The purpose of this paper is to present the fixed points and endpoints of set-valued contractions concerning with the stronger Meir-Keeler cone-type mappings in cone metric spcaes. Our results generalize the recent results of Kadelburg and Radenović, 2011; Wardowski, 2009.

1. Introduction and Preliminaries

Throughout this paper, by ℝ+, ℝ we denote the sets of all nonnegative real numbers and all real numbers, respectively, while ℕ is the set of all natural numbers. Let (X,d) be a metric space, D a subset of X, and f:D→X a map. We say f is contractive if there exists α∈[0,1) such that, for all x,y∈D,
(1.1)d(fx,fy)≤α⋅d(x,y).
The well-known Banach fixed-point theorem asserts that if D=X, f is contractive and (X,d) is complete, then f has a unique fixed-point in X. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, Kannan [2] and Chatterjea [3] introduced two conditions that can replace (1.1) in Banach's theorem as follows:

Kannan [2]: There exists α∈[0,1) such that, for all x,y∈X,
(1.2)d(fx,fy)≤α2[d(x,fx)+d(y,fy)].

Chatterjea [3]: There exists α∈[0,1) such that for all x,y∈X,
(1.3)d(fx,fy)≤α2[d(x,fy)+d(y,fx)].

After these three conditions, many papers have been written generalizing some of the conditions (1.1), (1.2), and (1.3). In 1969, Boyd and Wong [4] showed the following fixed-point theorem.

Theorem 1.1 (see [<xref ref-type="bibr" rid="B12">4</xref>]).

Let (X,d) be a complete metric space and f:X→X a map. Suppose there exists a function ϕ:[0,∞)→[0,∞) satisfying ϕ(0)=0, ϕ(t)<t for all t>0 and ϕ is right upper semicontinuous such that
(1.4)d(fx,fy)≤ϕ(d(x,y))∀x,y∈X.
Then f has a unique fixed-point in X.

Later, Meir-Keeler [5], using a result of Chu and Diaz [6], extended Boyd-Wong's result to mappings satisfying the following more general condition:
(1.5)∀η>0∃δ>0suchthatη≤d(x,y)<η+δ⟹d(fx,fy)<η,
and Meir-Keeler proved the following very interesting fixed-point theorem which is a generalization of the Banach contraction principle.

Let (X,d) be a complete metric space and let f be a Meir-Keeler contraction; that is, for every η>0, there exists δ>0 such that d(x,y)<η+δ implies d(fx,fy)<η for all x,y∈X. Then f has a unique fixed-point.

Subsequently, some authors worked on this notion of Meir-Keeler contraction (e.g., [7–10]). In this paper, we introduce a new notion of the stronger Meir-Keeler-type mapping ψ:[0,∞)→[0,b), b∈(0,1], as follows.

Definition 1.3.

A mapping ψ:[0,∞)→[0,b), b∈(0,1] is called a stronger Meir-Keeler-type mapping in a metric space X, if for each η>0, there exists δ>0 such that for x,y∈X with η≤d(x,y)<η+δ, there exists γη∈[0,b) such that ψ(d(x,y))<γη.

Example 1.4.

Let (X,d) be a metric space, and define ψ:[0,∞)→[0,b), b∈(0,1] by
(1.6)ψ(d(x,y))=bd(x,y)d(x,y)+1,forx,y∈X.
Then ψ is a stronger Meir-Keeler-type mapping.

The existence of fixed-points for various multi valued contractive mappings had been studied by many authors under different conditions. In 1969, Nadler [11] extended the famous Banach contraction principle from single-valued mapping to multi valued mapping and proved the following fixed-point theorem for multi valued contraction. Let N(X) denote the collection of all nonempty subsets of X, B(X) the collection of all nonempty bounded subsets of X, C(X) the collection of all nonempty closed subsets of X, CB(X) the collection of all nonempty closed and bounded subsets of X, and K(X) the collection of all nonempty sequentially compact subsets of X.

Theorem 1.5 (see [<xref ref-type="bibr" rid="B32">11</xref>]).

Let (X,d) be a complete metric space, and, T be a mapping from X into CB(X). Assume that there exists c∈[0,1) such that
(1.7)H(Tx,Ty)≤cd(x,y)∀x,y∈X.
Then T has a fixed-point in X.

In 1989, Mizoguchi-Takahashi [12] proved the following fixed-point theorem.

