Research Article

# Nonlinear Contraction Theorems in Fuzzy Spaces

ABSTRACT

In this study, fuzzy metric and normed space are considered and some fixed point theorems in these spaces are proved. In this study at first two fixed point theorems in nonlinear case in the fuzzy metric spaces are proved then an nonlinear contraction theorem in the fuzzy normed spaces is proved.

 Services Related Articles in ASCI Similar Articles in this Journal Search in Google Scholar View Citation Report Citation Science Alert

 How to cite this article: M. Mohamadi, R. Saadati, A. Shahmari and S.M. Vaezpour, 2009. Nonlinear Contraction Theorems in Fuzzy Spaces. Journal of Applied Sciences, 9: 1397-1400. DOI: 10.3923/jas.2009.1397.1400 URL: https://scialert.net/abstract/?doi=jas.2009.1397.1400

INTRODUCTION

The concept of fuzzy sets was introduced initially by Zadeh (1965). To use this concept in topology and analysis many researchers have developed a theory of fuzzy sets and applications. George and Veeramani (1994) introduced the concept of fuzzy metric spaces. In this study we state some of the basic facts about fuzzy metric and normed spaces.

Definition 1: A binary operation *[0.1]x[0,1]→[0,1] is ontinuous t-norm if * is satisfying the following conditions:

 • * is commutative and associative • * is continuous • a*1 = a for all aε[0,1] • a*b≤c*d whenever a≤c and b≤d and a,b,c,d ε[0,1]

Two typical examples of continuous t-norm are a*b = ab and a*b = min (a,b).

Definition 2: A triple (X, M, *) is called a fuzzy metric space if X is an arbitrary (non-empty) set, * is a continuous t-norm, and M is a fuzzy set on X2x(0,∞), satisfying the following conditions for each x, y zεX and t.s>0:

 • M (x, y, t)>0 • M (x, y, t) = 1 if and only if x = y • M (x, y, t) = M (y, x, t) • M (x, y, t) * M (y, z, t+s)≤M(y, z, t+s) • M (x, y):]0,∞)→]0,1] is continuous

Example 1: Let (X.d) be a metric space, define a*b = ab or a*b = min (a,b) and

which is called the standard fuzzy metric induced by metric d.

Let (X, M, *) be a fuzzy metric space . For t>0, the open ball B(x, r, t) with center x ε X and radius 0<r<1 is defined by:

Let (X, M, *) be a fuzzy metric space. Let τ be the set of all A⊂X with x ε A if and only if there exist t>0 and 0<r<1 such that B(x, r, t) ⊂ A.

Then τ is a topology on x (induced by the fuzzy metric M). This topology is Hausdorff and first countable. A sequence {xn} in X converges to x if and only M(xn, x, t)→1 as n→∞, for each t>0. It is called a Cauchy sequence if for each 0<ε<1 and t>0, there exits n0 ε N such that M(xn, xm, t)>1-ε for each n, m≥n0. The fuzzy metric space (x, M, *) is said to be complete if every Cauchy sequence is convergent.

Lemma 1: Let (X, M, *) be fuzzy metric space. Then, M (x, y, t) is non-decreasing with respect to t, for all x, y in X.

Definition 3: The triple (X, N, *) is said to be a fuzzy normed space if X is a vector space, * is a continuous t-norm and N is a fuzzy set on Xx(0,∞) satisfying the following conditions for every x, y ε X and t, s>0:

 • N (x, t)>0 • N (x, t) = 1 if and only if x = 0 • • N (x, t)*N(y, s)≤N(x+y, t+s) • N (x) : (0, ∞)→[0, 1] is continuous

where, in (c), α is in the scalar field of X. In this case N is called a fuzzy norm.

Lemma 2: Let (X, N,*) be a fuzzy normed space. If define:

 M (x, y, t) = N (x-y, t)

then, M is a fuzzy metric on X, which is said to be induced by the fuzzy norm N.

The fuzzy normed space (X, N, *) is said to be a fuzzy Banach space whenever X is complete with respect to the fuzzy metric induced by fuzzy norm.

Lemma 3: Let (X, M, *) be fuzzy metric space and define

Xλ,M:X2→R+ ∪ {0} by

 Eλ,M (x, y) = inf{t>0:M(x, y, t)>1-λ}

for each λε]0,1[ and x, y ε X. Then

(i) For any με]0,1[ there exists λε]0,1[ such that:

 Eμ,M (x1, xn)≤Eλ, M(x1, x2)+...+Eμ,M (xn-1, xn)

for any x1,...,xn εX

(ii) The sequence {xn}nεN is convergent with respect to fuzzy metric M if and only if Eλ,M(xn, x)→0. Also the sequence {xn} is a Cauchy sequence with respect to fuzzy metric M if and only if it is a Cauchy sequence with Eλ,M.

