# The Existence of Cone Critical Point and Common Fixed Point with Applications

Authors: Wei-Shih Du
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

We first establish some new critical point theorems for nonlinear dynamical
systems in cone metric spaces or usual metric spaces, and then we present some applications
to generalizations of Dancš-Hegedüs-Medvegyev's principle and the existence theorem related with
Ekeland's variational principle, Caristi's common fixed point theorem for multivalued maps, Takahashi's
nonconvex minimization theorem, and common fuzzy fixed point theorem. We also obtain
some fixed point theorems for weakly contractive maps in the setting of cone metric spaces and focus
our research on the equivalence between scalar versions and vectorial versions of some results of fixed
point and others.


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## Body

## 1. Introduction

In 1983, Dancš et al. [1] proved the following interesting existence theorem of critical point (or stationary point) for a nonlinear dynamical system.Dancš-Hegedüs-Medvegyev's Principle [1]
Let(X,d) be a complete metric space. Let Γ:X→2X be a multivalued map with nonempty values. Suppose that the following conditions are satisfied:(i)
for eachx∈X, we have x∈Γ(x), and Γ(x) is closed;(ii)
x, y∈X with y∈Γ(x) implies Γ(y)⊆Γ(x);(iii)
for eachn∈ℕ and each xn+1∈Γ(xn), we have limn→∞d(xn,xn+1)=0. Then there exists v∈X such that Γ(v)={v}.Dancš-Hegedüs-Medvegyev's Principle has been popularly investigated and applied in various fields of applied mathematical analysis and nonlinear analysis, see, for example, [2, 3] and references therein. It is well known that the celebrated Ekeland's variational principle can be deduced by the detour of using Dancš-Hegedüs-Medvegyev's principle, and it is equivalent to the Caristi's fixed point theorem, to the Takahashi's nonconvex minimization theorem, to the drop theorem, and to the petal theorem. Many generalizations in various different directions of these results in metric (or quasi-metric) spaces and more general in topological vector spaces have been studied by several authors in the past; for detail, one can refer to [2–12].LetE be a topological vector space (t.v.s. for short) with its zero vector θE. A nonempty subset K of E is called a convex cone if K+K⊆K and λK⊆K for λ≥0. A convex cone K is said to be pointed if K∩(-K)={θE}. For a given proper, pointed, and convex cone K in E, we can define a partial ordering ≾K with respect to K by (1.1)x≾Ky⟺y-x∈K.x≺Ky will stand for x≾Ky and x≠y, while x≪Ky will stand for y-x∈intK, where intK denotes the interior of K.In the following, unless otherwise specified, we always assume thatY is a locally convex Hausdorff t.v.s. with its zero vector θ, K a proper, closed, convex, and pointed cone in Y with intK≠∅, e∈intK and ≾K a partial ordering with respect to K. Denote by ℝ and ℕ the set of real numbers and the set of positive integers, respectively.Fixed point theory inK-metric and K-normed spaces was studied and developed by Perov [13], Kvedaras et al. [14], Perov and Kibenko [15], Mukhamadiev and Stetsenko [16], Vandergraft [17], Zabrejko [18], and references therein. In 2007, Huang and Zhang [19] reintroduced such spaces under the name of cone metric spaces and investigated fixed point theorems in such spaces in the same work. Since then, the cone metric fixed point theory is prompted to study by many authors; for detail, see [20–29] and references therein.Very recently, in order to improve and extend the concept of cone metric space in the sense of Huang and Zhang, Du [23] first introduced the concepts of TVS-cone metric and TVS-cone metricspace as follows.Definition 1.1 (see [23]).
LetX be a nonempty set. A vector-valued function p:X×X→Y is said to be a TVS-cone metric, if the following conditions hold: (C1)
θ≾Kp(x,y) for all x, y∈X and p(x,y)=θ if and only if x=y;(C2)
p(x,y)=p(y,x) for all x, y∈X;(C3)
p(x,z)≾Kp(x,y)+p(y,z) for all x,y,z∈X. The pair (X,p) is then called a TVS-cone metric space.Definition 1.2 (see [23]).
Let(X,p) be a TVS-cone metric space, x∈X, and, {xn}n∈ℕ let be a sequence in X. (i)
{xn} is said to TVS-cone converge tox if, for every c∈Y with θ≪Kc, there exists a natural number ℕ0 such that p(xn,x)≪Kc for all n≥ℕ0. We denote this by cone-limn→∞xn=x or xn→conex as n→∞ and call x the TVS-cone limit of {xn}.(ii)
{xn} is said to be a TVS-cone Cauchy sequence if, for every c∈Y with θ≪Kc, there is a natural number ℕ0 such that p(xn,xm)≪Kc for all n,  m≥ℕ0.(iii)
(X,p) is said to be TVS-cone complete if every TVS-cone Cauchy sequence in  X is TVS-cone convergent.In [23], the author proved the following important results.Theorem 1.3 (see [23]).
Let(X,p) be a TVS-cone metric spaces. Then dp:X×X→[0,∞) defined by dp:=ξe∘p is a metric, where the nonlinear scalarization function ξe:Y→ℝ is defined by
(1.2)ξe(y)=inf⁡{r∈R:y∈re-K},∀y∈Y.Example 1.4.
LetX=[0,1], Y=ℝ2, K=ℝ+2={(x,y)∈ℝ2:x,y≥0}, e=(1,1), and θ=(0,0). Define p:X×X→Y by
(1.3)p(x,y)=(|x-y|,5|x-y|).
Then (X,p) is a TVS-cone complete metric space. It is easy to verify that
(1.4)dp(x,y)=ξe(p(x,y))=inf⁡{r∈R:p(x,y)∈re-K}=5|x-y|,
so dp is a metric on X, and (X,dp) is a complete metric space.Theorem 1.5 (see [23]).
Let(X,p) be a TVS-cone metric  space, let x∈X, and let {xn}n∈ℕ be a sequence in X. Then the following statements hold. (a)
If{xn}TVS-cone  converges to  (i.e.,xn→conexasn→∞), then dp(xn,x)→0 as n→∞(i.e.,xn→dpxasn→∞).(b)
If{xn} is a TVS-cone  Cauchy sequence in (X,p), then {xn} is a Cauchy sequence (in usual sense) in (X,dp).The paper is organized as follows. In Section2, we first establish some new critical point theorems for nonlinear dynamical systems in cone metric spaces or usual metric spaces, and then we present some applications to generalizations of Dancš-Hegedüs-Medvegyev's principle and the existence theorem related with Ekeland's variational principle, Caristi's common fixed point theorem for multivalued maps, Takahashi's nonconvex minimization theorem, and the common fuzzy fixed point theorem. Section 3 is dedicated to the study of fixed point theorems for weakly contractive maps in the setting of cone metric spaces. In Section 4, we focus our research on the equivalence between scalar versions and vectorial vesions of some results of fixed point and others.