Theorem 1.6 (see [<xref ref-type="bibr" rid="B30">12</xref>]).

Let (X,d) be a complete metric space and T a map from X into CB(X). Assume that
(1.8)H(Tx,Ty)≤ξ(d(x,y))⋅d(x,y)
for all x,y∈X, where ξ:[0,∞)→[0,1) satisfies limsup
s→t+ξ(s)<1 for all t∈[0,∞). Then T has a fixed-point in X.

Remark 1.7.

It is clear that if the function ξ:[0,∞)→[0,b), b∈(0,1] satisfies
(1.9)limsups→t+ξ(s)<b,∀t∈[0,∞),
then ξ is also a stronger Meir-Keeler-type function.

In 2003, Rus [13] introduced the notion of endpoints (strict fixed-points) and proved some results on strict fixed-point theory on multi valued operator. An element x∈X is said to be a fixed-point of T:X→N(X), if x∈Tx. If {x}=Tx, then x is called an endpoint of T. We will denote the sets of all fixed-points and endpoints of T by Fix(T) and End(T), respectively. The following theorems were the main results of Rus [13].

Theorem 1.8 (see [<xref ref-type="bibr" rid="B40">13</xref>]).

Let (X,d) be a complete metric space and T:X→B(X) a multi valued operator. Suppose that

x∈Tx for all x∈X

there exist a comparison function φ:ℝ+→ℝ+ (see [14]) and a Picard sequence xn+1∈Txn, n∈ℕ such that
(1.10)δ(Txn+1)≤φ(δ(Txn)),n∈N,
where δ(A)=sup{d(x,y):x,y∈A}.

Then xn→x* as n→∞ and x* is a unique strict fixed-point of T.

Recently, Wardowski [15] proved the following theorems concerning fixed-points and endpoints of set-valued contractions in complete cone metric spaces.

Definition 1.9.

A function I:X→ℝ is called lower semicontinuous, if, for any sequence {xn}⊂X and x∈X,
(1.11)xn⟶ximpliesf(x)≤liminfn→∞f(xn).

If (X,d) is a cone metric space and T:X→C(X), then, for x∈X, we denote
(1.12)D(x,Tx)={d(x,z):z∈Tx},S(x,Tx)={y∈D(x,Tx):‖y‖=inf{‖z‖:z∈D(x,Tx)}}.

Theorem 1.10 (see [<xref ref-type="bibr" rid="B44">15</xref>]).

Let (X,d) be a complete cone metric space, let P be a normal cone with normal constant K, and let T:X→C(X). Assume that a function I:X→ℝ defined by I(x)=infy∈Tx∥d(x,y)∥, x∈X is lower semicontinuous. If there exist λ∈[0,1), b∈(λ,1] such that
(1.13)∀x∈X∃y∈Tx∃v∈D(y,Ty)∀u∈D(x,Tx),[bd(x,y)≼u]∧[v≼λd(x,y)],
then Fix(T)≠ϕ.

Theorem 1.11 (see [<xref ref-type="bibr" rid="B44">15</xref>]).

Let (X,d) be a complete cone metric space, let P be a normal cone with normal constant K, and let T:X→K(X). Assume that a function I:X→ℝ defined by I(x)=infy∈Tx∥d(x,y)∥, x∈X is lower semicontinuous. The following hold

If there exist λ∈[0,1), b∈(λ,1] such that
(1.14)∀x∈X∃y∈Tx∃v∈S(y,Ty)∀u∈S(x,Tx),[bd(x,y)≼u]∧[v≼λd(x,y)],
then Fix(T)≠ϕ.

If there exist λ∈[0,1), b∈(λ,1] such that
(1.15)∀x∈X∀y∈Tx∃v∈S(y,Ty)∀u∈S(x,Tx),[bd(x,y)≼u]∧[v≼λd(x,y)],
then Fix(T)=End(T)≠ϕ.

In 1997, Zabrejko [16] introduced the K-metric and K-normed linear spaces and showed the existence and uniqueness of fixed-points for operators which act in K-metric or K-normed linear spaces. Huang and Zhang [17] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space, and they showed some fixed-point theorems of contractive type mappings on cone metric spaces. The category of cone metric spaces is larger than metric spaces. Subsequently, many authors like Abbas and Jungck [18] have generalized the results of Huang and Zhang [17] and studied the existence of common fixed-points of a pair of self-mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Rezapour and Hamlbarani [19] studied the existence of common fixed-points of a pair of self- and non-self-mappings satisfying a contractive type condition in the situation in which the cone does not need to be normal. Many authors studied this subject and many results on fixed-point theory are proved (see, e.g., [12, 19–39]).

