Volume 2010, Article ID 782680,11pages doi:10.1155/2010/782680
Research Article
Ordered Non-Archimedean Fuzzy Metric Spaces and Some Fixed Point Results
Ishak Altun
1and Dorel Mihet¸
21Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey
2Departament of Mathematics, Faculty of Mathematics and Computer Science, West University of Timis¸oara, Bv. V. Parvan 4, 300223 Timis¸oara, Romania
Correspondence should be addressed to Ishak Altun,[email protected] Received 2 July 2009; Accepted 9 February 2010
Academic Editor: Mohamed A. Khamsi
Copyrightq2010 I. Altun and D. Mihet¸. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In the present paper we provide two different kinds of fixed point theorems on ordered nonArchimedean fuzzy metric spaces. First, two fixed point theorems are proved for fuzzy order ψ-contractive type mappings. Then a common fixed point theorem is given for noncontractive type mappings. Kirk’s problem on an extension of Caristi’s theorem is also discussed.
1. Introduction and Preliminaries
After the definition of the concept of fuzzy metric space by some authors 1–3, the fixed point theory on these spaces has been developing see, e.g.,4–9. Generally, this theory on fuzzy metric space is done for contractive or contractive-type mappingssee2,10–13 and references therein. In this paper we introduce the concept of fuzzy orderψ-contractive mappings and give two fixed point theorems on ordered non-Archimedean fuzzy metric spaces for fuzzy order ψ-contractive type mappings. Then, using an idea in 14, we will provide a common fixed point theorem for weakly increasing single-valued mappings in a complete fuzzy metric space endowed with a partial order induced by an appropriate function. Some fixed point results on ordered probabilistic metric spaces can be found in15.
For the sake of completeness, we briefly recall some notions from the theory of fuzzy metric spaces used in this paper.
Definition 1.1see16. A binary operation∗:0,1×0,1 → 0,1is called a continuous t-norm if0,1,∗is an Abelian topological monoid with the unit 1 such thata∗b≤ c∗d whenevera≤candb≤dfor alla, b, c, d∈0,1.
A continuous t-norm∗is of Hadˇzi´c-type if there exists a strictly increasing sequence {bn} ⊂0,1such thatbn∗bnbnfor alln∈N.
Definition 1.2see3. A fuzzy metric space in the sense of Kramosil and Mich´alekis a tripleX, M,∗, whereXis a nonempty set,∗is a continuoust-norm andMis a fuzzy set on X2×0,∞, satisfying the following properties:
KM-1Mx, y,0 0, for allx, y∈X,
KM-2Mx, y, t 1, for allt >0 if and only ifxy, KM-3Mx, y, t My, x, t,for allx, y∈Xandt >0,
KM-4Mx, y,·:0,∞ → 0,1is left continuous, for allx, y∈X,
KM-5Mx, z, ts≥Mx, y, t∗My, z, s,for allx, y, z∈X,for allt, s >0.
If, in the above definition, the triangular inequalityKM-5is replaced by Mx, z,max{t, s}≥M
x, y, t
∗M y, z, s
, ∀x, y, z∈X, ∀t, s >0, NA then the tripleX, M,∗is called a non-Archimedean fuzzy metric space. It is easy to check that the triangular inequalityNAimpliesKM-5, that is, every non-Archimedean fuzzy metric space is itself a fuzzy metric space.
Example 1.3. Let X, d be an ordinary metric space and let θ be a nondecreasing and continuous function from0,∞into0,1such that limt→ ∞θt 1. Some examples of these functions areθt t/t1,θt 1−e−tandθt e−1/t. Leta∗b≤abfor alla, b ∈0,1.
For eacht∈0,∞, define
M x, y, t
θtdx,y 1.1
for allx, y∈X. It is easy to see thatX, M,∗is a non-Archimedean fuzzy metric space.
Definition 1.4see 1,16. LetX, M,∗ be a fuzzy metric space. A sequence {xn} in X is called an M-Cauchy sequence, if for eachε ∈ 0,1andt > 0 there exists n0 ∈ Nsuch that Mxn, xm, t > 1−εfor allm, n ≥ n0. A sequence{xn}in a fuzzy metric spaceX, M,∗is said to be convergent tox ∈X if limn→ ∞Mxn, x, t 1 for allt > 0. A fuzzy metric space X, M,∗is called M-complete if everyM-Cauchy sequence is convergent.
