tx. Thus (1/n)y0+(1-1/n)a∈A𝒰∩Tnx. Therefore, A𝒰∩Tnx≠∅ for every x∈A𝒰. Let K be a closed subset of E with δ(K)>0. As before, δ(TnK)≤(1-1/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 {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).
LetE 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]).
LetX 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 Section3.1.Definition 3.14.
Let𝒰 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{xn} is an apfs for T in E and x∈E, then limn→𝒰H(Txn,Tx)≤limn→𝒰∥xn-x∥.Remark 3.15.
(i)  LetE 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)  everyT-invariant, closed, and convex subset possesses an afps,
thenT 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 x∈A𝒰. The compactness of Txn implies that for each n we can take yn∈Txn so that
(3.25)‖xn-yn‖=dist⁡(xn,Txn).
Since Tx is compact, select zn∈Tx for each n such that
(3.26)‖zn-yn‖=dist⁡(yn,Tx).
Let limn→𝒰zn=z∈Tx. Note that
(3.27)‖xn-z‖≤‖xn-yn‖+‖yn-zn‖+‖zn-z‖.
We obtain
(3.28)lim⁡n→U⁡‖xn-z‖≤lim⁡n→U⁡‖yn-zn‖=lim⁡n→U⁡dist⁡(yn,Tx)≤lim⁡n→U⁡H(Txn,Tx)≤lim⁡n→U⁡‖xn-x‖=rU(E,{xn})
proving that z∈A𝒰 and hence A𝒰∩Tx≠∅ for all x∈A𝒰, 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 T:E→E satisfies condition (L) on E provided that it fulfills the following two conditions. (1)
If a setD⊂E 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 x∈E,(3.29)limsup⁡n→∞∥xn-Tx∥≤limsup⁡n→∞∥xn-x∥.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.
LetX 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{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 yn∈Txn such that
(3.30)‖xn-yn‖=dist⁡(xn,Txn).
If x∈A𝒰, since Tx is compact, select zn∈Tx for each n such that
(3.31)‖zn-yn‖=dist⁡(yn,Tx).
Let limn→𝒰zn=z∈Tx. Note that
(3.32)‖xn-z‖≤‖xn-yn‖+‖yn-zn‖+‖zn-z‖.
Thus
(3.33)lim⁡n→U⁡‖xn-z‖≤lim⁡n→U⁡‖yn-zn‖=lim⁡n→U⁡dist⁡(yn,Tx)≤lim⁡n→U⁡H(Txn,Tx)≤lim⁡n→U⁡‖xn-x‖=rU(E,{xn})
proving that z∈A𝒰 and hence A𝒰∩Tx≠∅ for all x∈A𝒰. By assumption, A(E,{xn}) is nonempty and compact which implies that A𝒰 is also nonempty and compact. Now define a mapping F:A𝒰→KC(A𝒰) by Fx∶=A𝒰∩Tx for all x∈A𝒰. Thus F is upper semicontinuous. Indeed, let {un}⊂A be such that limn→∞un=u, and let vn∈Fun be such that limn→∞vn=v. Since T is upper semicontinuous and A𝒰 is compact, we have v∈Tu and v∈A𝒰, that is v∈Fu. 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 Theorem3.16 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]).
LetE 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]).
LetE 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 (Cλ).
García-Falset et al. [9] introduced the following mappings.
Let E be a nonempty subset of a Banach spaceX. For λ∈(0,1), we say that a mapping T:E→X satisfies Condition (Cλ) on E if, for each x,y∈E,
(3.34)λ‖x-Tx‖≤‖x-y‖implies‖Tx-Ty‖≤‖x-y‖.
It is natural to define a multivalued version of Condition (Cλ) (see [18]).
LetE 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 (Cλ) for some λ∈(0,1) if, for each x,y∈E,
(3.35)λdist⁡(x,Tx)≤‖x-y‖impliesH(Tx,Ty)≤‖x-y‖.
Clearly, T satisfies (2) of condition (**).Proposition 3.20.
LetE be a nonempty bounded closed and convex subset of a Banach space X. If T:E→CB(E) satisfies condition (Cλ) for some λ∈(0,1), then T satisfies condition (**).Proof.
We only show thatE contains an afps for T. But this follows from [37, Lemma  2.8].Generalized Nonexpansive Mappings
LetE 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 ∑i=15αi=1,
(3.36)‖Tx-Ty‖≤α1‖x-y‖+α2‖x-Tx‖+α3‖y-Ty‖+α4‖x-Ty‖+α5‖y-Tx‖,
for each x,y∈E.We will use the following equivalent condition.For some nonnegative constantsα,β,γ with α+2β+2γ≤1,(3.37)‖Tx-Ty‖≤α‖x-y‖+β(‖x-Tx‖+‖y-Ty‖)+γ(‖x-Ty‖+‖y-Tx‖),for allx,y∈E.We introduce a multivalued version of these mappings.LetT: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 holds(3.38)H(Tx,Ty)≤α‖x-y‖+β(dist⁡(x,Tx)+dist⁡(y,Ty))+γ(dist⁡(x,Ty)+dist⁡(y,Tx)).Proposition 3.21.
