ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version) doi:http://dx.doi.org/10.12732/ijam.v32i2.5
COMMON FIXED POINT THEOREM IN b-MENGER
SPACES WITH A FULLY CONVEX STRUCTURE
Abderrahim Mbarki1§, Rachid Oubrahim2 1National School of Applied Sciences
Mohammed First University Oujda, 60000, MOROCCO
2ANO Laboratory, Faculty of Sciences
Mohammed First University Oujda, 60000, MOROCCO
Abstract: We prove, in b-Menger spaces [9] the existence of common fixed point for nonexpansive mappings in fully convex b-Menger space by using the normal structure property. We provide examples to analyze and illustrate our main results.
AMS Subject Classification: 54E70, 54H25, 47S50, 34B15
Key Words: b-Menger space, fully convex structure, normal structure, non-expansive mapping, common fixed point
1. Introduction
Let (M, d) be a metric space and suppose A⊂ M. A mapping f :A → M is called nonexpansive if its Lipschitz constantk(f) does not exceed 1. Thus this class of mappings includes the contractions and strictly contractive mappings; moreover it contains all isometries (including the identity). Explicitly, f :A→ M is nonexpansive if
d(f a, f b)≤d(a, b), a, b∈A.
Received: October 27, 2018 c 2019 Academic Publications
If M is a Banach space with norm k.k and A is a nonempty subset of M, in this context a mappingf :A→A is nonexpansive if
kf a−f bk ≤ ka−bk, a, b∈A.
The nonexpansive mappingf may frequently fail to have a fixed point in a set-ting which permits the existence of a decreasing sequence{Ai}i∈INof nonempty,
closed, convex and f-invariant (f(Ai) ⊂ Ai, i ∈IN)) sets having empty inter-section. However, Kirk [7] observed that the presence of a geometric property called ’normal structure’ (M.S. Brodskii, D.P. Milman, On the center of a con-vex set, Dokl. Akad. Nauk. SSSR, 59 (1948), 837-840) guarantees that the nonexpansive f such that A is nonempty, weakly compact convex subset of a Banach spaceM, has a fixed point inM.
Since its publication in 1965, many have tried to extend it to metric spaces. But because of its strong connection to the linear convexity structure of linear spaces, it was hard to come up with a nice and flexible extension. For example, Takahashi [15] was may be the first one to give a metric analogue to Kirk’s theorem. His approach was based on defining a convexity in metric spaces extremely similar to the linear convexity also known as Menger convexity [1]-[3], [8].
In 1987 Hadˇzi´c [6] offered an extension of Takahachi’s structure to Menger spaces and proved fixed point theorem for nonexpansive mappings in proba-bilistic metric spaces with a convex structure.
Following Hadˇzi´c’s approach, Jeˇsi´c et al. [14] have introduced, strictly con-vex and normal structure in Menger spaces and proved fixed point theorem for nonexpansive mappings.
Recently, Mbarki et al. [9] introduced the probabilistic b-metric spaces (b-Menger spaces) as a generalization of probabilistic metric spaces (Menger spaces) and they studied some topological properties and showed the fixed point property for nonlinear contractions in these spaces.
2. Preliminaries
In this section, we collect some basic definitions, notation and conventions of the theory of probabilistic b-metric spaces, which will be used throughout the paper.
Definition 1. A distance distribution function (briefly, a d.d.f.) is a nondecreasing function F defined on IR+∪ {∞} that satisfies f(0) = 0 and
f(∞) = 1, and is left continuous on (0,∞). The set of all d.d.f’s will be noted by ∆+; and the set of all F in ∆+ for which lim
t→∞f(t) = 1 byD +.
A simple example of distribution function is Heavyside function in D+
H(t) =
0 if t≤0,
1 if t >0.
A commutative, associative and nondecreasing mapping T : [0,1]2 → [0,1] is
called a t-norm if and only if
1. T(a,1) =a, for all a∈[0,1], 2. T(0,0) = 0.
As examples we mention the three typical examples of continuous t-norms as follows:
Tp(a, b) =ab, TM(a, b) = min(a, b) and TL(a, b) = max{a+b−1,0}. We define the operatorTn recursively byT1(a
1, a2) =T(a1, a2) andTn(a1, a2, ..., an+1) =T(Tn−1(a1, a2, ..., an), an+1) for alln≥1 anda1, a2, ..., an+1 ∈[0,1].
