Abstract
Mustafa and Sims [Fixed Point Theory Appl. 2009, Article ID 917175, 10, (2009)] generalized a concept of a metric space and proved fixed point theorems for mappings satisfying different contractive conditions. In this article, we extend and generalize the results obtained by Mustafa and Sims and prove common fixed point theorems for three maps in these spaces. It is worth mentioning that our results do not rely on continuity and commutativity of any mappings involved therein. We also introduce the notation of a generalized probabilistic metric space and obtain common fixed point theorem in the frame work of such spaces.
2000 Mathematics Subject Classification: 47H10.
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1. Introduction and Preliminaries
The study of fixed points of mappings satisfying certain contractive conditions has been at the center of vigorous research activity. Mustafa and Sims [1] generalized the concept of a metric space. Based on the notion of generalized metric spaces, Mustafa et al. [2–5] obtained some fixed point theorems for mappings satisfying different contractive conditions. Abbas and Rhoades [6] motivated the study of a common fixed point theory in generalized metric spaces. Recently, Saadati et al. [7] proved some fixed point results for contractive mappings in partially ordered G-metric spaces.
The purpose of this article is to initiate the study of common fixed point for three mappings in complete G-metric space. It is worth mentioning that our results do not rely on the notion of continuity, weakly commuting, or compatibility of mappings involved therein. We generalize various results of Mustafa et al. [3, 5].
Consistent with Mustafa and Sims [1], the following definitions and results will be needed in the sequel.
Definition 1.1. Let X be a nonempty set. Suppose that a mapping G : X × X × X → R + satisfies:
-
(a)
G(x, y, z) = 0 if and only if x = y = z,
-
(b)
0 < G(x, y, z) for all x, y ∈ X, with x ≠ y,
-
(c)
G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X, with z ≠ y,
-
(d)
G(x, y, z) = G(x, z, y) = G(y, z, x) = ⋯ (symmetry in all three variables), and
-
(e)
G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a ∈ X.
Then G is called a G-metric on X and (X, G) is called a G-metric space.
Definition 1.2. A G-metric is said to be symmetric if G(x, y, y) = G(y, x, x) for all x, y ∈ X.
Definition 1.3. Let (X, G) be a G-metric space. We say that {x n } is
-
(i)
a G-Cauchy sequence if, for any ε > 0, there is an n 0 ∈ N (the set of all positive integers) such that for all n, m, l ≥ n 0, G(x n , x m , x l ) < ε;
-
(ii)
a G-Convergent sequence if, for any ε > 0, there is an x ∈ X and an n 0 ∈ N, such that for all n, m ≥ n 0, G(x, x n , x m ) < ε.
A G-metric space X is said to be complete if every G-Cauchy sequence in X is convergent in X. It is known that {x n } converges to x ∈ (X, G) if and only if G(x m , x n , x) → 0 as n, m → ∞.
Proposition 1.4. Every G-metric space (X, G) will define a metric space (X, d G ) by
Definition 1.5. Let (X, G) and (X′, G′) be G-metric spaces and let f : (X, G) → (X′, G′) be a function, then f is said to be G-continuous at a point a ∈ X if and only if, given ε > 0, there exists δ > 0 such that x, y ∈ X; and G(a, x, y) < δ implies G′(f(a), f(x), f(y)) < ε. A function f is G-continuous at X if and only if it is G-continuous at all a ∈ X.
2. Common Fixed Point Theorems
In this section, we obtain common fixed point theorems for three mappings defined on a generalized metric space. We begin with the following theorem which generalize [[5], Theorem 1].
Theorem 2.1. Let f, g, and h be self maps on a complete G-metric space X satisfying
where and
for all x, y, z ∈ X. Then f, g, and h have a unique common fixed point in X. Moreover, any fixed point of f is a fixed point g and h and conversely.
Proof. Suppose x 0 is an arbitrary point in X. Define {x n } by x 3n+1= fx 3n , x 3n+2= gx 3n+1, x 3n+3= hx 3n+2for n ≥ 0. We have
for n = 0, 1, 2, ..., where
In case max{G(x 3n , x 3n+1, x 3n+2), G(x 3n+1, x 3n+2, x 3n+3)} = G(x 3n , x 3n+1, x 3n+2), we obtain that
If max{G(x 3n , x 3n+1, x 3n+2), G(x 3n+1, x 3n+2, x 3n+3)} = G(x 3n+1, x 3n+2, x 3n+3), then
which implies that G(x 3n+1, x 3n+2, x 3n+3) = 0, and x 3n+1= x 3n+2= x 3n+3and the result follows immediately.