LetE 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{xn} be an afps for T in E and x∈E. By assumption we obtain
(3.39)H(Txn,Tx)≤α‖xn-x‖+β(dist⁡(xn,Txn)+dist⁡(x,Tx))+γ(dist⁡(xn,Tx)+dist⁡(x,Txn)).
Since dist(x,Tx)≤∥x-xn∥+dist(xn,Txn)+H(Txn,Tx),dist(xn,Tx)≤dist(xn,Txn)+H(Txn,Tx), dist(x,Txn)≤∥x-xn∥+dist(xn,Txn),
(3.40)dist⁡(x,Tx)≤lim⁡n→U⁡‖x-xn‖+lim⁡n→U⁡H(Txn,Tx),(3.41)lim⁡n→U⁡dist⁡(xn,Tx)≤lim⁡n→U⁡H(Txn,Tx),(3.42)lim⁡n→Udist⁡(x,Txn)≤lim⁡n→U∥x-xn∥.
By (3.39),
(3.43)lim⁡n→U⁡H(Txn,Tx)≤αlim⁡n→U⁡‖xn-x‖+βlim⁡n→U⁡‖x-xn‖+βlim⁡n→U⁡H(Txn,Tx)+γlim⁡n→U⁡H(Txn,Tx)+γlim⁡n→U⁡‖x-xn‖.
Thus
(3.44)(1-β-γ)lim⁡n→U⁡H(Txn,Tx)≤(α+β+γ)lim⁡n→U⁡‖x-xn‖,
and therefore,
(3.45)lim⁡n→U⁡H(Txn,Tx)≤lim⁡n→U⁡‖xn-x‖.Takahashi Generalized Nonexpansive Mappings
Definition 3.22.
LetE 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
(3.46)‖Tx-Ty‖2≤α‖x-y‖2+β(‖y-Tx‖2+‖x-Ty‖2)forx,y∈E.
The following are examples of Takahashi generalized nonexpansive mappings:(i)
nonexpansive mappingsT:∥Tx-Ty∥≤∥x-y∥;(ii)
nonspreading mappingsT [38]: 2∥Tx-Ty∥2≤∥y-Tx∥2+∥x-Ty∥2;(iii)
hybrid mappingsT [39]: 3∥Tx-Ty∥2≤∥y-Tx∥2+∥x-Ty∥2;(iv)
mappingsT [39]: 2∥Tx-Ty∥2≤∥x-y∥2+∥y-Tx∥2;(v)
mappingsT: 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.
LetE 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
(3.47)H2(Tx,Ty)≤α‖x-y‖2+β(dist⁡2(x,Ty)+dist⁡2(y,Tx)),
then T satisfies (2) of condition (**).
Proof.
Let{xn} be an afps for T in E and x∈E. By (3.41) and (3.42),
(3.48)lim⁡n→U⁡H2(Txn,Tx)≤αlim⁡n→U⁡‖xn-x‖2+βlim⁡n→U⁡dist⁡2(x,Txn)+βlim⁡n→U⁡dist⁡2(xn,Tx)≤αlim⁡n→U⁡‖xn-x‖2+βlim⁡n→U⁡‖xn-x‖2+βlim⁡n→U⁡H2(Txn,Tx).
Thus
(3.49)(1-β)lim⁡n→U⁡H2(Txn,Tx)≤(α+β)lim⁡n→U⁡‖x-xn‖2.
Therefore,
(3.50)lim⁡n→U⁡H(Txn,Tx)≤lim⁡n→U⁡‖xn-x‖.
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,x3]. Since 0 is a fixed point of T, the sequence {xn} given by xn≡0 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 xn≡0, if x∈(0,1/2], then
(3.51)lim⁡n→U⁡H(Txn,Tx)=x>x=lim⁡n→U⁡‖xn-x‖.
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 Theorem3.16 is still valid when “limn→u” 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 x∈A𝒰, then (3.33) becomes
(3.52)lim⁡n→U⁡‖xn-z‖≤lim⁡n→U⁡‖yn-zn‖=lim⁡n→U⁡dist(yn,Tx)≤lim⁡n→U⁡H(Txn,Tx)≤limsup⁡n→∞⁡H(Txn,Tx)≤limsup⁡n→∞⁡‖xn-x‖=r(E,{xn})=rU(E,{xn}).
Hence A𝒰∩Tx≠∅, and the rest of the proof follows.
It is observed that ifA𝒰=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)  IfT: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 Theorem3.16, 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.



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*Source: 826851-2011-07-21.xml*