Definition 1.5see7. LetX, M,∗be a fuzzy metric space. A sequence{xn}inXis called G-Cauchy if
nlim→ ∞Mxn, xn1, t 1 1.2
for all t > 0. The space X, M,∗ is called G-complete if every G-Cauchy sequence is convergent.
Lemma 1.6see11. Each M -complete non-Archimedean fuzzy metric spaceX, M, TwithTof Hadˇzi´c-type is G-complete.
Theorem 2.10in the next section is related to a partial order on a fuzzy metric space under theŁukasiewicz t-norm. We will refer to14.
Lemma 1.7see14. LetX, M,∗be a non-Archimedean fuzzy metric space witha∗b≥max{a b−1,0}andφ:X×0,∞ → R.Define the relation “” onXas follows:
xy⇐⇒M x, y, t
≥1φx, t−φ y, t
, ∀t >0. 1.3 Thenis a (partial) order onX,named the partial order induced byφ.
2. Main Results
The first two theorems in this section are related to Theorem 2.1 in17. We begin by giving the following definitions.
Definition 2.1. Letbe an order relation onX. A mappingf:X → Xis called nondecreasing w.r.tifxyimpliesfxfy.
Definition 2.2. LetX,be a partially ordered set, letX, M,∗be a fuzzy metric space, and let ψ be a function from0,1 to0,1. A mappingf : X → X is called a fuzzy order ψ- contractive mapping if the following implication holds:
x, y∈X, xy⇒ M
fx, fy, t
≥ψ M
x, y, t
∀t >0
. 2.1
Theorem 2.3. LetX,be a partially ordered set andX, M,∗be anM-complete non-Archimedean fuzzy metric space with∗ of Hadˇzi´c-type. Letψ : 0,1 → 0,1be a continuous, nondecreasing function and letf : X → X be a fuzzy orderψ-contractive and nondecreasing mapping w.r.t . Suppose that either
f is continuous, 2.2
or
xnx ∀n, whenever
{xn} ⊂X is nondecreasing sequence withxn−→x∈X 2.3
hold. If there existsx0∈Xsuch that
x0fx0, lim
n→ ∞ψn M
x0, fx0, t
1 2.4
for eacht >0, thenfhas a fixed point.
Proof. Letxnfxn−1forn∈ {1,2, . . .}. Sincex0fx0andfis nondecreasing w.r.t, we have x0x1x2 · · · xnxn1 · · · . 2.5
Then, it immediately follows by induction that
Mxn1, xn2, t≥ψMxn, xn1, t, n∈N, t >0, 2.6
hence
Mxn, xn1, t≥ψn M
x0, fx0, t
, n∈N, t >0. 2.7
By taking the limit asn → ∞we obtain
nlim→ ∞Mxn, xn1, t 1 2.8
for allt >0,that is,{xn}is G-Cauchy. SinceXisG-completeLemma 1.6, then there exists x∈Xsuch that limn→ ∞xnx.
Now, iff is continuous then it is clear thatfx x, while if the condition2.3hold then, for allt >0,
M
xn1, fx, t M
fxn, fx, t
≥ψMxn, x, t 2.9
and lettingn → ∞it follows
M
x, fx, t
≥ψ1 1, 2.10
hencefxx.
Theorem 2.4. LetX,be a partially ordered set, letX, M,∗be anM-complete non-Archimedean fuzzy metric space, and letψ : 0,1 → 0,1be a continuous mapping such thatψt > tfor all t∈0,1. Also, letf :X → Xbe a nondecreasing mapping w.r.t, with the property
M
fx, fy, t
≥ψ M
x, y, t
∀t >0, wheneverxy. 2.11
Suppose that either2.2or2.3holds. If there existsx0∈Xsuch that x0fx0, M
x0, fx0, t
>0 2.12
for allt >0, thenfhas a fixed point.
Proof. Letxn fxn−1forn∈ {1,2, . . .}. Then, as in the proof of the preceding theorem we can prove that
Mxn1, xn2, t≥ψMxn, xn1, t≥Mxn, xn1, t, n∈N, t >0. 2.13
Therefore, for every t > 0, {Mxn, xn1, t}n∈N is a nondecreasing sequence of numbers in 0,1. Let, for fixed t > 0, limn→ ∞Mxn, xn1, t l. Then we have l ∈ 0,1, since Mx0, x1, t>0. Also, since
Mxn1, xn2, t≥ψMxn, xn1, t 2.14
andψis continuous, we havel≥ψl. This impliesl1 and therefore, for allt >0,
nlim→ ∞Mxn, xn1, t 1. 2.15
Now we show that{xn}is an M-Cauchy sequence. Supposing this is not true, then there are ε∈0,1andt >0 such that for eachk ∈Nthere existmk, nk∈Nwithmk> nk≥k and
M
xmk, xnk, t
≤1−ε. 2.16
Let, for eachk,mkbe the least integer exceedingnksatisfying the inequality2.16, that is,
M
xmk−1, xnk, t
>1−ε. 2.17
Then, for eachk,
1−ε≥M
xmk, xnk, t
≥M
xmk−1, xnk, t
∗M
xmk−1, xmk, t
≥1−ε∗M
xmk−1, xmk, t .
2.18
Lettingk → ∞and using2.15, we have, fort >0,
k→ ∞limM
xmk, xnk, t
1−ε. 2.19
Then, sincexnkxmk, we have M
xmk, xnk, t
≥M
xmk, xmk1, t
∗M
xmk1, xnk1, t
∗M
xnk1, xnk, t
≥M
xmk, xmk1, t
∗ψ M
xmk, xnk, t
∗M
xnk1, xnk, t .
2.20 Lettingk → ∞and using2.15and2.19, we obtain
1−ε≥1∗ψ1−ε∗1ψ1−ε>1−ε, 2.21
which is a contradiction. Thus{xn}is an M-Cauchy sequence. SinceX isM-complete, then there existsx∈Xsuch that
nlim→ ∞xnx. 2.22
Iff is continuous, then fromxn fxn−1 n∈ Nit follows thatfx x.Also, if2.3holds, thensincexnxwe have
M
xn1, fx, t M
fxn, fx, t
≥ψMxn, x, t, n∈N, t >0. 2.23
Lettingn → ∞, we obtain that
M
x, fx, t
1 ∀t >0, 2.24
hencefxx.
Example 2.5. LetX 0,∞. Consider the following relation onX:
xy⇐⇒
xy orx, y∈1,4, x≤y
. 2.25
It is easy to see thatis a partial order onX. Leta∗baband
M x, y, t
min x, y max
x, y, ∀t >0. 2.26
ThenX, M,∗is an M-complete non-Archimedean fuzzy metric spacesee18satisfying Mx, y, t>0 for allt >0. Define a self mapfofXas follows:
fx
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
2x, 0< x <1 x5
3 , 1≤x≤4 2x−5, x >4.
2.27
Now, it is easy to see thatfis continuous and nondecreasing w.r.t. Also, forx01 we have 1x0fx02. Now we can see thatfis fuzzy orderψ-contractive withψt √
t.
Indeed, letx, y∈Xwithxy. Now ifxy, then
M
fx, fy, t
1≥ψ1 ψ M
x, y, t
. 2.28
Ifx, y∈1,4withx≤y, then
M
fx, fy, t
min fx, fy max
fx, fy min
x5/3, y5
/3 max
x5/3, y5
/3 x5
y5
≥ x
y ψ
M
x, y, t .
2.29
Thereforefis fuzzy orderψ-contractive withψt √
t. Hence all conditions ofTheorem 2.4 are satisfied and sofhas a fixed point onX.
In order to state our next theorem, we give the concept of weakly comparable mappings on an ordered space.
Definition 2.6. LetX,be an ordered space. Two mappings f, g : X → X are said to be weakly comparable iffxgfxandgxfgxfor allx∈X.
Note that two weakly comparable mappings need not to be nondecreasing.
Example 2.7. LetX 0,∞and≤be usual ordering. Letf, g:X → Xdefined by
fx
⎧⎨
⎩
x if 0≤x≤1,
0 if 1< x <∞, gx
⎧⎨
⎩
√x if 0≤x≤1,
0 if 1< x <∞. 2.30 Then it is obvious thatfx ≤ gfxand gx ≤ fgxfor allx ∈ X. Thus f andg are weakly comparable mappings. Note that bothfandgare not nondecreasing.
Example 2.8. Let X 1,∞×1,∞ and be coordinate-wise ordering, that is, x, y z, w ⇔ x ≤ z and y ≤ w. Let f, g : X → X be defined by fx, y 2x,3y and gx, y x2, y2, then fx, y 2x,3y gfx, y g2x,3y 4x2,9y2 and gx, y x2, y2 fgx, y fx2, y2 2x2,3y2. Thusf andg are weakly comparable mappings.
Example 2.9. LetX R2andbe lexicographical ordering, that is,x, yz, w⇔x < z orifxz,theny≤w. Letf, g:X → Xbe defined by
f x, y
max
x, y ,min
x, y , g
x, y
max
x, y ,xy
2
, 2.31
thenfx, y gfx, yandgx, y fgx, yfor allx, y ∈ X.Thusf and g are weakly comparable mappings. Note that,1,42,3butf1,4 4,13,2 f2,3,thenfis not nondecreasing. Similarlygis not nondecreasing.
Theorem 2.10. LetX, M,∗be an M -complete non-Archimedean fuzzy metric space witha∗b≥ max{ab−1,0}, φ:X×0,∞ → Rbe a bounded-from-above function, and letbe the partial order induced byφ.Iff, g : X → X are two continuous and weakly comparable mappings, thenf andghave a common fixed point inX.
Proof. LetX0be an arbitrary point ofXand let us define a sequence{xn}inXas follows:
x2n1fx2n, x2n2ngx2n1 forn∈ {0,1, . . .}. 2.32
Note that, sincefandgare weakly comparable, we have x1fx0gfx0gx1x2,
x2gx1fgx1 fx2x3. 2.33
By continuing this process we get
x1x2 · · · xnxn1 · · · , 2.34 that is, the sequence{xn}is nondecreasing. By the definition ofwe haveφx0, t≤φx1, t≤ φx2, t ≤ · · · for allt >0, that is, for even t >0, the sequence{φxn, t}is a nondecreasing sequence inR. Sinceφis bounded from above,{φxn, t}is convergent and hence it is Cauchy.
Then, for all ε > 0 there exists n0 ∈ Nsuch that for all m > n > n0 and t > 0 we have
|φxm, t−φxn, t|φxm, t−φxn, t< ε. Therefore, sincexnxm, we have Mxn, xm, t≥1φxn, t−φxm, t
1−
φxm, t−φxn, t
>1−ε.
2.35
This shows that the sequence{xn}is M-Cauchy. SinceXis M-complete, it converges to a point z∈X. Asx2n1 → zandx2n2 → z, by the continuity offandgwe getfzgzz.
Corollary 2.11Caristi fixed point theorem in non-Archimedean fuzzy metric spaces. Let X, M,∗be an M -complete non-Archimedean fuzzy metric space witha∗b≥max{ab−1,0},let φ :X×0,∞ → Rbe a bounded-from-above function andf :X → Xbe a continuous mapping, such that
M
x, fx, t
≥1φx, t−φ fx, t
2.36 for allx∈Xandt >0.Thenfhas a fixed point inX.
Proof. We take in the above theoremg1Xand note that the weak comparability offandg reduces to2.36.
The generalization suggested by Kirk of Caristi’s fixed point theorem 19 is well known. A similar theorem in the setting of non-Archimedean fuzzy metric spaces is stated in the final part of our paper.
In what followsν :0,1 → 0,1is nondecreasing, subadditive mappingi.e.,νa b≤νa νbfor alla, b∈0,1, withν0 0.
Theorem 2.12. LetX, M,∗be a non-Archimedean fuzzy metric space witha∗b≥max{ab−1,0}
andφ:X×0,∞ → R.Define the relation “” onXthrough xy⇐⇒φ
y, t
−φx, t≥ν 1−M
x, y, t
, ∀t >0. 2.37 Then “” is a (partial) order onX.
Proof. Sinceν0 0, then for allx∈Xandt >0,
0φx, t−φx, t≥ν1−Mx, x, t 0, 2.38
that is, “” is reflexive.
Letx, y∈Xbe such thatxyandyx.Then for allt >0, φ
y, t
−φx, t≥ν 1−M
x, y, t , φx, t−φ
y, t
≥ν 1−M
x, y, t
, 2.39
implying thatMx, y, t 1 for allt >0,that is,xy. Thus “” is antisymmetric.
Now forx, y, z∈X, letxyandyz. Then, for givent >0, φ
y, t
−φx, t≥ν 1−M
x, y, t
, 2.40
φz, t−φ y, t
≥ν 1−M
z, y, t
. 2.41
By using2.40and2.41we get φz, t−φx, t≥ν
1−M
x, y, t ν
1−M y, z, t
≥ν 1−M
x, y, t
1−M y, z, t
. 2.42
On the other hand, from the triangular inequalityNA, the inequality Mx, z, t≥M
x, y, t M
y, z, t
−1 2.43
holds. This implies
1−M x, y, t
1−M y, z, t
≥1−Mx, z, t. 2.44
Asνis nondecreasing, it follows that ν
1−M x, y, t
1−M y, z, t
≥ν1−Mx, z, t 2.45
and therefore
φz, t−φx, t≥ν1−Mx, z, t. 2.46
This shows thatxz, that is, “” is transitive.
From the above theorem we can immediately obtain the following generalization of Corollary 2.11.
Corollary 2.13. LetX, M,∗be an M -complete non-Archimedean fuzzy metric space witha∗b≥ max{ab−1,0},letφ:X×0,∞ → Rbe a bounded-from-above function andf :X → Xbe a continuous mapping, such that
φ fx, t
−φx, t≥ν 1−M
x, fx, t
2.47
for allx∈Xandt >0.Ifνsatisfies the property
∀ε >0 ∃δ >0 :νx< δ⇒x < ε, 2.48
thenfhas a fixed point inX.
The reader is referred to the nice paper20for some discussion of Kirk’s problem on an extension of Caristi’s fixed point theorem.
References
1 A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy Sets and Systems, vol.
64, no. 3, pp. 395–399, 1994.
2 O. Kaleva and S. Seikkala, “On fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 12, no. 3, pp. 215–229, 1984.
3 I. Kramosil and J. Mich´alek, “Fuzzy metrics and statistical metric spaces,” Kybernetika, vol. 11, no. 5, pp. 336–344, 1975.
4 S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung, and S. M. Kang, “Coincidence point theorems and minimization theorems in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 88, no. 1, pp. 119–127, 1997.
5 Y. J. Cho, “Fixed points in fuzzy metric spaces,” Journal of Fuzzy Mathematics, vol. 5, no. 4, pp. 949–962, 1997.
6 J. X. Fang, “On fixed point theorems in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 46, no. 1, pp.
107–113, 1992.
7 M. Grabiec, “Fixed points in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 27, no. 3, pp. 385–389, 1988.
8 V. Gregori and A. Sapena, “On fixed-point theorems in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 125, no. 2, pp. 245–252, 2002.
9 O. Hadˇzi´c and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, vol. 536 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
10 D. Mihet¸, “On fuzzy contractive mappings in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 158, no. 8, pp. 915–921, 2007.
11 D. Mihet¸, “Fuzzyψ-contractive mappings in non-Archimedean fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 159, no. 6, pp. 739–744, 2008.
12 D. Mihet¸, “Fuzzy quasi-metric versions of a theorem of Gregori and Sapena,” Iranian Journal of Fuzzy Systems, vol. 7, no. 1, pp. 59–64, 2010.
13 S. N. Mishra, N. Sharma, and S. L. Singh, “Common fixed points of maps on fuzzy metric spaces,”
International Journal of Mathematics and Mathematical Sciences, vol. 17, no. 2, pp. 253–258, 1994.
14 I. Altun, “Some fixed point theorems for single and multi valued mappings on ordered non- archimedean fuzzy metric spaces,” Iranian Journal of Fuzzy Systems, vol. 7, no. 1, pp. 91–96, 2010.
15 L. B. ´Ciri´c, D. Mihet¸, and R. Saadati, “Monotone generalized contractions in partially ordered probabilistic metric spaces,” Topology and Its Applications, vol. 156, no. 17, pp. 2838–2844, 2009.
16 B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313–
334, 1960.
17 R. P. Agarwal, M. A. El-Gebeily, and D. O’Regan, “Generalized contractions in partially ordered metric spaces,” Applicable Analysis, vol. 87, no. 1, pp. 109–116, 2008.
18 V. Radu, “Some remarks on the probabilistic contractions on fuzzy Menger spaces,” Automation Computers Applied Mathematics, vol. 11, no. 1, pp. 125–131, 2002.
19 J. Caristi, “Fixed point theory and inwardness conditions,” in Applied Nonlinear Analysis, pp. 479–483, Academic Press, New York, NY, USA, 1979.
20 M. A. Khamsi, “Remarks on Caristi’s fixed point theorem,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 71, no. 1-2, pp. 227–231, 2009.