Proof: For (i), for every μ ε]0,1[, there is a λε]0,1[ such that (1-λ)*...*(1-λ)>(1-μ). By the triangular inequality:

for every δ>0, which implies that:

Definition 4: The fuzzy metric space (X, M, *) said that has the property (C), if it satisfies the following condition:

M (x, y, t) = C, for all t>0 implies C = 1

Lemma 4: Let the function φ(t) satisfies the following condition:

(φ) φ(t):[0,∞] is nondecreasing and

for all t>0, when φn(t) denotes the n-th iterative function of φ (t), then φ(t)<t for all t>0.

THE MAIN RESULTS

Theorem 1: Let {An} be a sequence of mappings Ai of a complete fuzzy metric space (X, M, *), which this space has the property (C), into itself such that, for any two mappings Ai, Aj:

for some m, x, yεX and for all t>0.

Here, φi,j:[0, ∞)→[0, ∞) is a function such that φi,j (t)<φ(t) for i, j = 1,2,... and the function φ(t): [0, ∞)→on to [0,∞) is strictly increasing and satisfies condition φ. Then the sequence {An} has a unique common fixed point in X.

Proof: Let x0 be an arbitrary point in X and define a sequence {xn} in X by:

Then

and:

and so on. By induction,

which implies

for every λε]0,1[.

Now, showed that {xn} is a Cauchy sequence. For every με]0,1[, there exists γε]0,1[such that:

as m, n > 1. Since, X is complete, there is x ε X such that

Now is proved that x is a periodic point of Ai for any i = 1, 2, .... Notice:

as n→∞. Thus and is got

To show uniqueness, assume that y ≠ x is another periodic point of Ai.

Then:

On the other hand, by Lemma 1 implies that:

Hence, M(x, y, t) = C for all t>0. Since, M has the property (C), it follows:

That C = 1, i.e., X = Y. Also:

i.e., Ai (x) is also a periodic point of Ai. Therefore, x = Ai (x), i.e., x is a unique common fixed periodic point of the mappings An for n = 1,2... . This completes the proof.

Theorem 2: Let (X, M, *) be a complete fuzzy metric space, let (X, M, *) has the property (C) and let f, g: X→X be maps that satisfy the following conditions:

 • g (x) ⊆ f(X) • f is continuous • M (g(x), g(y), φ(t))≥M(f(x), f(y),t) for all x, y ε X where, the function is strictly increasing and satisfies condition φ

Then f and g have a unique common fixed point provided f and g commute.

Proof: Let x0 εX. By (a) there is x1 such that f(xl) = g (x0). By induction, we can define a sequence {xn}n such that f(xn) = g(xn-1). By induction again:

for n = 12,..., which implies that:

for every λε]0,1[.

Now, have been showed that {f(xn)} is a Cauchy sequence. For every με]0,1[, there exists γε]0,1[ such that, for m≥n,

as m,n→∞. Since, X is complete, there exists y ε X such that limn→∞f(xn) = y. So, g(xn-1) = f(xn) tends to y. It can be seen from (c) that the continuity of f implies that to g.

Thus {g(f(xn))}n converges to g(y). However, g(f(xn)) = f(g(xn)) by the commutativity of f and g. Thus f(g (xn)) converges to f (y). Because the limits are unique, f(y) = g(y). So, f(f(y)) = f(g(y)) by commutativity and:

On the other hand, Lemma 1 implies that:

Hence, M (g(y), g(g(y), t) = C for all t>0. Since, M has the property (C), it follows that C = 1, i.e., g(y) = g(g(y)). Thus, g (y) = g(g(y)) = f (g(y)). So, g(y) is a common fixed point of f and g.

If y and z are two fixed points common to f and g, then:

On the other hand, by Lemma 1

Hence, M(y, z, t) = C for all t>0. Since, M has the property (C), it follows that C = 1, i.e., y = z.

Theorem 3: Let W be a closed and convex subset of a fuzzy Banach space (V, N, *) and f: W→W a mapping which satisfies the condition:

 (1)

for all x, yεW and for all t>0. The function is strictly increasing and satisfies condition φ. Then f has at least a fixed point.

Proof: Let x0 in W be arbitrary and let a sequence {xn} be defined by:

xn+1 = [xn + f(xn)]/2 (n = 0,1,2,...)

For this sequence

and hence,

 (2)

Therefore, for x = xn-1 and y = xn the condition (Eq. 1) states:

By condition (Eq. 2):

Hence,

By Lemma 2 as proof of Theorem 1 is concluded that {xn} is Cauchy sequence in W and converges to some uεW. Since:

Hence,

 limn→∞fxn = u

Now, let us put in Eq. 1 x = u and y = xn and use Eq. 2. Then:

If now n tend to infinity one has

 N (u-fu,φ(t)) = 1

which implies that fu = u and this theorem is established.

REFERENCES
1:  George, A. and P. Veeramani, 1994. On some result in fuzzy metric space. Fuzzy Sets Syst., 64: 395-399.
CrossRef  |

2:  Zadeh, L.A., 1965. Fuzzy sets. Inform. Control, 8: 338-353.