## 2. Critical Point Theorems in Cone Metric Spaces

LetX be a nonempty set. A fuzzy set in X is a function of X into [0,1]. Let ℱ(X) be the family of all fuzzy sets in X. A fuzzy map on X is a map from X into ℱ(X). This enables us to regard each fuzzy map as a two-variable function of X×X into [0,1]. Let F be a fuzzy map on X. An element x of X is said to be a fuzzy fixed point of F if F(x,x)=1 (see, e.g., [4, 5, 30–32]). Let Γ:X→2X be a multivalued map. A point x∈X is called to be a critical point (or stationary point) [1–3, 7, 32] of Γ if Γ(v)={v}.Recall that the nonlinear scalarization functionξe:Y→ℝ is defined by (2.1)ξe(y)=inf⁡{r∈R:y∈re-K},∀y∈Y.Lemma 2.1 (see [6, 23, 29, 33]).
For eachr∈ℝ and y∈Y, the following statements are satisfied: (i)
ξe(y)≤r⇔y∈re-K,(ii)
ξe(y)>r⇔y∉re-K,(iii)
ξe(y)≥r⇔y∉re-intK,(iv)
ξe(y)<r⇔y∈re-intK,(v)
ξe(·) is positively homogeneous and continuous on Y,(vi)
ify1∈y2+K (i.e., y2≾Ky1), then ξe(y2)≤ξe(y1),(vii)
ξe(y1+y2)≤ξe(y1)+ξe(y2) for all y1,y2∈Y.Remark 2.2.
Notice that the reverse statement of (vi) in Lemma2.1 (i.e.,  ξe(y2)≤ξe(y1)⇒y1∈y2+K or y2≾Ky1) does not hold in general. For example, let Y=ℝ2, let K=ℝ+2={(x,y)∈ℝ2:x,y≥0}, and let e=(1,1). Then K is a proper, closed, convex, and pointed cone in Y with intK={(x,y)∈ℝ2:x,y>0}≠∅ and e∈intK. For r=1, it is easy to see that y1=(5,-6)∉re-intK, and y2=(0,0)∈re-intK. By applying (iii) and (iv) of Lemma 2.1, we have ξe(y2)<1≤ξe(y1) while y1∉y2+K.Definition 2.3.
LetA be a nonempty subset of a TVS-cone metric space (X,p). (i)
TheTVS-cone closure of A, denoted tvsc-cl(A), is defined by
(2.2)tvsc-cl(A)={x∈X:∃{xn}⊂Asuchthatxn⟶cone⁡xasn⟶∞}.
Obviously,A⊆tvsc-cl(A).(ii)
A is said to be TVS-cone closed if A=tvsc-cl(A).(iii)
A is said to be TVS-cone open if the complement X∖A of A is TVS-cone closed.
IfY=ℝ, K=[0,∞)⊂ℝ and e=1, then p≡d is a metric in usual sense, and the closure of A is denoted by cld(A).Theorem 2.4.
Let(X,p) be a TVS-cone metric  space and let
(2.3)Tp={U⊆X:UisTVS-coneopenin(X,p)}.
Then 𝒯p is a topology on (X,p) induced by p.Proof.
Clearly,∅ and X are TVS-cone closed in (X,p). Thus, X and ∅ are TVS-cone open in (X,p), and hence ∅, X∈𝒯p. Let U1, U2∈𝒯p. Then V1=X∖U1 and V2=X∖U2 are TVS-cone closed in (X,p). We claim that U1∩U2∈𝒯p. Let v∈tvsc-cl(V1∪V2). Then ∃{vn}⊂V1∪V2 such that vn→conev as n→∞. Without loss of generality, we may assume that there exists a subsequence {vnk} of {vn}∩V1. Since vnk→conev as k→∞, we have v∈tvsc-cl(V1)=V1⊆V1∪V2. So tvsc-cl(V1∪V2)⊆V1∪V2, and; hence, V1∪V2  is TVS-cone closed in (X,p). From
(2.4)U1∩U2=X∖(V1∪V2),
we see that U1∩U2 is TVS-cone open in (X,p) and U1∩U2∈𝒯p.
LetI be any index set, and let {Ui}i∈I⊂𝒯p. We show that ⋃i∈IUi∈𝒯p. For each i∈I, set Vi=X∖Ui. Thus, Vi is TVS-cone closed in X for all i∈I. Let w∈tvsc-cl(⋂i∈IVi). Then ∃{wn}⊂⋂i∈IVi such that wn→conew as n→∞. For each i∈I, since {wn}⊂𝒱i and wn→conew, w∈tvsc-cl(Vi)=Vi. Hence, w∈⋂i∈IVi. So tvsc-cl(⋂i∈IVi)⊆⋂i∈IVi, and then ⋂i∈IVi is TVS-cone closed in (X,p). Since
(2.5)⋃i∈IUi=X∖⋂i∈IVi,⋃i∈IUi is TVS-cone open in (X,p), and ⋃i∈IUi∈𝒯p.
Therefore, by above, we prove that𝒯p is a topology on (X,p).The following result is simple, but it is very useful in this paper.Lemma 2.5.
LetE be a t.v.s., K a convex cone with intK≠∅ in E, and let a, b, c∈E. Then the following statements hold. (i)
intK+K⊆intK.(ii)
Ifa≾Kb and b≾Kc, then a≾Kc.(iii)
Ifa≾Kb and b≪Kc, then a≪Kc.(iv)
Ifa≪Kb and b≾Kc, then a≪Kc.(v)
Ifa≪Kb and b≪Kc, then a≪Kc.(vi)
Ifa≾Kc and b≾Kc, then a+b≾K2c.(vii)
Ifa≾Kc and b≪Kc, then a+b≪K2c.(viii)
Ifa≪Kc and b≪Kc, then a+b≪K2c.Proof.
The conclusion (i) follows from the facts that the setintK+K is open in E, and K is a convex cone. By the transitivity of partial ordering ≾K, we have the conclusion (ii). To see (iii), since a≾Kb⇔b-a∈K and b≪kc⇔c-b∈intK, it follows from (i) that
(2.6)c-a=(c-b)+(b-a)∈int⁡K+K⊆int⁡K,
which means that a≪kc. The proofs of conclusions (iv)–(viii) are similar to (iii).Definition 2.6.
Let(X,p) be a TVS-cone metric space. A nonempty subset C of X is said to be TVS-cone compact if every sequence in C has a TVS-cone convergent subsequence whose TVS-cone limit is an element of C.