Definition 1.12 (see [<xref ref-type="bibr" rid="B20">17</xref>]).

Let E be a real Banach space and P a nonempty subset of E. P≠{θ}, where θ denotes the zero element of E, is called a cone if and only if:

P is closed;

a,b∈ℝ+, a,b>0, x,y∈P⇒ax+by∈P;

x∈P and -x∈P⇒x=θ.

For given a cone P⊂E, we define a partial ordering with respect to P by x≼y or x≽y if and only if y-x∈P for all x,y∈E. The real Banach space E equipped with the partial order induced by P is denoted by (E,≼). We shall write x≺y to indicate that x≼y but x≠y, while x≪y or y≫x will stand for y-x∈int(P), where int(P) denotes the interior of P.

Proposition 1.13 (see [<xref ref-type="bibr" rid="B37">40</xref>]).

Suppose P is a cone in a real Banach space E. Then one has the following.

If e≼f and f≪g, then e≪g.

If e≪f and f≼g, then e≪g.

If e≪f and f≪g, then e≪g.

If a∈P and a≪e for each e∈int(P), then a=θ.

Proposition 1.14 (see [<xref ref-type="bibr" rid="B23">41</xref>]).

Suppose e∈int(P), θ≼an and an→θ. Then there exists n0∈ℕ such that an≪e for all n≥n0.

The cone P is called normal if there exists a real number K>0 such that, for all x,y∈E,
(1.16)θ≼x≼yimplies∥x∥≤K∥y∥.

The least positive number K, satisfying the above is called the normal constant of P.

The cone P is called regular if every increasing sequence which is bounded from above is convergent, that is, if {xn} is a sequence such that
(1.17)x1≼x2≼⋯≼xn≼⋯≼y,
for some y∈E, then there is x∈E such that ∥xn-x∥→0 as n→∞. Equivalently, the cone P is regular if and only if every decreasing sequence which is bounded from below is convergent. It is well known that a regular cone is a normal cone.

Definition 1.15 (see [<xref ref-type="bibr" rid="B20">17</xref>]).

Let X be a nonempty set, E a real Banach space, and P a cone in E. Suppose that the mapping d:X×X→(E,≼) satisfies

θ≼d(x,y), for all x,y∈X;

d(x,y)=θ if and only if x=y;

d(x,y)=d(y,x), for all x,y∈X;

d(x,y)+d(y,z)≽d(x,z), for all x,y,z∈X.

Then d is called a cone metric on X, and (X,d) is called a cone metric space.
Definition 1.16 (see [<xref ref-type="bibr" rid="B20">17</xref>]).

Let (X,d) be a cone metric space, and let {xn} be a sequence in X and x∈X. If for every c∈E with θ≪c, there is n0∈ℕ such that
(1.18)d(xn,x)≪c,∀n>n0,
then {xn} is said to be convergent and {xn} converges to x.

Definition 1.17 (see [<xref ref-type="bibr" rid="B20">17</xref>]).

Let (X,d) be a cone metric space, and let {xn} be a sequence in X. We say that {xn} is a Cauchy sequence if, for any c∈E with θ≪c, there is n0∈ℕ such that
(1.19)d(xn,xm)≪c,∀n,m>n0,

Definition 1.18 (see [<xref ref-type="bibr" rid="B20">17</xref>]).

Let (X,d) be a cone metric space. If every Cauchy sequence is convergent in X, then X is called a complete cone metric space.

Remark 1.19 (see [<xref ref-type="bibr" rid="B20">17</xref>]).

If P is a normal cone, then {xn} converges to x if and only if d(xn,x)→θ as n→∞. Further, in the case {xn} is a Cauchy sequence if and only if d(xn,xm)→θ as m,n→∞.

The purpose of this paper is to present the fixed-points and endpoints of set-valued contractions concerning with the stronger Meir-Keeler cone-type mappings in cone metric spcaes. Our results generalize the recent results of Kadelburg and Radenović [42–44] and Wardowski [15].

2. Main Results

In this section, we first introduce the following notion of stronger Meir-Keeler cone-type mapping.

Definition 2.1.

Let (X,d) be a cone metric space with cone P, and let
(2.1)ψ:int(P)∪{θ}⟶[0,b),where b∈(0,1].
Then the function ψ is called a stronger Meir-Keeler cone-type mapping, if, for each η∈int(P), there exists δ≫θ such that, for x,y∈X with η≼d(x,y)≪δ+η, there exists γη∈[0,b) such that ψ(d(x,y))<γη.

We now are in a position to present the following fixed-point theorem.

Theorem 2.2.

Let (X,d) be a complete cone metric space, and let P be a normal cone in E and T:X→C(X). Assume that a function I:X→ℝ defined by I(x)=infy∈Tx∥d(x,y)∥, x∈X is lower semicontinuous. The following holds.

There exists a stronger Meir-Keeler cone-type mapping
(2.2)ψ:int(P)∪{θ}⟶[0,b),whereb∈(0,1]
such that
(2.3)∀x∈X∃y∈Tx∃v∈D(y,Ty)∀u∈D(x,Tx),[bd(x,y)≼u]∧[v≼ψ(d(x,y))d(x,y)].

Then Fix(T)≠ϕ.
Proof.

Given x1∈X, we define the sequence {xn} recursively as follows. Take any u1∈𝒟(x1,Tx1). By (*), there exist x2∈Tx1 and u2∈𝒟(x2,Tx2) such that
(2.4)bd(x1,x2)≼u1,u2≼ψ(d(x1,x2))d(x1,x2),ψ(d(x1,x2))<b.
Thus,
(2.5)u1-u2≽[b-ψ(d(x1,x2))]d(x1,x2),u2≼ψ(d(x1,x2))bu1.
If d(x1,x2)=θ, then x1=x2∈Tx1, and we are done. Assume that d(x1,x2)≫θ. Put d(x1,x2)=η1. By the definition of the stronger Meir-Keeler cone-type mapping ψ, corresponding to η1 use, there exists δ1≫θ and γη1∈[0,b) with η1≼d(x1,x2)≪δ1+η1 such that ψ(d(x1,x2))<γη1. Taking into account (2.5), we have that
(2.6)u1-u2≽[b-γη1]d(x1,x2),u2≼γη1bu1.

By the same process, for x2∈X, there exist x3∈Tx2 and u3∈𝒟(x3,Tx3) such that
(2.7)bd(x2,x3)≼u2,u3≼ψ(d(x2,x3))d(x2,x3),ψ(d(x2,x3))<b.
Thus,
(2.8)u2-u3≽[b-ψ(d(x2,x3))]d(x2,x3),(2.9)u3≼ψ(d(x2,x3))bu2≼ψ(d(x1,x2))⋅ψ(d(x2,x3))b2u1.
Now, put d(x2,x3)=η2. By the definition of the stronger Meir-Keeler cone-type mapping ψ, corresponding to η2 use, there exists δ2≫θ and γη2∈[0,b) with η2≼d(x2,x3)≪δ2+η2 such that ψ(d(x2,x3))<γη2. And we put α1=γη1 and α2=max{γη1,γη2}. Taking into account (2.9), we have that
(2.10)u2-u3≽[b-γη2]d(x2,x3)≽[b-α2]d(x2,x3),(2.11)u3≼γη2bu2≼γη1⋅γη2b2u1≼α22b2u1.

We continue in this manner. Inductively, for xn∈X, there exist xn+1∈Txn and un+1∈𝒟(xn+1,Txn+1) such that
(2.12)bd(xn,xn+1)≼un,un+1≼ψ(d(xn,xn+1))d(xn,xn+1),ψ(d(xn,xn+1))<b.
Thus,
(2.13)un-un+1≽[b-ψ(d(xn,xn+1))]d(xn,xn+1),(2.14)un+1≼ψ(d(xn,xn+1))bun≼⋯≼ψ(d(x1,x2))⋅ψ(d(x2,x3))⋯ψ(d(xn,xn+1))bnu1.
Put d(xn,xn+1)=ηn. By the definition of the stronger Meir-Keeler cone-type mapping ψ, corresponding to ηn use, there exist δn≫θ and γηn∈[0,b) with ηn≼d(xn,xn+1)≪δn+ηn such that ψ(d(xn,xn+1))<γηn. And we put αn=γηn and αn=max{γη1,γη2,…,γηn}. Taking into account (2.14), we have that
(2.15)un-un+1≽[b-αn]d(xn,xn+1),(2.16)un+1≼αnnbnu1.