Hence,
Similarly it can be shown that
and
Therefore, for all n,
Now, for any l, m, n with l > m > n,
The same holds if l = m > n and if l > m = n we have
Consequently G(x n , x m , x l ) → 0 as n, m, l → ∞. Hence {x n } is a G-Cauchy sequence. By G-completeness of X, there exists u ∈ X such that {x n } converges to u as n → ∞. We claim that fu = u. If not, then consider
where
On taking limit n → ∞, we obtain that
where
Thus
a contradiction. Hence, fu = u. Similarly it can be shown that gu = u and hu = u. To prove the uniqueness, suppose that if v is another common fixed point of f, g, and h, then
where
If U(u, v, v) = G(u, v, v), then
which gives that G(u, v, v) = 0, and u = v. Also for U(u, v, v) = G(v, u, u) we obtain
which gives that G(u, v, v) = 0 and u = v. Hence, u is a unique common fixed point of f, g, and h.
Now suppose that for some p in X, we have f(p) = p. We claim that p = g(p) = h(p), if not then in case when p ≠ g(p) and p ≠ h(p), we obtain
where
Now U(p, p, p) = G(p, gp, gp) gives
a contradiction. For U(p, p, p) = G(p, hp, hp), we obtain
a contradiction. Similarly when p ≠ g(p) and p = h(p) or when p ≠ h(p) and p = g(p), we arrive at a contradiction following the similar arguments to those given above. Hence, in all cases, we conclude that p = gp = hp. The same conclusion holds if p = gp or p = hp. □
Example 2.2. Let X = {0, 1, 2, 3} be a set equipped with G-metric defined by
and f, g, h : X → X be defined by
It may be verified that the mappings satisfy contractive condition (2.1) with contractivity factor equal to . Moreover, 0 is a common fixed point of mappings f, g, and h.
Corollary 2.3. Let f, g, and h be self maps on a complete G-metric space X satisfying
for all x, y, z ∈ X, where . Then f, g, and h have a unique common fixed point in X. Moreover, any fixed point of f is a fixed point g and h and conversely.
Proof. It follows from Theorem 2.1, that f m , g m and h m have a unique common fixed point p. Now f(p) = f(f m (p)) = f m+1(p) = f m (f(p)), g(p) = g(g m (p)) = g m+1(p) = g m (g(p)) and h(p) = h(h m (p)) = h m+1(p) = h m (h(p)) implies that f(p), g(p) and h(p) are also fixed points for f m , g m and h m . Now we claim that p = g(p) = h(p), if not then in case when p ≠ g(p) and p ≠ h(p), we obtain
which is a contradiction. Similarly when p ≠ g(p) and p = h(p) or when p ≠ h(p) and p = g(p), we arrive at a contradiction following the similar arguments to those given above. Hence in all cases, we conclude that, f(p) = g(p) = h(p) = p. It is obvious that every fixed point of f is a fixed point of g and h and conversely. □
Theorem 2.4. Let f, g, and h be self maps on a complete G-metric space X satisfying
where and
for all x, y, z ∈ X. Then f, g, and h have a unique common fixed point in X. Moreover, any fixed point of f is a fixed point g and h and conversely.
Proof. Suppose x 0 is an arbitrary point in X. Define {x n } by x 3n+1= fx 3n , x 3n+2= gx 3n+1, x 3n+3= hx 3n+2. We have
for n = 0, 1, 2, ..., where
Now if U(x 3n , x 3n+1, x 3n+2) = G(x 3n , x 3n+1, x 3n+2), then
Also if U(x 3n , x 3n+1, x 3n+2) = G(x 3n+1, x 3n+2, x 3n+3), then
which implies that G(x 3n+1, x 3n+2, x 3n+3) = 0, and x 3n+1= x 3n+2= x 3n+3and the result follows immediately.
Finally U(x 3n , x 3n+1, x 3n+2) = G(x 3n+2, x 3n+1, x 3n+1) + G(x 3n , x 3n+3, x 3n+3), implies
which further implies that
Thus,
where . Obviously 0 < λ < 1.
Hence,
Similarly it can be shown that
and
Therefore, for all n,
Following similar arguments to those given in Theorem 2.1, G(x n , x m , x l ) → 0 as n, m, l → ∞. Hence, {x n } is a G-Cauchy sequence. By G-completeness of X, there exists u ∈ X such that {x n } converges to u as n → ∞. We claim that fu = u. If not, then consider
where
On taking limit n → ∞, we obtain that
where
Thus
gives a contradiction. Hence, fu = u. Similarly it can be shown that gu = u and hu = u. To prove the uniqueness, suppose that if v is another common fixed point of f, g, and h, then
where
Hence,
which gives that G(u, v, v) = 0, and u = v. Therefore, u is a unique common fixed point of f, g, and h.