IfX is TVS-cone compact, then we say that (X,p) is a TVS-cone compact metric space.Theorem 2.7.
LetC be a nonempty subset of a TVS-cone  metric space (X,p). Then the following statements hold. (a)
IfC is a closed set in the metric space (X,dp), then C is TVS-cone  closed in (X,p) and tvsc-cl(C)=cldp(C), where dp:=ξe∘p.(b)
IfC is TVS-cone  compact, then it is TVS-cone closed.(c)
IfC is TVS-cone  closed and (X,p) is TVS-cone  complete, then (C,p) is also TVS -cone complete.(d)
IfC is TVS-cone  compact, then (C,p) is TVS-cone complete.(e)
IfC is TVS-cone  compact, then C is (sequentially) compact in the metric space (X,dp).Proof.
Applying Theorem1.3, dp is a metric on X. Let C be a closed set in the metric space (X,dp). By Theorem 1.5, we have
(2.7)tvsc-cl(C)={x∈X:∃{xn}⊂Csuchthatxn⟶cone⁡xasn⟶∞}⊆{x∈X:∃{xn}⊂Csuchthatxn⟶dpxasn⟶∞}=cldp(C)=C,
which implies that C is TVS-cone closed in (X,p) and tvsc-cl(C)=cldp(C). Hence, the conclusion (a) holds.
Next, assume thatC is TVS-cone compact in (X,p). Let x∈tvsc-cl(C). Then there exists {xn}⊂C such that xn→conex as n→∞. By the TVS-cone compactness of C, there exist {xnj}⊂{xn} and w∈C such that xnj→conew as j→∞. Applying Theorem 1.5, dp(xn,x)→0 as n→∞ and dp(xnj,w)→0 as j→∞. By the uniqueness of limit, x=w∈C, and; hence, tvsc-cl(C)⊆C. So C is TVS-cone closed in (X,p), and (b) is proved.
To see (c), let{xn} be a TVS-cone Cauchy sequence in (C,p)⊆(X,p). Since (X,p) is TVS-cone complete, there exists v∈X such that xn→conev as n→∞. Hence, v∈tvsc-cl(C)=C, which show that (C,p) is TVS-cone complete.
Let us verify (d). Givenc∈Y with θ≪Kc, and let {zn} be a TVS-cone Cauchy sequence in (C,p). Then there exists  ν1∈ℕ such that p(zn,zm)≪K(1/2)c for all m,  n≥ν1. Since C is TVS-cone compact, there exists a subsequence {znt} of {zn}, and ẑ∈C such that znt→coneẑ as t→∞. For c, there exists ν2∈ℕ such that p(znt,ẑ)≪K(1/2)c  for all t≥ν2. Let ν=max{ν1,ν2}. For any n≥ν, since
(2.8)p(zn,ẑ)≾Kp(zn,znν)+p(znν,ẑ),p(zn,znν)+p(znν,ẑ)≪Kc,
by (iii) of Lemma 2.5, we have p(zn,ẑ)≪Kc. So {xn} is TVS-cone convergent to x. Therefore, (C,p) is TVS-cone complete.
The conclusion (e) is obvious. The proof is completed.LetC be a subset of a TVS-cone metric space (X,p). We denote (2.9)diamdp(C):={0,ifC=∅,sup⁡{dp(x,y):x,y∈C},ifC≠∅.It is obvious thatA⊆B in (X,p) implies diamdp(A)≤diamdp(B).Now, we first introduce the concepts of fitting nest.Definition 2.8.
A sequence{An}n∈ℕ of subsets of a TVS-cone metric space (X,p) is said to be a fitting nestif it satisfies the following properties: (FN1)
An+1⊆An for each n∈ℕ,(FN2)
for anyc∈Y with θ≪Kc, there exists nc∈ℕ such that p(x,y)≪Kc for all x,y∈Anc.Remark 2.9.
(a) It is easy to observe that ifY=ℝ, K=[0,∞)⊂ℝ, and e=1, then p is a metric, and Assumption (FN2) is equivalent to limn→∞diam(An)=0 if Assumption (FN1) holds, where diam(An) is the diameter of An. Indeed, “limn→∞diam(An)=0⇒ (FN2)” is obvious. Conversely, if (FN2) holds, then, by (FN1), for any ε>0, there exists nε∈ℕ such that diam(An)≤ε for all n≥nε. This show that limn→∞diam(An)=0.
(b) Let(X,d) be a metric space. Then a sequence {An}n∈ℕ in (X,d) is a fitting nest  ⇔An+1⊆An for each n∈ℕ and limn→∞diam(An)=0.The following intersection theorem inTVS-cone metric spaces is one of the main results of this paper.Theorem 2.10.
Let{An} be a fitting nest in a TVS-cone metric  space (X,p). Then the following statements hold.     (a)
limn→∞diamdp(An)=0.(b)
IfX is TVS-cone  complete and An is TVS-cone  closed in (X,p) for all n∈ℕ, then ⋂n=1∞An contains precisely one point.Proof.
(a) Letε>0 be given. Then θ≪Kεe. By (FN2) and (iv) of Lemma 2.1, there exists n0∈ℕ such       that
(2.10)p(x,y)≪Kεe⟺dp(x,y)=ξe∘p(x,y)<ε,
for all x,y∈An0, which implies diamdp(An0)≤ε. By (FN1), we obtain
(2.11)diamdp(An)≤diamdp(An0)≤ε,∀n≥n0.
Hence, limn→∞diamdp(An)=0.
(b) Givenc∈Y with θ≪Kc. By (FN2), there exists nc∈ℕ such that p(a,b)≪Kc for all a,b∈Anc. For each n∈ℕ, choose xn∈An. Then, for m,n∈ℕ with m≥n≥nc; since xm∈Am⊆An⊆Anc from (FN1), we have
(2.12)p(xm,xn)≪Kc.
Hence, {xn} is a TVS-cone Cauchy sequence in (X,p). By the TVS-cone completeness of (X,p), there exists w∈X, such that {xn}TVS-cone converges to w. For any n∈ℕ, from the TVS-coneclosedness of An and xm→conew as m→∞, we have
(2.13)w∈tvsc-cl(An)=An.
So w∈⋂n=1∞An, and hence ⋂n=1∞An≠∅. Finally, we claim ⋂n=1∞An={w}. For each z∈⋂n=1∞An, applying (a), we have
(2.14)dp(z,w)≤diamdp(An)⟶0asn⟶∞.
Hence, dp(z,w)=0 or, equivalency, z=w, which gives the required result (b).Theorem 2.11.
Let(X,p) be a TVS-cone  complete metric space. Then there exists a nonempty proper subset ℳ of X, such that ℳ contains infinite points of X, and (ℳ,dp) is a complete metric space, where dp:=ξe∘p.Proof.
Let{An} be a fitting nest in (X,p). Following the same argument as in the proof of (b), we can obtain a sequence {xn} satisfying (1)
xn∈An,(2)
{xn} is a TVS-cone Cauchy sequence in (X,p),(3)
{xn}TVS-coneconverges to some point w in X.
Applying Theorem1.5 with (2) and (3), we know that {xn} is a Cauchy sequence in (X,dp), and dp(xn,w)→0 or xn→dpw as n→∞. Let ℳ={xn}n∈ℕ∪{w}. Therefore, (ℳ,dp) is a complete metric space.The celebrated Cantor intersection theorem [2] in metric spaces can be proved by Theorem 2.10 and Remark 2.9.Corollary 2.12 (Cantor).
Let(X,d) be a metric space, and let {An}n∈ℕ be a sequence of closed subsets of X satisfying An+1⊆An for each n∈ℕ and limn→∞diam(An)=0. Then ⋂n=1∞An contains precisely one point.The following existence theorems relate with critical point and common fuzzy fixed point for a nonlinear dynamical system inTVS-cone complete metric spaces or complete metric spaces.Theorem 2.13.
Let(X,p) be a TVS-cone  complete metric space, let g:X→X a map, and let Γ:X→2X be a multivalued map with nonempty values. Let I be any index set. For each i∈I, let Fi be a fuzzy map on X. Suppose that the following conditions are satisfied. (H1)
For eachx∈X, Γ(x) is TVS-cone  closed in (X,p).(H2)
x,y∈X with y∈Γ(x) implies g(y)∈Γ(x) and Γ(y)⊆Γ(x).(H3)
For any{xn}⊂X with g(xn+1)∈Γ(xn) for each n∈ℕ, it satisfies the following:For anyc∈Y with θ≪Kc, there exists nc∈ℕ such that p(s,t)≪Kc for all s,t∈Γ(xnc).(H4)
For any(i,x)∈I×X, there exists y(i,x)∈Γ(x) such that Fi(x,y(i,x))=1. Then there exists v∈X such that(a)
Fi(v,v)=1 for all i∈I.(b)
Γ(v)={g(v)}={v}.Proof.
Letu∈X be given. Define a sequence {xn} by x1=u and xn+1∈Γ(xn) for n∈ℕ. Hence, g(xn+1)∈Γ(xn) for all n∈ℕ from (H2). For each n∈ℕ, let An=Γ(xn). By (H2) and (H3), we know that {An} is a fitting nest in (X,p). By (H1), An is TVS-cone closed in (X,p) for all n∈ℕ. Applying Theorem 2.10, there exists v∈X such that ⋂n=1∞Γ(xn)=⋂n=1∞An={v}. Since v∈Γ(xn) for all n∈ℕ, by (H2), we obtain
(2.15)∅≠Γ(v)⊆⋂n=1∞Γ(xn)={v},
which implies Γ(v)={v}. For each i∈I, by (H4), Fi(v,v)=1. By (H2) again, we have g(v)∈Γ(v)={v}. Therefore, Γ(v)={g(v)}={v}. The proof is completed.Remark 2.14.
LetY=ℝ, let K=[0,∞), and let e=1, then p is a metric, and Assumption (H3) in Theorem 2.13 is equivalent to
(H3)ℝ   for any {xn}⊂X with g(xn+1)∈Γ(xn) for each n∈ℕ, we have limn→∞diam(Γ(xn))=0.The following critical point theorems are immediate from Theorem2.13.Theorem 2.15.
Let(X,d) be a complete metric space, let g:X→X be a map and let Γ:X→2X be a multivalued map with nonempty values. Let I be any index set. For each i∈I, let Fi be a fuzzy map on X. Suppose that the following conditions are satisfied. (H1)ℝ
For eachx∈X, Γ(x) is closed.(H2)
x,y∈X with y∈Γ(x) implies g(y)∈Γ(x) and Γ(y)⊆Γ(x).(H3)ℝ
For any{xn}⊂X with g(xn+1)∈Γ(xn) for each n∈ℕ, we have limn→∞diam(Γ(xn))=0.(H4)
For any(i,x)∈I×X, there exists y(i,x)∈Γ(x) such that Fi(x,y(i,x))=1. Then there exists v∈X such that (a)
Fi(v,v)=1 for all i∈I.(b)
Γ(v)={g(v)}={v}.Corollary 2.16.
Let(X,p) be a TVS- cone complete metric space, and let Γ:X→2X be a multivalued map with nonempty values. Suppose that the following conditions are satisfied. (i)
For eachx∈X, Γ(x) is TVS-cone  closed in (X,p).(ii)
x,y∈X with y∈Γ(x) implies Γ(y)⊆Γ(x).(iii)
For any{xn}⊂X with xn+1∈Γ(xn), for each n∈ℕ, it satisfies the following:For anyc∈Y with θ≪Kc, there exists nc∈ℕ such that p(s,t)≪Kc for all s,t∈Γ(xnc).
Then there existsv∈X such that Γ(v)={v}.Proof.
LetF be a fuzzy map on X defined by F(x,y)=1 for all x, y∈X, and let g≡id be an identity map. Therefore, the conclusion follows from Theorem 2.13.Corollary 2.17.
Let(X,d) be a complete metric space, and let Γ:X→2X be a multivalued map with nonempty values. Suppose that the following conditions are satisfied. (i)
For eachx∈X, Γ(x) is closed.(ii)
x,y∈X with y∈Γ(x) implies Γ(y)⊆Γ(x).(iii)
For any{xn}⊂X with xn+1∈Γ(xn), for each n∈ℕ, we have limn→∞diam(Γ(xn))=0.
Then there existsv∈X such that Γ(v)={v}.Theorem 2.18.
Let(X,d) be a complete metric space, and let Γ:X→2X be a multivalued map with nonempty values such that x,y∈X with y∈Γ(x) implies Γ(y)⊆Γ(x).  Then the following statements holds. (1)
If a sequence{xn} in X satisfies xn+1∈Γ(xn) for each n∈ℕ and limn→∞diam(Γ(xn))=0, then limn→∞d(xn,xn+1)=0.(2)
If(X,d) has the following property (𝒫):
((P))forany{xn}⊂Xwithxn+1∈Γ(xn)foreachn∈N,wehavelim⁡n→∞⁡d(xn,xn+1)=0,
then there exists a sequence {zn} in X satisfying  zn+1∈Γ(zn) for each n∈ℕ and limn→∞diam(Γ(zn))=0.Proof.
(1) Let{xn} in X with xn+1∈Γ(xn) for each n∈ℕ, and let limn→∞diam(Γ(xn))=0. For any n∈ℕ, by our hypothesis, Γ(xn+1)⊆Γ(xn), and; hence,
(2.16)xm+1∈Γ(xm)⊆Γ(xn),∀m∈Nwithm≥n.
So d(xn+1,xn+2)≤diam(Γ(xn)) for each n∈ℕ. Since limn→∞diam(Γ(xn))=0, we have
(2.17)lim⁡n→∞⁡d(xn+1,xn+2)=lim⁡n→∞⁡diam⁡(Γ(xn))=0.
That is, limn→∞d(xn,xn+1)=0.
(2) Suppose that(X,d) has the property (𝒫). Define a function τ:X→ℝ by
(2.18)τ(x)=sup⁡y∈Γ(x)d(x,y).
We first note that τ(w)<∞ for some w∈X. Indeed, on the contrary, assume that τ(x)=∞ for all x∈X. Let û∈X be given. Set u1=û. Since τ(u1)=∞, τ(u1)=supy∈Γ(u1)d(u1,y)>1, and then there exists u2∈Γ(u1) such that d(u1,u2)>1. Since τ(u2)>2, there exists u3∈Γ(u2) such that d(u2,u3)>2. Continuing in the process, we can obtain a sequence {un}⊂X, such that, for each n∈ℕ,
(𝒦1)
un+1∈Γ(un),(𝒦2)
d(un,un+1)>n. By (𝒫), we have limn→∞d(un,un+1)=0. On the other hand, by (𝒦2), we also obtain limn→∞d(un,un+1)=∞, a contradiction. Therefore, there exists w∈X such that τ(w)<∞. Let z1=w. Since
(2.19)∞>τ(z1)=sup⁡y∈Γ(z1)d(z1,y)≥12diam⁡(Γ(z1)),
we have 0≤diam(Γ(z1))<∞, and there exists z2∈Γ(z1) such that
(2.20)d(z1,z2)>12diam⁡(Γ(z1))-12.
Since Γ(z2)⊆Γ(z1), we have τ(z2)≤τ(z1)<∞ and 0≤diam(Γ(z2))<∞. So there exists z3∈Γ(z2) such that
(2.21)d(z2,z3)>12diam⁡(Γ(z2))-122.
Continuing in this way, we can construct a sequence {zn} in X satisfying, for each n∈ℕ, (𝒦3)
zn+1∈Γ(zn),(𝒦4)
d(zn,zn+1)>(1/2)diam(Γ(zn))-(1/2n). From (𝒫), we have limn→∞d(zn,zn+1)=0. By (𝒦4), it follows that limn→∞diam(Γ(zn))=0.Remark 2.19.
In general, under the same assumptions of Theorem2.18, limn→∞d(xn,xn+1)=0 does not always imply limn→∞diam(Γ(xn))=0. For example, let X=[0,5] with the metric d(x,y)=|x-y|. Then (X,d) is a complete metric space. For each x∈X, define Γ:X→2X by Γ(x)=[x,5]. Thus, x, y∈X with y∈Γ(x) implies Γ(y)⊆Γ(x). Choose {xn} in X with xn=2; for all n∈ℕ, we have limn→∞diam(Γ(xn))=3≠0 while limn→∞d(xn,xn+1)=0.The following result is also a generalized Dancš-Hegedüs-Medvegyev's principle with common fuzzy fixed point. Notice that we do not assumex∈Γ(x) for all x∈X.Theorem 2.20.
Let(X,d) be a complete metric space. Let Γ:X→2X be a multivalued map with nonempty values. Let I be any index set. For each i∈I, let Fi be a fuzzy map on X. Suppose that the following conditions are satisfied. (D1)
For eachx∈X, Γ(x) is closed.(D2)
x, y∈X with y∈Γ(x) implies Γ(y)⊆Γ(x).(D3)
(Property  (𝒫)) for any {xn}⊂X with xn+1∈Γ(xn) for each n∈ℕ, we have limn→∞d(xn,xn+1)=0.(D4)
For any(i,x)∈I×X, there exists y(i,x)∈Γ(x) such that Fi(x,y(i,x))=1. Then there exists v∈X such that (a)
Fi(v,v)=1 for all i∈I,(b)
Γ(v)={v}.Proof.
By conclusion (2) of Theorem2.18, there exists a sequence {zn} in X satisfying zn+1∈Γ(zn) for each n∈ℕ and limn→∞diam(Γ(zn))=0. For each n∈ℕ, let An=Γ(zn). By (D2) and limn→∞diam(Γ(zn))=0, we see that {An} is a fitting nest in (X,d). Applying Theorem 2.10, there exists v∈X such that ⋂n=1∞Γ(zn)=⋂n=1∞An={v}. Since v∈Γ(zn) for all n∈ℕ, by (D2) again, we obtain
(2.22)∅≠Γ(v)⊆⋂n=1∞Γ(zn)={v},
which implies Γ(v)={v}. For each i∈I, by (D4), Fi(v,v)=1. The proof is completed.The following existence theorem relate with common fixed point for multivalued maps and critical point for a nonlinear dynamical system inTVS-cone complete metric spaces.Theorem 2.21.
Let(X,p), g, and Γ be the same as in Theorem 2.13. Assume that conditions (H1), (H2), and (H3) in Theorem 2.13 hold. Let I be any index set. For each i∈I, let Ti:X→2X be a multivalued map with nonempty values. Suppose that, for each (i,x)∈I×X, there exists y(i,x)∈Ti(x)∩Γ(x). Then there exists v∈X such that (a)
v is a common fixed point for the family {Ti}i∈I (i.e., v∈Ti(v) for all i∈I),(b)
Γ(v)={g(v)}={v}.Proof.
For eachi∈I, define a fuzzy map Fi on X by
(2.23)Fi(x,y)=χTi(x)(y),
where χA is the characteristic function for an arbitrary set A⊂X. Note that y∈Ti(x)⇔Fi(x,y)=1 for i∈I. Then, for any (i,x)∈I×X, there exists yx∈Γ(x) such that Fi(x,yx)=1. So (H4) in Theorem 2.13 holds. Therefore, the result follows from Theorem 2.13.Remark 2.22.
(a) Theorems2.20 and 2.21 all generalize and improve the primitive Dancš-Hegedüs-Medvegyev's principle.
(b) Corollary2.17 is a special case of Theorem 2.20 or Theorem 2.21.The following result is a special case of [32, Theorem  4.1], but it can also be proved by applying Theorem 2.20 (please follow a similar argument as in the proof of [32, Theorem  4.1]).Theorem 2.23.
Let(X,d) be a complete metric space, let f:X→(-∞,∞] be a proper l.s.c. and bounded from below function, and let φ:(-∞,∞]→(0,∞) be a nondecreasing function. Let I be any index set. For each i∈I, let Fi be a fuzzy map on X. Suppose that, for each (i,x)∈I×X, there exists y(i,x)∈X such that Fi(x,y(i,x))=1 and d(x,y(i,x))≤φ(f(x))(f(x)-f(y(i,x))). Then, for each u∈X with f(u)<∞, there exists v∈X such that (a)
d(u,v)≤φ(f(u))(f(u)-f(v)),(b)
d(v,x)>φ(f(v))(f(v)-f(x))  for all x∈X with x≠v,(c)
Fi(v,v)=1 for all i∈I.Moreover, if further assume that
(H) for anyx∈X with f(x)>infz∈Xf(z), there exists y∈X with y≠x such thatd(x,y)≤φ(f(x))(f(x)-f(y)),then f(v)=infz∈Xf(z).By using Theorem2.23, one can immediately obtain the following existence theorem related to generalized Ekeland's variational principle, generalized Takahashi's nonconvex minimization theorem, and generalized Caristi's common fixed point theorem for multivalued maps.Theorem 2.24.
Let(X,d), f, and φ be be the same as in Theorem 2.23. Let I be any index set. For each i∈I, let Ti:X→2X be a multivalued map with nonempty values such that, for each (i,x)∈I×X, there exists y(i,x)∈Ti(x) such that d(x,y(i,x))≤φ(f(x))(f(x)-f(y(i,x))). Then, for each u∈X with f(u)<∞, there exists v∈X such that (a)
d(u,v)≤φ(f(u))(f(u)-f(v)).(b)
d(v,x)>φ(f(v))(f(v)-f(x))  for all x∈X with x≠v.(c)
v is a common fixed point for the family {Ti}i∈I.Moreover, if further assume that
(H) for anyx∈X with f(x)>infz∈Xf(z), there exists y∈X with y≠x such that d(x,y)≤φ(f(x))(f(x)-f(y)), then  f(v)=infz∈Xf(z).


## 3. Fixed Point Theorems in Cone Metric Spaces

In this section, motivated by the recent results of Abbas and Rhoades [21], we will present some generalizations of those in TVS-cone complete metric spaces.Theorem 3.1.
Let(X,p) be a TVS-cone  complete metric space. Suppose that T,S:X→X are two self-maps of X satisfying
(3.1)p(Tx,Sy)≾Kαp(x,y)+β[p(x,Tx)+p(y,Sy)]+γ[p(x,Sy)+p(y,Tx)],
for all x, y∈X, where α, β, γ≥0 and α+2β+2γ<1. Then the following statements hold:(a)
There exists a nonempty proper subsetW of X, such that W contains infinite points of X and (W,dp) is a complete metric space, where dp:=ξe∘p.(b)
T and S have a unique common fixed point in X (in fact, the unique common fixed point of T and S belongs to W). Moreover, for each x0∈X, the mixed iterative sequence {xn}n∈ℕ∪{0}, defined by x2n+1=Tx2n and x2n+2=Sx2n+1 for n∈ℕ∪{0}, TVS-cone converges to the common fixed point.(c)
Any fixed point ofT is a fixed point of S, and conversely.Proof.
SinceY is a locally convex Hausdorff’s t.v.s. with its zero vector θ, let τ denote the topology of Y, and let 𝒰τ be the base at θ consisting of all absolutely convex neighborhood of θ. Let
(3.2)L={l:lbeaMinkowskifunctionalofUforU∈Uτ}.
Then ℒ is a family of seminorms on Y. For each ℓ∈ℒ, let
(3.3)V(l)={y∈Y:l(y)<1},
and let
(3.4)UL={U:U=r1V(l1)∩r2V(l2)∩⋯∩rnV(ln),rk>0,lk∈L,1≤k≤n,n∈N}.
Then 𝒰ℒ is a base at θ, and the topology Γℒ generated by 𝒰ℒ is the weakest topology for Y such that all seminorms in ℒ are continuous and τ=Γℒ. Moreover, given any neighborhood 𝒪θ of θ, there exists U∈𝒰ℒ such that θ∈U⊂𝒪θ (see, e.g., [34, Theorem  12.4 in II.12, Page 113] or the proofs of [28, Theorem  3.1] and [29, Theorem  2.1]).
Letx0∈X be given. First, from our hypothesis, we have
(3.5)0≤α+β+γ<1-β-γ.
If x0 is a fixed point of T, then, by using (3.1),
(3.6)p(x0,Sx0)=p(Tx0,Sx0)≾Kαp(x0,x0)+β[p(x0,Tx0)+p(x0,Sx0)]+γ[p(x0,Sx0)+p(x0,Tx0)],
implies that
(3.7)p(x0,Sx0)≾K(β+γ)p(x0,Sx0),
or
(3.8)(1-β-γ)p(x0,Sx0)∈-K.
Since 1-β-γ>0 and K is pointed, we have p(x0,Sx0)∈K∩(-K)={θ}. So p(x0,Sx0)=θ, and; hence, Sx0=x0=Tx0; that is, x0 is a common fixed point of T and S. Otherwise, if Tx0≠x0, we will define the mixed iterative sequence {xn} by x2n+1=Tx2n and x2n+2=Sx2n+1 for n∈ℕ∪{0}. Then p(x1,x0)≠θ. We claim that {xn} is a TVS-cone Cauchy’s sequence in (X,p). Let λ=(α+β+γ)/(1-β-γ).
By (3.5), we know λ∈[0,1). For each n∈ℕ∪{0}, we have
(3.9)p(x2n+1,x2n+2)=p(Tx2n,Sx2n+1)≾Kαp(x2n,x2n+1)+β[p(x2n,Tx2n)+p(x2n+1,Sx2n+1)]+γ[d(x2n,Sx2n+1)+d(x2n+1,Tx2n)]≾K(α+β+γ)p(x2n,x2n+1)+(β+γ)p(x2n+1,x2n+2),
which implies that
(3.10)p(x2n+1,x2n+2)≾Kλp(x2n,x2n+1).
Similarly, we also obtain
(3.11)p(x2n+2,x2n+3)≾Kλp(x2n+1,x2n+2).
Hence, for each n∈ℕ,
(3.12)p(xn+1,xn)≾Kλp(xn,xn-1)≾K⋯≾Kλnp(x1,x0).
Therefore, for m,n∈ℕ with m>n,
(3.13)p(xm,xn)≾K∑j=nm-1p(xj+1,xj)≾Kλn1-λp(x1,x0).
Given c∈Y with θ≪Kc (i.e., c∈intK=int(intK)), there exists a neighborhood Nθ of θ such that c+Nθ⊆intK. Therefore, there exists Uc∈𝒰ℒ with Uc⊆Nθ such that c+Uc⊆c+Nθ⊆intK, where
(3.14)Uc=r1V(l1)∩r2V(l2)∩⋯∩rsV(ls),
for some ri>0, ℓi∈ℒ and 1≤i≤s. Let δ=min{ri:1≤i≤s}>0, and let ρ=max{ℓi(p(x1,x0)):1≤i≤s}. If ρ=0, since each ℓi is a seminorm, we have ℓi(p(x1,x0))=0 and
(3.15)li(-λn1-λp(x1,x0))=λn1-λli(p(x1,x0))=0<ri,
for all 1≤i≤s and all n∈ℕ. If ρ>0, since λ∈[0,1), limn→∞(λn/(1-λ))=0, and hence there exists n0∈ℕ such that λn/(1-λ)<δ/ρ for all n≥n0. So, for each i∈{1,2,…,s} and any n≥n0, we obtain
(3.16)li(-λn1-λp(x1,x0))=λn1-λli(p(x1,x0))<δρli(p(x1,x0))≤δ≤ri.
Therefore, for any n≥n0, (-λn/(1-λ))p(x1,x0)∈riV(ℓi) for all 1≤i≤s, and hence (-λn/(1-λ))p(x1,x0)∈Uc. So we obtain
(3.17)c-λn1-λp(x1,x0)∈c+Uc⊆int⁡K,
or
(3.18)λn1-λp(x1,x0)≪Kc,
for all n≥n0. For m, n∈ℕ with m>n≥n0, by (3.13), (3.18), and Lemma 2.5, it follows that
(3.19)p(xm,xn)≪Kcform,n∈Nwithm>n≥n0.
Hence, {xn} is a TVS-cone Cauchy sequence in (X,p). By the TVS-cone completeness of (X,p), there exists v∈X, such that {xn}TVS-cone converges to v. On the other hand, applying Theorem 1.5, {xn} is a Cauchy sequence in (X,dp), and dp(xn,v)→0 or xn→dpv as n→∞. Let W={xn}n∈ℕ∪{v}. Then (W,dp) is a complete metric space, and the conclusion (a) holds.
To see (b), it suffices to show thatv is the unique common fixed point of T and S. By (v) and (vi) of Lemma 2.1, the assumption (3.1) implies
(3.20)dp(Tx,Sy)≤αdp(x,y)+β[dp(x,Tx)+dp(y,Sy)]+γ[dp(x,Sy)+dp(y,Tx)],∀x,y∈X.
For any n∈ℕ∪{0}, by (3.20),
(3.21)dp(v,Sv)≤dp(v,x2n+1)+dp(Tx2n,Sv)≤dp(v,x2n+1)+αdp(x2n,v)+β[dp(x2n,x2n+1)+dp(v,Sv)]+γ[dp(x2n,Sv)+dp(v,x2n+1)]≤dp(v,x2n+1)+αdp(x2n,v)+β[dp(x2n,x2n+1)+dp(v,Sv)]+γ[dp(x2n,v)+dp(v,Sv)+dp(v,x2n+1)],
which implies that
(3.22)dp(v,Sv)≤11-β-γ{dp(v,x2n+1)+αdp(x2n,v)+β[dp(x2n,x2n+1)]+γ[dp(x2n,v)+dp(v,x2n+1)]}.
Since dp is a metric and xn→dpv as n→∞, the right-hand side of (3.22) approaches zero as n→∞. Hence, dp(v,Sv)=0 or Sv=v. Also, since
(3.23)dp(Tv,v)=dp(Tv,Sv)≤αdp(v,v)+β[dp(v,Tv)+dp(v,Sv)]+γ[dp(v,Sv)+dp(v,Tv)]=(β+γ)dp(Tv,v),
this implies that
(3.24)(1-β-γ)dp(Tv,v)≤0.
Since 1-β-γ>0, we have dp(Tv,v)=0 or Tv=v. Therefore, v is a common fixed point of T and S. Suppose that there exists w∈X such that Tw=Sw=w. Since
(3.25)dp(v,w)=dp(Tv,Sw)≤αdp(v,w)+β[dp(v,Tv)+dp(w,Sw)]+γ[dp(v,Sw)+dp(w,Tv)]=(α+2γ)dp(v,w),
and 1-α-2γ>2β≥0, it follows that dp(v,w)=0, and hence w=v. So the uniqueness of common fixed point in X of T, and S is proved. Following a similar argument as above, one can verify conclusion (c). The proof is completed.The following result is immediate from Theorem3.1.Corollary 3.2.
Let(X,d) be a complete metric space. Suppose that T,S:X→X are two self-maps of X satisfying
(3.26)d(Tx,Sy)≤αd(x,y)+β[d(x,Tx)+d(y,Sy)]+γ[d(x,Sy)+d(y,Tx)],
for all x, y∈X, where α, β, γ≥0 and α+2β+2γ<1. Then the following statements hold. (a)
T and S have a unique common fixed point in X. Moreover, for each x0∈X, the mixed iterative sequence {xn}n∈ℕ∪{0}, defined by x2n+1=Tx2n and x2n+2=Sx2n+1 for n∈ℕ∪{0}, converges to the common fixed point;(b)
any fixed point ofT is a fixed point of S, and conversely.Remark 3.3.
[21, Theorem  2.1] is a special case of Theorem 3.1.The following result improves and extends [23, Theorem  2.3], and it is immediate from Theorem 3.1.Theorem 3.4.
Let(X,p) be a TVS-cone  complete metric space, and let the map T:X→X be a cone-contraction; that is, T satisfies the contractive condition
(3.27)p(Tx,Ty)≾Kαp(x,y)
for all x,y∈X, where α∈[0,1) is a constant. Then the following statements hold:(a)
there exists a nonempty proper subsetM of X, such that M contains infinite points of X, and (M,dp) is a complete metric space, where dp:=ξe∘p;(b)
T has a unique fixed point in X (in fact, the unique fixed point of  T belongs to M). Moreover, for each x∈X, the iterative sequence {Tnx}n∈ℕTVS -cone converges to the fixed point.Proof.
SetS=T and β=γ=0 in Theorem 3.1.Remark 3.5.
(a) It is obvious that the classical Banach’s contraction principle is a special case of Theorem3.4;
(b) Theorem3.4 generalizes and improves [19, Theorem 1] and [24, Theorem  2.3].
(c) In fact, following a very similar argument as in the proof of Theorem3.1 under the assumptions of Theorem 3.4, we can obtain an important fact that there exists a nonempty subset M of X, such that (M,dp) is a complete metric space and TM⊆M. Since
(3.28)p(Tx,Ty)≾Kαp(x,y)⟹dp(Tx,Ty)≤αdp(x,y),∀x,y∈X,
one can apply the Banach’s contraction principle to prove that T has a unique fixed point in X. So the classical Banach contraction principle and [23, Theorem  2.3] are equivalent if we are only asked to find the fixed point of T.
(d) Another proof of Theorem2.11 is given hereunder. Let φ:X→X be any cone-contraction. Take v0∈X and let vn=φvn-1=φnv0 for n∈ℕ. Following a similar argument as in the proof of Theorem 3.1, there exists v̂∈X, such that vn→dpv̂ as n→∞. Let ℳ={vn}n∈ℕ∪{v̂}. Then (ℳ,dp) is a complete metric space.Theorem 3.6.
Let(X,p) be a TVS-cone  complete metric space, and let the map T:X→X satisfy
(3.29)p(Tx,Ty)≾Kβ[p(x,Tx)+p(y,Ty)]
for allx,y∈X, where β∈[0,1/2) is a constant. Then the following statements hold: (a)
there exists a nonempty proper subsetM of X, such that M contains infinite points of X, and (M,dp) is a complete metric space, where dp:=ξe∘p;(b)
T has a unique fixed point in X (in fact, the unique fixed point of T belongs to M). Moreover, for each x∈X, the iterative sequence {Tnx}n∈ℕTVS-cone converges to the fixed point.Proof.
SetS=T and α=γ=0 in Theorem 3.1.Remark 3.7.
Theorem3.6 generalizes the fixed point theorems of Kannan's type [21, 24, 35].Theorem 3.8.
Let(X,p) be a TVS-cone  complete metric space, and let the map T:X→X satisfy
(3.30)p(Tx,Ty)≾Kγ[p(x,Ty)+p(y,Tx)],
for all x, y∈X, where γ∈[0,1/2) is a constant. Then the following statements hold:(a)
there exists a nonempty proper subsetM of X, such that M contains infinite points of X and (M,dp) is a complete metric space, where dp:=ξe∘p;(b)
T has a unique fixed point in X (in fact, the unique fixed point of T belongs to M). Moreover, for each x∈X, the iterative sequence {Tnx}n∈ℕTVS -cone converges to the fixed point.Proof.
SetS=T and α=β=0 in Theorem 3.1.Remark 3.9.
Theorem3.8 improves the fixed point theorems of Chatterjea's type [21, 24, 36].


## 4. Some Equivalences

In this final section, we introduce the following new concepts.Definition 4.1.
LetX be a nonempty set with a TVS-cone metricp,x∈X, and let {xn}n∈ℕ be a sequence in X. (i)
{xn}w-cone converges tox if for any ε>0, there is a natural number ℕ0 such that
(4.1)p(xn,x)≪Kεe,∀n≥N0.
We denote this by w-conelimn→∞xn=x or xn→w-conex as n→∞ and call x the w-cone limit of {xn};(ii)
{xn} is a w-cone Cauchy’s sequence; if for any ε>0, there is a natural number ℕ0 such that p(xn,xm)≪Kεe for all n, m≥ℕ0;(iii)
(X,p) is w-cone complete if every w-cone Cauchy sequence in X is w-cone convergent.We establish the following crucial and useful properties.Theorem 4.2.
LetX be a nonempty set with a TVS-cone metric p, x∈X, and let {xn}n∈ℕ be a sequence in X. Then the following statements hold. (a)
{xn}w-cone  converges to xif and only if  dp(xn,x)→0 as n→∞.(b)
{xn} is a w-cone Cauchy sequence in(X,p)if and only if {xn} is a Cauchy sequence (in usual sense) in (X,dp).(c)
(X,p) is w-cone complete if and only if (X,dp) is a complete metric space.Proof.
Letε>0 be given. If  {xn}w-cone converges to x, then, from (iv) of Lemma 2.1, there exists n0∈ℕ such that
(4.2)p(xn,x)≪Kεe⟺dp(xn,x)=ξe∘p(xn,x)<ε
for all n≥n0. Hence, dp(xn,x)→0 as n→∞. Conversely, if dp(xn,x)→0 as n→∞, then, by (4.2), we show that {xn}w-cone converges to x. Therefore, (a) holds.
To see (b), let{xn} be a w-cone Cauchy sequence in (X,p). Then there exists n1∈ℕ such that
(4.3)p(xn,xm)≪Kεe⟺dp(xn,xm)<ε,
for all n, m≥n1. So {xn} is a Cauchy sequence in (X,dp). The converse holds obviously.
Conclusion (c) is immediate from conclusions (a) and (b).Theorem 4.3.
Let(X,p) be a w-cone complete metric space. Suppose that T,S:X→X are two self-maps of X satisfying
(4.4)p(Tx,Sy)≾Kαp(x,y)+β[p(x,Tx)+p(y,Sy)]+γ[p(x,Sy)+p(y,Tx)]
for allx,y∈X, where α, β, γ≥0 and α+2β+2γ<1. Then the following statements hold: (a)
T and S have a unique common fixed point in X. Moreover, for each x0∈X, the mixed iterative sequence {xn}n∈ℕ∪{0}, defined by x2n+1=Tx2n and x2n+2=Sx2n+1 for n∈ℕ∪{0}, w-cone converges to the common fixed point;(b)
any fixed point ofT is a fixed point of S, and conversely.Proof.
Applying (c) of Theorem4.2, (X,dp) is a complete metric space. By Lemma 2.1, the assumption (4.4) implies
(4.5)dp(Tx,Sy)≤αdp(x,y)+β[dp(x,Tx)+dp(y,Sy)]+γ[dp(x,Sy)+dp(y,Tx)],∀x,y∈X.
Therefore, the conclusion follows from Corollary 3.2 and Theorem 4.2.It is obvious that Theorem4.3 implies Corollary 3.2, so we obtain the following equivalence between Theorem 4.3 and Corollary 3.2.Theorem 4.4.
Theorem4.3 and Corollary 3.2 are equivalent.Remark 4.5.
Using the same argument as above, we can prove the equivalences between scalar version and vectorial version of the Banach contraction principle, Kannan's fixed point theorem, Chatterjea's fixed point theorem, and others (e.g., [20, 21, 24, 25]) in w-cone complete metric spaces.The following result tell us the relationship between theTVS-cone completeness and the w-cone completeness.Theorem 4.6.
If(X,p) is TVS-cone  complete, then there exists a nonempty proper subset M of X, such that M contains infinite points of X, and (M,p) is a w-cone complete metric space.Proof.
Applying Theorem2.11, there exists a nonempty proper subset M of X, such that M contains infinite points of X, and (M,dp) is a complete metric space. By using (c) of Theorem 4.2, (M,p) is a w-cone complete metric space.



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*Source: 985797-2011-10-25.xml*