Let m,n∈ℕ such that m>n. Taking into account (2.15) and (2.16), we obtain that
(2.17)d(xn,xm)≼∑i=nm-1d(xi,xi+1)≼∑i=nm-11[b-αi](ui-ui+1)≼∑i=nm-11[b-αm](ui-ui+1)≼1[b-αm](un-um)≼1[b-αm]αnnbnu1.
Let e≫θ be given. Since αn<b and αm<b, we get
(2.18)limn→∞1[b-αm]αnnbnu1=θ.
Thus, there exists n0∈ℕ such that, for all m,n≥n0,
(2.19)1[b-αm]αnnbnu1≪e,
and we also conclude that d(xn,xm)≪e for all m,n≥n0. This implies that {xn} is a Cauchy sequence. Since X is complete, there exists ν∈X such that
(2.20)limn→∞xn=ν.

By the definition of 𝒟(x,Tx), since un∈𝒟(xn,Txn), there exists a sequence {zn} such that zn∈Txn, un=d(xn,zn) for all n∈ℕ. From the convergence of the sequence {un} and from the lower semicontinuity of the function I, we obtain that
(2.21)infy∈Tν‖d(ν,y)‖≤liminfn→∞‖d(xn,y)‖≤liminfn→∞‖d(xn,zn)‖=0.
Thus,
(2.22)infy∈Tx‖d(ν,y)‖=0.
We claim that ν∈Tν. To prove this, on the contrary, assume that ν∉Tν. Then by (2.22), there exists a sequence {yn}⊂Tν such that limn→∞∥d(ν,yn)∥=0, and hence limn→∞d(yn,ν)=θ. Thus, for any m,n≥0,
(2.23)d(ym,yn)≼d(ym,ν)+d(ν,yn).
Let τ≫θ be given. Then there exists N∈ℕ such that for all m,n>N, d(ym,p)≪1/2τ and d(yn,p)≪1/2τ, and so
(2.24)d(ym,yn)≼d(ym,ν)+d(ν,yn)≪12τ+12τ=τ.
This implies that {yn} is a Cauchy sequence in X. Since X is complete, there exists μ∈X such that
(2.25)limn→∞yn=μ.
By the closedness of the Tν, we deduce that μ∈Tν. Then, for any n∈ℕ,
(2.26)d(μ,ν)≼d(ν,yn)+d(yn,μ).
Let ω≫θ be given. Since limn→∞d(yn,ν)=θ and limn→∞yn=μ, we can deduce that there exists M∈ℕ such that, for all n>M, d(yn,ν)≪1/2ω and d(yn,μ)≪1/2ω, and hence
(2.27)d(μ,ν)≪12ω+12ω=ω.
By Proposition 1.13, we obtain that d(μ,ν)=θ that is, μ=ν, which is a contradiction. Therefore, ν∈Tν.

Applying Remark 1.7, it is easy to establish the following corollary.

Corollary 2.3.

Let (X,d) be a complete cone metric space, P a normal cone in E, and T:X→C(X). Assume that a function I:X→ℝ defined by :I(x)=infy∈Tx∥d(x,y)∥, x∈X is lower semicontinuous. The following holds.

There exists a mapping
(2.28)ψ:int(P)∪{θ}⟶[0,b),whereb∈(0,1]
such that
(2.29)limsups→tψ(s)<b,∀t∈int(P)∪{θ},(2.30)∀x∈X∃y∈Tx∃v∈D(y,Ty)∀u∈D(x,Tx),[bd(x,y)≼u]∧[v≼ψ(d(x,y))d(x,y)].

Then Fix(T)≠ϕ.
Corollary 2.4.

Let (X,d) be a complete cone metric space, P a normal cone in E, and T:X→X. Assume that a function I:X→ℝ defined by I(x)=infy∈Tx∥d(x,y)∥, x∈X is lower semicontinuous. The following holds.

There exists a stronger Meir-Keeler cone-type mapping
(2.31)ψ:int(P)∪{θ}⟶[0,b),whereb∈(0,1],
such that
(2.32)d(Tx,T2x)≼ψ(d(x,Tx))d(x,Tx),∀x∈X.