Definition 2. [9] A b-Menger space is a quadruple (M, F, T, s) where M
is a nonempty set, F is a function fromM ×M into ∆+,T is a t-norm,s≥1
is a real number, and the following conditions are satisfied: For all p, r, q ∈M
and x, y >0, 1. Fpp=H,
2. Fpr =H ⇒ p=r, 3. Fpr =Frp,
It should be noted that a Menger space is ab-Menger space withs= 1.
Definition 3. Let{xn}be a sequence in a probabilistic semimetric space (M, F) (i.e., (i), (ii) and (iii) of Definition 2 are satisfied).
1. A sequence {xn} in M is said to be convergent to x in M if, for every
ǫ > 0 and δ ∈ (0,1), there exists a positive integer N(ǫ, δ) such that
Fxnx(ǫ)>1−δ, whenevern≥N(ǫ, δ).
2. A sequence {xn} in M is called Cauchy sequence if, for every for every
ǫ > 0 and δ ∈ (0,1), there exists a positive integer N(ǫ, δ) such that
Fxnxm(ǫ)>1−δ, whenever n, m≥N(ǫ, δ).
3. Results and discussion
3.1. Some topological properties of b-Menger space
Different kinds of topologies can be introduced in ab-Menger space (M, F, T, s). The strong topology is introduced by a strong neighborhood system℘=S
p∈M℘p, where℘p={Np(t) : t >0},and
Np(t) ={q∈M : Fpq(t)>1−t} fort >0 and p∈M. In [9] the following results are proved.
Lemma 4. Let(M, F, T, s)be ab-Menger space withT is continuous, then the familyℑconsisting of∅ and all unions of elements of this strong neighbor-hood system determines a Hausdorff topology for M. Also, the function F is in general not continuous.
It is special interest (ε, λ)-topology on (M, F, T, s) which is introduced by a family of (ε, λ)-neighborhood {Np(ε, λ)}p∈M, ε>0, λ∈[0,1], where
Np(ε, λ) ={q∈M : Fpq(ε)>1−λ}. Since Np(t, t) =Np(t) for t >0, and
the strong neighborhood system is equivalent to the (ε, λ)-neighborhood system. Hence we reach the same conclusions as in Lemma 4, with (ε, λ)-topology in place of strong topology.
In this topology the function g is continuous in p ∈ M if and only if for every sequencepn→p it holds that g(pn)→g(p).
Here and in the sequel, when we speak about ab-Menger space (M, F, T, s), we always assume thatT is a continuous t-norm andRanF ⊂D+.
Definition 5. LetAbe a subset nonempty of ab-Menger space (M, F, T, s), the closure A of A is the set of allp inM such that, for anyε >0, λ ∈(0,1), there is aq inA such that
Fpq(ε)>1−λ.
As direct consequence of Definition 5 we have the following.
Lemma 6. LetAbe a subset nonempty of ab-Menger space (M, F, T, s). Then
1. p∈A if and only if for anyε >0, λ∈(0,1), Np(ε, λ)∩A6=∅.
2. p∈A if and only if there exists a sequence{pn} in A such thatpn→p.
3. A is a closed set if and only ifA=A.
4. A is a closed set. Moreover,A is the smallest closed set containing A.
The set Np[ε, λ] = {q ∈ M : Fpq(ε) ≥ 1−λ}, is called closed (ε, λ )-neighbourhood of a point p∈M.
Definition 7. We say that a b-Menger space has the property (P1) if the closed (ε, λ)-neighbourhoods are closed sets.
Proposition 8. Let(M, F, T, s)be ab-Menger space, if for anyq ∈M, t >
0 and sequencepn→p inM we have
Fpq(t)≥lim inf
n Fpnq(t). (3.1)
Proof. Letp∈Nq[ε, λ], by Lemma 6 there exists a sequence{pn} ⊆Nq[ε, λ] such thatpn→p, and so
Fpnq(ε)≥1−λfor all n∈IN,
we deduce that
Fpq(ε) ≥ lim inf
n Fpnq(ε)
≥ 1−λ..
Consequently, p∈Nq[ε, λ] and henceNq[ε, λ] is closed set.
Example 9. LetM = IR, i∈IN∗− {1}. Define F :M×M →∆+ by
Fxy(t) =H(t− |x−y|i).
Firstly, from [9], (M, F, TM,2i−1) is a b-Menger space. Next, let pn → p in (M, F, TM,2i−1). Clearly, pn → p in (M,| . |). Now, letq ∈ M, t >0, we can write
t− |pj −q|i ≥ inf
n≥j(t− |pn−q|
i), ∀j∈IN∗,
and passing to limit asj → ∞in the previous inequality, we get
t− |p−q|i ≥sup j
inf
n≥j(t− |pn−q| i),
which implies
H(t− |p−q|i)≥H(sup j
xj). (3.2)
wherexj = infn≥j(t− |pn−q|i).Since {xj} ↑supjxj. Using the fact thatH is non-decreasing left continuous function, we obtain
H(sup j
xj) = lim
j H(xj) = supj H(xj). (3.3) On the other hand,
H(xj) ≥ H( inf
n≥j(t− |pn−q| i))
≥ H(t−sup n≥j
|pn−q|i)
≥ inf
It follows by (3.2) and (3.3) that
H(t− |p−q|i) ≥ H(sup j
xj)
≥ sup j
H(xj)
≥ sup j
inf
n≥jH(t− |pn−q| i).
Therefore, condition (3.1) is satisfied, then (M, F, TM,2i−1) has the property (P1).
Definition 10. A subsetAof a b-Menger space is called compact if every open cover ofA has a finite subcover.
Consequently, a subsetAof ab-Menger space is compact if and only if, from any family of closed subsets of Awhose intersection is empty, we can extract a finite subfamily whose intersection is empty.
Remark 11. Since (M, F, T, s) is a Hausdorff topological space, then every compact subsetA is closed set.
3.2. Some geometrical properties and fully convex b-Menger space
Egbert [5], in 1968 introduced a probabilistic generalization of the notion of diameter of nonempty set in metric space.
Definition 12. Let (M, F, T, s) be a b-Menger space and A ⊂M. The probabilistic diameter of setA is given by
DA(t) = sup ε<t
inf
a,b∈AFab(ε), and the diameter of the set Ais defined by
DA= sup t>0supε<t
inf
a,b∈AFab(ε). IfDA= 1 the setA will be called probabilistic bounded.
Example 13. Let (M, d, s) be ab-metric space. DefineF :M×M →∆+ by
It is easy to check that for a nonempty subsetA ofM we have
DA(t) =H(t−diam(A)),
where
diam(A) = sup{d(p, q) : p, q∈A}.
Indeed, lett >0, ifH(t−diam(A)) = 1 then diam(A)< t, we deduce that
H(ε−diam(A)) = 1 for everydiam(A)< ε < t, and since,
inf
a,b∈AH(ε−d(p, q))≥H(ε−diam(A)), for everyε < t. Consequently, we get
DA(t) = sup ε<t
inf
a,b∈AH(ε−d(p, q))
≥ sup ε<t
H(ε−diam(A))
≥ sup
diam(A)<ε<t
H(ε−diam(A))
≥ 1,
and hence, DA(t)≥H(t−diam(A)). On the other hand, suppose that H(t− diam(A)) = 0,thendiam(A)≥t.Now, letε < t,so forζ =diam(A)−εthere exista, b∈Asuch thatd(a, b)> diam(A)−ζ >0,which impliesH(ε−d(a, b)) = 0, and so infa,b∈AH(ε−d(a, b)) = 0. Since the choice of ε < t was arbitrary, this is true for all ε < t,which gives
DA(t) = sup ε<t
inf
a,b∈AH(ε−d(a, b)) = 0. Hence
DA(t)≤H(t−diam(A)).
We conclude that DA(t) =H(t−diam(A)). So, A is probabilistic bounded in (M, F, TM, s) if and only if A is bounded in (M, d, s).
From the definition of functionals sup and inf, we have the following lemma.
Lemma 15. Every compact subset A of a b-Menger space (M, F, T, s) is probabilistic bounded.
Proof. Let A be a compact subset of a b-Menger space M. Let fix ε and
λ ∈ (0,1). Since T2 is continuous in (1,1,1) there exists δ ∈ (0,1) such that
T2(1−δ,1−δ,1−δ)>1−λ.Now, we will consider an (ε, δ)-cover {Na(ε, δ) : a ∈ A}. Since A is compact, there exist a1, a2, ..., an ∈ A such that A ⊂
Sn
i=1Nai(ε, λ). Since Ran(F) ⊂D
+ then for all i, j there exists ε
ij > ε such thatFaiaj(εij)>1−δ.If we taket= (2s
2+s)ρwhereρ= max{ε
ij}, leta, b∈A. Then there existi, j∈ {1, ..., n} such thata∈Nai(ε, δ) andb∈Naj(ε, δ). Thus
we have Faai(ε)>1−δ and Fbaj(ε)>1−δ. It follows that
Fab(t) =Fab((2s2+s)ρ) ≥ T2(Faai(ρ), Faiaj(ρ), Fajb(ρ))
≥ T2(Faai(εij), Faiaj(εij), Fajb(εij))
≥ T2(Faai(ε), Faiaj(εij), Fajb(ε))
≥ T2(1−δ,1−δ,1−δ)>1−λ.