Now suppose that for some p in X, we have f(p) = p. We claim that p = g(p) = h(p), if not then in case when p ≠ g(p) and p ≠ h(p), we obtain
where
If U(p, p, p) = G(p, gp, gp), then
a contradiction.
Also for U(p, p, p) = G(p, gp, gp) + G(p, hp, hp), we obtain
a contradiction. If U(p, p, p) = G(p, hp, hp), then
a contradiction. Similarly when p ≠ g(p) and p = h(p) or when p ≠ h(p) and p = g(p), we arrive at a contradiction following the similar arguments to those given above. Hence, in all cases, we conclude that p = gp = hp. □
Corollary 2.5. Let f, g, and h be self maps on a complete G-metric space X satisfying
where and
for all x, y, z ∈ X. Then f, g, and h have a unique common fixed point in X. Moreover, any fixed point of f is a fixed point g and h and conversely.
Proof. It follows from Theorem 2.4 that f m , g m , and h m have a unique common fixed point p. Now f(p) = f(f m (p)) = f m+1(p) = f m (f(p)), g(p) = g(g m (p)) = g m+1(p) = g m (g(p)) and h(p) = h(h m (p)) = h m+1(p) = h m (h(p)) implies that f(p), g(p) and h(p) are also fixed points for f m , g m and h m .
We claim that p = g(p) = h(p), if not then in case when p ≠ g(p) and p ≠ h(p), we obtain
a contradiction. Similarly when p ≠ g(p) and p = h(p) or when p ≠ h(p) and p = g(p), we arrive at a contradiction following the similar arguments to those given above. Hence, in all cases, we conclude that, f(p) = g(p) = h(p) = p. □
Theorem 2.6. Let f, g, and h be self maps on a complete G-metric space X satisfying
where and
for all x, y, z ∈ X. Then f, g, and h have a common fixed point in X. Moreover, any fixed point of f is a fixed point g and h and conversely.
Proof. Suppose x 0 is an arbitrary point in X. Define {x n } by x 3n+1= fx 3n , x 3n+2= gx 3n+1, x 3n+3= hx 3n+2. We have
for n = 0, 1, 2, ..., where
Now if U(x 3n , x 3n+1, x 3n+2) = G(x 3n , x 3n+1, x 3n+1) + G(x 3n+2, x 3n+1, x 3n+1), then
Also if U(x 3n , x 3n+1, x 3n+2) = G(x 3n , x 3n+2, x 3n+2) + G(x 3n+1, x 3n+2, x 3n+2), then
Finally for U(x 3n , x 3n+1, x 3n+2) = G(x 3n , x 3n+3, x 3n+3) + G(x 3n+1, x 3n+3, x 3n+3) + G(x 3n+2, x 3n+3, x 3n+3), implies
implies that
Thus,
where . Obviously 0 < λ < 1.
Hence,
Similarly it can be shown that
and
Therefore, for all n,
Following similar arguments to those given in Theorem 2.1, G(x n , x m , x l ) → 0 as n, m, l → ∞. Hence, {x n } is a G-Cauchy sequence. By G-completeness of X, there exists u ∈ X such that {x n } converges to u as n → ∞. We claim that fu = gu = u. If not, then consider
where
On taking limit as n → ∞, we obtain that
where
Now for U(u, u, u) = 3G(fu, fu, fu), then
a contradiction. Hence, fu = gu = u. Also for U(u, u, u) = 3G(u, gu, gu),
a contradiction. Hence, fu = gu = u. Similarly it can be shown that gu = u and hu = u.
Now suppose that for some p in X, we have f(p) = p. We claim that p = g(p) = h(p), if not then in case when p ≠ g(p) and p ≠ h(p), we obtain
where
If U(p, p, p) = 3G(p, gp, gp), then
a contradiction. Also, U(p, p, p) = 3G(p, hp, hp) gives
a contradiction. Similarly when p ≠ g(p) and p = h(p) or when p ≠ h(p) and p = g(p), we arrive at a contradiction following the similar arguments to those given above. Hence in all cases, we conclude that p = gp = hp. □
Remark 2.7. Let f, g, and h be self maps on a complete G-metric space X satisfying (2.5). Then f, g and h have a unique common fixed point in X provided that .
Proof. