some fixed point theorems in fuzzy metric spaces from ... · introduction metrics from fuzzy...

IntroductionMetrics from fuzzy metrics

Some remarks on fixed point results

Some fixed point theorems in fuzzy metricspaces from Banach’s principle

P. Tirado

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Index

1 Introduction

2 Metrics from fuzzy metrics

3 Some remarks on fixed point results

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Here we present the concept of p-metric that use to obtainsome well-known fixed point theorems in fuzzy metric spacesfrom the classical Banach’s principle

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a⊕1 b = min{1,a+b}

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⊕p : [0,1]× [0,1] :→ [0,1]

⊕p(a,b) = min{1,(ap +bp)1/p}, p > 0.

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p-sums (Yager t-conorms)

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(X ,d)→ (X ,d1), where d1 = min{1,d}

τd ≡ τd1

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Definition of p-metric spaceA p-metric space is a triple (X ,D,⊕p) such that X is anonempty set, ⊕p is a p-sum and D is a fuzzy set in X ×X suchthat for all x ,y ,z ∈ X :(i) D(x ,y) = 0 if and only if x = y(ii) D(x ,y) = D(y ,x)(iii) D(x ,z)≤ D(x ,y)⊕p D(y ,z)

We will say that (D,⊕p) is a p-metric on X .

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For each x ∈ X and r > 0 we can define the open ballBD(x , r) = {y ∈ X : D(x ,y)< r} and it is obvious thatBD(x , r1)⊆ BD(x , r2) provided that r1 ≤ r2. Consequently, wemay define a topology τD on X as τD = {A⊆ X : for each x ∈ Athere exists r > 0 such that BD(x , r)⊆ A}.

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If (D,⊕p) is a p-metric on X then d(x ,y) = Dp(x ,y) for allx ,y ∈ X is a metric on X and that τd = τD. Reciprocally, if d is a1−bounded metric on X then D(x ,y) = d1/p(x ,y) for allx ,y ∈ X is a p-metric on X for ⊕p, p > 0.

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t-norms and t-conorms

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A t-norm is a binary operation ∗ : [0,1]× [0,1]→ [0,1] satisfyingthe following conditions:(i) ∗ is associative and commutative; (ii)a∗1 = a for every a ∈ [0,1], (iii) a∗b ≤ c ∗d whenever a≤ c andb ≤ d with a,b,c,d ∈ [0,1].

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A t-conorm is a binary operation � : [0,1]× [0,1]→ [0,1]satisfying the following conditions:(i) � is associative andcommutative; (ii) a�0 = a for every a ∈ [0,1], (iii) a�b ≤ c �dwhenever a≤ c and b ≤ d with a,b,c,d ∈ [0,1].

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If ∗ is a (continuous) t-norm we can define a (continuous)t-conom �∗as follows: a�∗ b = 1− [(1−a)∗ (1−b)] for alla,b ∈ [0,1]

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p-sums are continuous t-conorms (Yager continuous t-conorms)

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Zadeh. L, Similarity relations and fuzzy orderings. InformationSciences, 3, 159-176, 1971.

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Definition of similarity relationA similarity relation on a set X is a pair (E ,∗) such that ∗ is at-norm and E is a fuzzy set in X ×X such that for all x ,y ,z ∈ X :

(E1) E(x ,y) = 1 if and only if x = y

(E2) E(x ,y) = E(y ,x)

(E3) E(x ,z)≥ E(x ,y)∗E(y ,z).

If we define D(x ,y) = 1−E(x ,y) for all x ,y ∈ X , thenD(x ,z)≤ D(x ,y)�∗D(y ,z). So if �∗ ≤⊕p for some p > 0 then(X ,D,⊕p) is a p-metric space.

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I. Kramosil and J. Michalek, Fuzzy metrics and Statisticalmetric spaces, Kibernetica v.2 n 2, 1975

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DefinitionA fuzzy metric on a set X is a pair (M,∗) such that ∗ is acontinuous t-norm and M is a fuzzy set in X ×X × [0,∞) suchthat for all x ,y ,z ∈ X :

(FM1) M(x ,y ,0) = 0;

(FM2) x = y if and only if M(x ,y , t) = 1 for all t > 0;

(FM3) M(x ,y , t) = M(y ,x , t);

(FM4) M(x ,z, t +s)≥M(x ,y , t)∗M(y ,z,s) for all t ,s ≥ 0;

(FM5) M(x ,y , ) : R+→ [0,1] is left continuous.

By a fuzzy metric space we mean a triple (X ,M,∗) such that Xis a set and (M,∗) is a fuzzy metric on X .

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Given a fuzzy metric (M,∗) on a set X we can define a openball for each x ∈ X , t > 0 and ε ∈ (0,1) asBM(x ,ε, t) = {y ∈ X : M(x ,y , t)> 1− ε}. Consequently, we maydefine a topology τM on X as τM = {A⊆ X : for each x ∈ A thereexists ε ∈ (0,1) and t > 0 such that BM(x ,ε, t)⊆ A}.

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A. George, P. Veeramani, On some results in fuzzy metricspaces. Fuzzy Sets and Systems, 64 (1994), 395-399.

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Definition of Cauchy sequenceA Cauchy sequence in a fuzzy metric space (X ,M,∗) is asequence {xn}n∈N in X such that for each ε ∈ (0,1) and t > 0there exists an n0 ∈ N satisfying M(xn,xm, t)> 1− ε whenevern,m ≥ n0.

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Definition of complete fuzzy metric spaceA fuzzy metric space (X ,M,∗) is said to be complete if everyCauchy sequence {xn}n∈N converges with respect to thetopology τM , i.e, if there exists y ∈ X such that for each t > 0,limn M(y ,xn, t) = 1.

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Index

1 Introduction



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V. Gregori, S. Romaguera, Some properties of fuzzy metricspaces. Fuzzy Sets and Systems, 115 (2000), 485-489.

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V. Gregori, S. Romaguera, Characterizing completable fuzzymetric spaces, Fuzzy Sets and Systems, 144 (2004), 411-420.

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Definition of stationary fuzzy metric spaceA fuzzy metric space (X ,M,∗) is said to be stationary if M doesnot depend on t

A similarity relation (E ,∗) on a set X is a stationary fuzzy metricspace (X ,E ,∗) by defining E(x ,y ,0) = 0 when ∗ is continuous

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If (X ,M,∗) is a stationary fuzzy metric space such that �∗ ≤⊕pfor some p > 0 then (X ,D,⊕p) is a p-metric space whereD(x ,y) = 1−M(x ,y) for all x ,y ∈ X . Reciprocally, if (X ,D,⊕p)is a p-metric space then (X ,M,∗) is a stationary fuzzy metricspace, where M(x ,y) = 1−D(x ,y) for all x ,y ∈ X ,M(x ,y ,0) = 0 and �∗ ≤⊕p.

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If(X ,M,∗) is a fuzzy metric space such that �∗ ≤⊕p for somep > 0 then d(x ,y) = Dp(x ,y) = (1−M(x ,y))p for all x ,y ∈ X isa metric on X and that τd = τD = τM .In particular if �∗ ≤⊕1 thend(x ,y) = 1−M(x ,y) is a metric on X .

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V. Radu, On the triangle inequality in PM-spaces. STPA, WestUniversity of Timisoara 39 (1978)

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Theorem 1Let (X ,M,∗) be a fuzzy metric space such that �∗ ≤⊕1. Foreach x ,y ∈ X put

dR(x ,y) = sup{t ≥ 0 : 1−M(x ,y , t)≥ t}.Then dR is a metric on X such that

dR(x ,y)< ε ⇔M(x ,y ,ε)> 1− ε,for all ε ∈ (0,1).

Therefore, the topologies induced by (M,∗) and dR coincide onX . In particular, (X ,M,∗) is complete if and only if (X ,dR) iscomplete.

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ExampleLet (X ,M,∗) be a fuzzy metric space such that �∗ ≤⊕p forsome p ∈ (0,1). The function d : X ×X → R+, defined as

d(x ,y) = sup{t ≥ 0 : (1−M(x ,y , t))p ≥ t}is a metric on X such thatd(x ,y)< ε ⇔M(x ,y ,ε)> 1− ε1/p,for all ε ∈ (0,1).

Therefore, the topologies induced by (M,∗) and d coincide onX . In particular, (X ,M,∗) is complete if and only if (X ,d) iscomplete.

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F.Castro-Company, S. Romaguera and P. Tirado, On theconstruction of metrics from fuzzy metrics and its application tothe fixed point theory of multivalued mappings, Fixed PointTheory and Applications (2015) 2015:226.

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Theorem 2Let (X ,M,∗) be a fuzzy metric space. Suppose that there existsa function α : R+→ R+ satisfying the following conditions:

(c1) α is strictly increasing on [0,1];(c2) 0 < α(t)≤ t for all t ∈ (0,1) and α(t)> 1 for all t > 1;(c3) α(t +s)≥ α(t)�∗α(s);

Then the function dα : X ×X → R+ defined as

dα(x ,y) = sup{t ≥ 0 : M(x ,y , t)≤ 1−α(t)},

is a metric on X such that dα(x ,y)≤ 1 for all x ,y ∈ X .

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If, in addition, the function α is left continuous on (0,1], then

dα(x ,y)< ε ⇔M(x ,y ,ε)> 1−α(ε),

for all ε ∈ (0,1). Thus the topologies induced by (M,∗) and dα

coincide on X . Moreover, (X ,M,∗) is complete if and only if(X ,dα) is complete.

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Index

1 Introduction



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F. Castro-Company, P. Tirado, On Yager and Hamacher t-normsand fuzzy metric spaces, International Journal of Intelligentsystems, 29 (2014), 1179-1180.

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Theorem 3Let (X ,M,∗) be a complete fuzzy metric space such that�∗ ≤⊕p for some p > 0. If T is a self-map on X such that thereis k ∈ (0,1) satisfying

M(Tx ,Ty , t)≥ 1−k +kM(x ,y , t)

for all x ,y ∈ X and t > 0, then T has a unique fixed point.

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1−M(Tx ,Ty , t)≤ k(1−M(x ,y , t))⇔ [1−M(Tx ,Ty , t)]p ≤ [k(1−M(x ,y , t))]p

⇔ [1−M(Tx ,Ty , t)]p ≤ kp[(1−M(x ,y , t))]p. So we can write

sup{t ≥ 0 : (1−M(Tx ,Ty , t))p ≥ t} ≤ kp sup{t ≥ 0 :(1−M(x ,y , t))p ≥ t}

i.e, following the notation in Example 1

d(Tx ,Ty)≤ kpd(x ,y). Since (X ,d) is complete, by the Banachcontraction principle T has a unique fixed point.

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A. George, P. Veeramani, On some results in fuzzy metricspaces. Fuzzy Sets and Systems, 64 (1994), 395-399

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Definition GV-fuzzy metricA GV-fuzzy metric on a set X is a pair (M,∗) such that ∗ is acontinuous t-norm and M is a fuzzy set in X ×X × (0,∞) suchthat for all x ,y ,z ∈ X and t ,s > 0 :

(GV1) M(x ,y , t)> 0;

(GV2) x = y if and only if M(x ,y , t) = 1;

(GV3) M(x ,y , t) = M(y ,x , t);

(GV4) M(x ,z, t +s)≥M(x ,y , t)∗M(y ,z,s) ;

(GV5) M(x ,y , ) : (0,∞)→ (0,1] is continuous.

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It is interesting to remark the fact that every GV-fuzzy metricspace (X ,M,∗) can be considered as a fuzzy metric space inthe sense of Kramosil and Michalek, simply puttingM(x ,y ,0) = 0 for all x ,y ∈ X , so the previous results remainvalid for GV-fuzzy metric spaces.

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V. Gregori, A. Sapena, On fixed point theorems in fuzzy metricspaces, Fuzzy Sets and Systems 125 (2002), 245-253

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fuzzy-contractive selp-mapLet (X ,M,∗) be a GV-fuzzy metric space and T : X → X aself-map. We will say that T is fuzzy contractive if there existsk ∈ (0,1) such that

1M(Tx ,Ty ,t) −1≤ k( 1

M(x ,y ,t) −1)

for all x ,y ∈ X and t > 0.

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Theorem 4Let (X ,M,∧) be a complete GV-fuzzy metric space.Then everyfuzzy contractive self-map T on X has a unique fixed point.

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Let α(t) = tt+1 for t ∈ [0,1] and 2 for t > 1, then α satisties the

conditions of Theorem 2, then the function dα : X ×X → R+

defined as

dα(x ,y) = sup{t ≥ 0 : M(x ,y , t)≤ 1− tt+1}, or , equivalently

dα(x ,y) = sup{t ≥ 0 : 1M(x ,y ,t) −1≥ t}

is a metric on X , thus (X ,dα) is a complete metric space.

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Let T a fuzzy contractive self-map, then1

M(Tx ,Ty ,t) −1≤ k( 1M(x ,y ,t) −1)

so, we have

sup{t ≥ 0 : 1M(Tx ,Ty ,t) −1≥ t} ≤ k sup{t ≥ 0 : 1

M(x ,y ,t) −1≥ t}, i.e

dα(Tx ,Ty)≤ kdα(x ,y). By the Banach contraction principle, Thas a unique fixed point.

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Some fixed point theorems in fuzzy metricspaces from Banach’s principle

P. Tirado

WATS 2016

some fixed point theorems in fuzzy metric spaces from ... · introduction metrics from fuzzy...

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