Then Fix(T)≠ϕ.
Example 2.5.

Let E=ℝ2, P={(x,y)∈E,x,y≥0}, X={a,b,c}⊂E where a=(0,0), b=(0,1), c=(1,0), d:X×X→E defined by d(a,b)=(1,2/3), d(c,a)=(4/3,1), d(b,c)=(7/3,5/3), with d(x,y)=d(y,x) for all x,y∈X and d(x,y)=θ if and only if x=y. Let T:X→X be defined by Ta=b, Tb=b, and Tc=a. Then T2a=b, T2b=b, T2c=b. Take ψ(t)=5/6 for all t∈int(P)∪{θ}. Calculation shows that, for x,y∈X,
(2.33)d(Tx,T2x)≼56d(x,Tx)=ψ(d(x,Tx))d(x,Tx)
that is, according to Corollary 2.4 we have that Fix(T)≠ϕ.

Theorem 2.6.

Let (X,d) be a complete cone metric space, P a normal cone in E, and T:X→K(X). Assume that a function I:X→ℝ defined by I(x)=infy∈Tx∥d(x,y)∥, x∈X, is lower semicontinuous. The following hold:

There exists a stronger Meir-Keeler cone-type mapping
(2.34)ψ:int(P)∪{θ}⟶[0,b),whereb∈(0,1]
such that
(2.35)∀x∈X∃y∈Tx∃v∈S(y,Ty)∀u∈S(x,Tx),[bd(x,y)≼u]∧[v≼ψ(d(x,y))d(x,y)].

Then Fix(T)≠ϕ.

There exists a stronger Meir-Keeler cone-type mapping
(2.36)ψ:int(P)∪{θ}⟶[0,b),whereb∈(0,1]
such that
(2.37)∀x∈X∀y∈Tx∃v∈S(y,Ty)∀u∈S(x,Tx),[bd(x,y)≼u]∧[v≼ψ(d(x,y))d(x,y)].

Then Fix(T)=End(T)≠ϕ.

Proof.

(A) By the same proof of the part (i) of Theorem 3.2 in [15], we have that 𝒮(x,Tx)≠ϕ for all x∈X. The remainder proof is similar to Theorem 2.2, we can deduce that Fix(T)≠ϕ.

(B) By (A), we obtain that the existence of ν∈X such that, ν∈Tν. Taking any y∈Tν, we have that, for all u∈𝒮(ν,Tν), bd(ν,y)≼u. Since ν∈Tν, we get θ∈𝒮(ν,Tν), and hence bd(ν,y)≼θ, which gives ν=y. Thus Tν={ν}.

Example 2.7.

Let X=[0,1], E=ℝ2, P={(x,y)∈E,x,y≥0}. Let d(x,y)=(|x-y|,0) for all x,y∈X, and let T:X→K(X), Tx={(1/2)x}. Then 𝒮(x,Tx)={((1/2)x,0):x∈X}, I(x)=(1/2)x. Let ψ(t)=2/3 for all t∈int(P)∪{θ}. It is easy to check that all conditions of Theorem 2.6 are satisfied and that Fix(T)=End(T)={0}≠ϕ.

Corollary 2.8.

Let (X,d) be a complete cone metric space, P a normal cone in E, and T:X→K(X). Assume that a function I:X→ℝ defined by I(x)=infy∈Tx∥d(x,y)∥, x∈X is lower semicontinuous. The following hold:

There exists a mapping
(2.38)ψ:int(P)∪{θ}⟶[0,b),whereb∈(0,1],
such that
(2.39)limsups→tψ(s)<b,∀t∈int(P)∪{θ},∀x∈X∃y∈Tx∃v∈S(y,Ty)∀u∈S(x,Tx),[bd(x,y)≼u]∧[v≼ψ(d(x,y))d(x,y)].

Then Fix(T)≠ϕ.

There exists a mapping
(2.40)ψ:int(P)∪{θ}⟶[0,b),whereb∈(0,1],
such that
(2.41)limsups→tψ(s)<b,∀t∈int(P)∪{θ},∀x∈X∀y∈Tx∃v∈S(y,Ty)∀u∈S(x,Tx),[bd(x,y)≼u]∧[v≼ψ(d(x,y))d(x,y)].

Then Fix(T)=End(T)≠ϕ.
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