ThereforeFab(t)>1−λfor all a, b∈A. Hence, by Lemma 14, we obtain that
A is probabilistic bounded set.
Remark 16. Note that ”RanF ⊂D+” is a necessary condition in Lemma 15. For example, considerM ={p, q} and Fpq(t) = 12H(t− |p−q|) + 12ε∞(t),
where
ε∞(t) =
1 if t=∞,
0 if otherwise.
Clearly, M is a compact space and is only semi-bounded.
We adopt the following Hadˇzi´c’s convex structure concept.
Definition 17. [6] Let (M, F, T, s) be a b-Menger space. A mapping
W : M×M ×[0,1]→ M, is said to be a convex structure on M if for every (x, y) ∈M ×M holds W(x, y,0) =y, W(x, y,1) = x and for all x, y, z ∈ M,
λ∈(0,1,) and t >0
FW(x,y,λ)z(2t)≥T(Fxz(
t λ);Fyz(
t
1−λ)).
structureW. Ifx∈X and λ∈[0,1], then W(x, x, λ) =x.
Proof. Let x ∈ X, then W(x, x,0) = x and W(x, x,1) = x. Now, let
λ∈(0,1), we have
FxW(x,x;λ)(t)≥T(Fxx( t
2λ);Fxx( t
2(1−λ))),
for all t >0. Hence, FxW(x,x;λ)(t) = 1, for allt >0, henceW(x, x, λ) =x.
Definition 19. Let (M, F, T, s) be ab-Menger space with a convex struc-ture W. A subset A ⊂ M is said to be convex set if for every a, b ∈ A and
λ∈[0,1] it follows that W(a, b, λ)∈A.
Definition 20. Let (M, F, T, s) be ab-Menger space with a convex struc-ture W. We denote by (C1) the condition: For every λ ∈ [0,1], t > 0 and a, b, c∈M we have
FW(a,b,λ)c(t)≥min{Fca(t), Fcb(t)}.
Lemma 21. Let (M, F, T, s) be a convex b-Menger space with a convex structure W satisfying (C1). Then the closed (ε, λ)-neighbourhoods Nx[ε, λ] are convex sets.
Proof. Leta, b∈Nx[ε, λ] be arbitrary points, then
Fax(ε)≥1−λand Fbx(ε)≥1−λ.
Now, using (C1) withδ∈(0,1), we obtain
FW(a,b,δ)x(ε)≥min{Fax(ε), Fbx(ε)} ≥min{1−λ,1−λ}= 1−λ, we deduceW(a, b, δ) ∈Nx[ε, λ]. Also, W(a, b,0) =band W(a, b,1) =abelong to Nx[ε, λ], and hence the proof is completed.
It is easy to check the following lemma.
Lemma 22. Let (M, F, T, s) be a b-Menger space and {Aµ} for µ ∈ Σ
be a family of convex subsets of M. Then the intersection A = T
µ∈ΣAµ is a
Definition 23. Let (M, F, T, s) be ab-Menger space with a convex struc-ture W andA⊆M. The closed convex shell of set A denoted byco(A), is the intersection of all closed, convex sets that contain A.
Definition 24. A convexb-Menger space (M, F, T, s) with a convex struc-tureW :M×M×[0,1]→Mwill be called fully convex if, for arbitrarya, b∈M,
a6=b there exists λ∈(0,1) such that W(a, b, λ)∈ {a, b}/ .
Example 25. Let n∈IN∗− {1}. DefineF : IR×IR→∆+ by Fab(t) =H(t− |a−b|n), t >0.
From [9], (IR, F, TM,2n−1) is ab-Menger space and in general it is not a Menger space. Define W : IR×IR×[0,1]→IR as
W(a, b;λ) =λa+ (1−λ)b.
We prove that W is a fully convex structure. Indeed, let a, b ∈IR. It is clear thatW(a, b; 1) =aand W(a, b; 0) =b.Now, letλ∈(0,1). Suppose that
M in(H(t
λ− |a−c|
n), H( t
1−λ − |b−c|
n)) = 1, then
t
λ >|a−c|
n
and t
1−λ >|b−c|
n
,
and so
|λa+ (1−λ)b−c|n = |λ(a−c) + (1−λ)(b−c)|n ≤ |λ|a−c|+ (1−λ)|b−c||n ≤ λ|a−c|n+ (1−λ)|b−c|n
(By convexity of f unction f(x) =xn)
< λt
λ+ (1−λ) t
1−λ
= 2t.
Therefore,
FW(a,b,λ)c(2t)≥T(Fac(
t λ);Fbc(
t
1−λ)).
Thus,W is a convex structure. On the other hand, leta, b∈IR such thata6=b. Suppose that for allλ∈(0,1),W(a, b;λ)∈ {a, b}. IfW(a, b;λ) =awe obtain
(1−λ)a= (1−λ)b,
3.3. Normal structure and fixed point theorem
We require the following definitions in our further discussion.
Definition 26. We say that a∈B is a diametral point of B if sup
ε<t inf
b∈BFab(ε) =DB(t), holds for all t >0.
Remark 27. A point a ∈ B is non-diametral of B if there exists r >0 such that
sup ε<r
inf
b∈BFab(ε)> DB(r).
Definition 28. Let (M, F, T, s) be ab-Menger space, we say that it pos-sesses a normal structure if, for every closed, probabilistic bounded and convex set B ⊂M, which consists of at least two different points, there exists a point
b∈B which is non-diametral.
Definition 29. Let (M, F, T, s) be ab-Menger space with a convex struc-tureW. We denote by (C2) the condition: For everyδ∈(0,1), a, b, c∈M and a6=b, there exists t0 >0 such that
FW(a,b,δ)c(t0)> min{Fca(t0), Fcb(t0)}.
Definition 30. The probabilistic b-metric F has the property (P2) if for
all compact subsetA of M, p∈A, there existsq0 ∈A such that
inf
q∈AFpq=Fpq0.
Before stating our main result, we need the following lemmas.
Lemma 31. Let (M, F, T, s) be a fully convex b-Menger space with a convex structure W satisfying (C2) and suppose that the mapping F has the property (P2). Let A ⊂ M be nonempty, convex and compact subset of M. Then Apossesses a normal structure.
B ⊂Aand two different pointsa1 anda2 inBthenB not have a non-diametral
point which gives by the use of Definition 26 that
sup ε<t
inf
b∈BFab(ε) =DB(t),
for every a ∈ B. In other hand, by the Definition 24 and the fact that M is fully convex, there exists δ ∈(0,1) such that W(a1, a2, δ) ∈ {a/ 1, a2}. But B is
a convex set it follows thatW(a1, a2, δ)∈B. Hence DB(t) = sup
ε<t inf
b∈BFW(a1,a2,δ)b(ε).
Thus the infimum of the above equality is attained sinceB is closed subset of the compact A andF possesses the property (P2). Let a3 ∈B such point. So,
we get
DB(t) = sup ε<t
Fa3W(a1,a2,δ)(ε).
In one side, since the mappingFa3W(a1,a2,δ)(.) is nondecreasing and left
contin-uous we obtain
DB(t) =Fa3W(a1,a2,δ)(t).
In the other side, with the fact that the structureW satisfies the condition (C2)
and the use of the infimum and supremum properties we have that, there exists
t0 >0 such that
DB(t0) = Fa3W(a1,a2,δ)(t0)
> min{Fa3a1(t0), Fa3a2(t0)}
which is a contradiction because we have DB ≤Fab for all a, b∈B.
Lemma 32. (Zorn’s Lemma)Let M be a nonempty partially ordered set in which every chain has a lower bound. ThenM has a minimal element.
We say that a subsetDofAis invariant underf, g:A→Aiff(D)∩g(D)⊂ D. The (f, g)-invariant subsets ofAwhich are nonempty, closed and convex are of particular interest since the search for common fixed points offandgmay be confined to such sets. Also, the intersection of any family of nonempty, closed, convex (f, g)-invariant subsets of Ais itself closed, convex and (f, g)-invariant, although it may be empty.
on A such that A is(f, g)-invariant. Denote by Γ a collection of all nonempty subset ofA, closed, convex and(f, g)-invariant. ThenΓhas a minimal element A0. Moreover, if A1 =Co(f(A0)∩g(A0)), thenA1 =A0.
Proof. We haveAis a compact set, then by Remark 11 the setA is closed. In other handA is (f, g)-invariant, also A is convex. So, A ∈Γ which implies that Γ 6= ∅. Now, we order this collection of Γ with inclusion: For D1, D2 ∈ A, D1 ≤ D2 provided D2 ⊂ D1, then (Γ,≤) is a partially ordered set. By
compactness, any arbitrary chain of the collection Γ such{Ai, i∈I}the subset
T
i∈IAi 6=∅, hence an upper bound relative to the order relation ≤. Finally, using Zorn’s Lemma we deduce that there exists a maximal element of Γ relative to ≤denoted A0. Moreover, we haveA1=Co(f(A0)∩g(A0)) andA0 is (f, g
)-invariant, then
A1 = Co(f(A0)∩g(A0)) =Co(f(A0)∩g(A0))
⊆ Co(A0) =A0=A0.
Thus A1 ⊆ A0. Furthermore, A1 ⊆ A0 ⇒ f(A1)∩g(A1) ⊆ f(A0)∩g(A0) ⊆ Co(f(A0)∩g(A0)) = A1 which claim that A1 is (f, g)-invariant. So, A1 ∈ Γ
and A0 ⊆A1. ThereforeA0=A1, which achieve the proof.
Lemma 34. Let (M, F, T, s) be a convex b-Menger space has the prop-erty (P1) with the convex structure satisfying the condition (C1), assume that there exists ζ ∈ (0.1) such that 1−ζ = supε<rinfb∈A0Fab(ε) for some r >
0 and some a ∈ A0 and consider U = (Tb∈A0Nb[r, ζ])
T
A0 and V =
(T
b∈f(A0)∩g(A0)Nb[r, ζ])
T
A0, withU 6=∅. Then U =V.
Proof. Since f(A0) ∩g(A0) ⊆ A0, we get immediately V ⊆ U. But to
have U ⊆ V, we will take an arbitrary c in V. So, by the definition of the set V we get c ∈ Nb[r, ζ], for every b ∈f(A0)∩g(A0). Then Fbc(r) ≥ 1−ζ, ∀b∈f(A0)∩g(A0), which implies that alsob∈Nc[r, ζ], for allb∈f(A0)∩g(A0).
Therefore
f(A0)∩g(A0)⊆Nc[r, ζ].
By Lemma 21, the Definition 7 and F satisfies (P1)-condition it follows that
the set Nc[r, ζ] is closed and convex, thus
A1=Co(f(A0)∩g(A0))⊆Nc[r, ζ].
Now, we use the Lemma 33 in order to get A0 ⊆ Nc[r, ζ] since A0 = A1. So,
Definition 35. [14] Let (M, F, T, s) be a b-Menger space and let f be a self-mapping on M. We say that f is a nonexpansive mapping if
Ff af b(t)≥Fab(t),
for all a, b∈M and t >0.
Now, we are ready to state and prove our main theorem.
Theorem 36. Let (M, F, T, s) be a fully convex b-Menger space has the property (P1) with a convex structure W satisfying (C1) and (C2), suppose that F has the property (P2), and let A ⊂ M be a nonempty, convex and compact subset of M. Let f and g be two self-mappings onA such that A is
(f, g)-invariant and satisfying
Ff agb(t)≥Fab(t), (3.4)
for all a, b ∈ A, a 6= b and for every t > 0. Then f and g have at least one common fixed point onA.
Proof. To prove this result, we will proceed by absurd. So, let A0 the
minimal element of Γ, such that Γ is the collection of all nonempty subsets of A, closed, convex and (f, g)-invariant (see Lemma 33). Indeed, we suppose thatA0 has at least two different points. Since the convex structureW satisfies (C2),F verifies (P2) andAis nonempty, convex and compact subset ofM, thus
using Lemma 31 we obtain that Apossesses a normal structure. Furthermore, by Lemma 33, we have thatA0 is closed subset of the compact setAthenA0 is also a compact set, so using Lemma 15 we conclude that A0 is a probabilistic
bounded set. Then, by what we obtained previously and Definition 28 it follows that there exists some non-diametral point a0 ∈ A0 which means by Remark 27 that
∃r >0, sup ε<r
inf b∈A0
Fa0b(ε)> DA0(r). (3.5)
In other hand, applying Definition 30 and the fact that A0 is a compact subset
of M and F having the property (P2), we have
∃b0∈A0, sup
ε<t inf b∈A0
Fa0b(ε) =Fa0b0(t), for all t >0 (3.6)
Then, from (3.6) we deduce
Also, we have that there exists r′ < r such that F
a0b0(r ′) > D
A0(r). Indeed,
suppose to the contrary, for allr′ < rthat Fa0b0(r
′)≤D
A0(r).
But Fa0b0 is left continuous, so, letting r
′→r, we get Fa0b0(r)≤DA0(r),
which lead to a contradiction with (3.7). So, there exists r′< r such that Fa0b0(r
′)> D
A0(r). (3.8)
Letζ a real number such that 1−ζ = supε<r′infb∈A0Fa0b(ε), then
sup ε<r′
Fa0b(ε)≥1−ζ for all b∈A0
by monotonicity and left continuity of the functionFa0b we get Fa0b(r
′)≥1−ζ
for all b∈A0, hence a0 ∈Nb[r′, ζ] for all b∈A0 therefore a0 ∈U and U 6=∅. It follows from Lemma 34, that U =V. The set U is closed and convex as an intersection of closed and convex sets. It remains to show thatf(U)∩g(U)⊆U
thereforeU ∈Γ to infer thatU =A0. For that, letc∈U andb∈f(A0)∩g(A0). Then there exists a∈A0 such that b=f(a) andb=g(a). Taking t=r′ in the
inequality (3.4), then
Ff(c)b(r′) =F
f(c)g(a)(r′)≥Fca(r′)≥1−ζ.
Thus, the definition of the set V implies f(c) ∈ V. It follows that f(U) ⊆V
and since V =U, we have thatf(U)⊆U. Also, since b=f(a) we get
Fg(c)b(r) =Fg(c)f(a)(r)≥Fca(r)≥1−ζ,
and by the definition of the setU we get g(c) ∈V. So, g(U)⊆V butV =U, theng(U)⊆U and thereforeU ∈Γ henceU =A0. This implies that
U ⊂ \
b∈U
Nb[r′, ζ].
From last we have that for every a, b∈ U it holds that Fab(r′) ≥ 1−ζ, then
infab∈UFab(r′)≥1−ζ.From the last and (3.8), it follows that
DU(r) = sup ε<r
inf
ab∈UFab(ε)≥r′sup≤ε<r inf
ab∈UFab(ε)≥1−ζ > DA0(r). This is a contradiction withU =A0. ThenA0 consists of only one point xand
sincef(A0)∩g(A0)⊆A0 this means thatf(x) =g(x) =x andx is a common
If we take f =g in Theorem 36, we get the following.
Corollary 37. Let(M, F, T, s) be a fully convex b-Menger space has the property (P1) with a convex structure W satisfying (C1) and (C2), suppose thatF has the property(P2). LetA⊂M be a nonempty, convex and compact subset ofM. Then any nonexpansive mappingf of AintoAhas a fixed point.
4. Applications
We require the following lemma in our further discussion.
Lemma 38. ([12])Let(M,k k) be a Banach space. The following condi-tions are equivalent
1. M is strictly convex.
2. If1< p <∞and x, y∈M, x6=y,thenkλx+ (1−λ)ykp < λkxkp+ (1−
λ)kykp for every 0< λ <1.
Example 39. LetM be a Banach space strictly convex and 1< p <∞. Define F :M×M →∆+ andW :M×M ×[0,1]→M by
Fab(t) =H(t− ka−bkp), and
W(a, b;λ) =λa+ (1−λ)b.
From [9] , (M, F, TM,2p−1) is ab-Menger space( but is not in general a standard probabilistic metric space). Proceeding as in Example 25, we prove that W is a fully convex structure. Consider the function f : IR→IR given by
f x=
(
x if x∈A,
2x otherwise.
where A be a nonempty, convex and compact subset of M. It is very essay to see that Na(t) = B(a, t−p) is the open ball of radius t−p centred at a in (M,k . k). So, a subset A is compact in (M, F, TM,2p−1) if and only if A is compact in (M,k . k). We claim thatF has the property (P2). Indeed, let B
in (M,k . k) and sincea 7−→ ka−bkp is continuous, then there exists a
0 ∈B
such that
ka0−bkp = sup
a∈B
ka−bkp.
We can check easily that
inf
a∈BFab =Fa0b.
Which prove our claims. By following the same lines of proof as in Example 9, we show that (M, F, TM,2p−1) has the property (P1). Next, we prove that
(M, F, TM,2p−1) satisfies the conditions (C2) and (C1). Indeed, let a, b, c ∈
M and λ∈[0,1]. By Lemma 38 we get
kW(a, b, λ)−ckp ≤λka−ckp+ (1−λ)kb−ckp,
so
kW(a, b, λ)−ckp ≤max{ka−ckp,kb−ckp}.
Therefor
kW(a, b, λ)−ckp≤ ka−ckp or kW(a, b, λ)−ckp ≤ kb−ckp, (4.1) then
FW(a,b,λ)c ≥Fac or FW(a,b,λ)c ≥Fbc. Hence, condition (C2) holds. Now, we prove that
kW(a, b, λ)−ckp<ka−ckp or kW(a, b, λ)−ckp <kb−ckp. (4.2) for alla, b, c∈M, λ∈(0,1) anda6=b.Suppose that (4.2) does not hold, which in view of (4.1) gives
kW(a, b, λ)−ckp =ka−ckp and kW(a, b, λ)−ckp =kb−ckp, (4.3) Taking kb−ck=r, we get
kW(a, b, λ)−ck
r =kλ
(a−c)
r + (1−λ)
(b−c)
r k= 1 , (4.4)
and
kb−ck
r =
ka−ck r = 1
Since (M,k .k) is strictly convex
kλ(a−c)
r + (1−λ)
(b−c)
which a contradiction with (4.4). Therefore, (4.2) holds. Now, if
kW(a, b, λ)−ckp <ka−ckp,
we take t0>0 such that
kW(a, b, λ)−ckp< t0<ka−ckp.
On the other hand, if
kW(a, b, λ)−ckp <kb−ckp, we take t0>0 such that
kW(a, b, λ)−ckp < t0 <kb−ckp.
We conclude that condition (C1) holds. Takinggis the identity function onM.
Clearly, all the conditions of Theorem 36 are satisfied and every point of the set Ais a common fixed point of f and g.
5. Conclusion
The geometric condition of normal structure plays a crucial role in the fixed point theory. In this paper, we use this condition to prove the existence of fixed point (common fixed point) for nonexpansive mappings in the framework of b-Menger spaces. Our main theorems extend and unify the existing results in the recent literature. Example is constructed to support our result.
References
[1] L.M. Blumenthal, Distance Geometries: A Study of the Development of
Abstract Metrics. University of Missouri Studies, (1938).
[2] L.M. Blumenthal,Theory and Applications of Distance Geometries, Clare-mont Press, Oxford, (1953).
[4] M.S. Brodskii, D.P. Milman, On the center of a convex set, Dokl. Akad.
Nauk. SSSR,59(1948), 837-840.
[5] R.J. Egbert, Products and quotients of probabilistic metric spaces,Pacific
J. Math,24(1968), 437-455.
[6] O. Hadˇzi´c, Common fixed point theorems in probabilistic metric spaces with a convex structure, Zb. Rad. Prirod-Mat.Fak. Ser. Mat., 18 (1987), 165-178.
[7] W.A. Kirk, A fixed point theorem for mappings which do not increase distances,Amer. Math. Monthly, 72(1965), 1004-1006.
[8] K. Menger, Untersuchungen¨uber allgemeine Metrik, Math. Ann., 100
(1928), 75-163.
[9] A. Mbarki, R. Oubrahim, Probabilisticb-metric spaces and nonlinear con-tractions, Fixed Point Theory Appl., 2017 (2017), 15-29.
[10] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci.,28(1942), 535-7. [11] K. Menger, G´eom´etrie g´en´erale, M´emor. Sci. Math., Gauthier-Villars,
Paris,124, (1954).
[12] S. Prus, Geometrical background of metric fixed point theory. In: W.A. Kirk, B. Sims (Eds), Handbook of Metric Fixed Point Theory, Springer, Dordrecht (2001).
[13] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathimatics,5 (1983).
[14] J.N. Sini´sa, R. M. Nikoli´c, N.A. Baba´cev, A common fixed point theorem in strictly convex Menger PM-spaces,Filomat,28, No 4 (2014), 735-743. [15] T. Wataru, A convexity in metric spaces and nonexpansive mappings, I.
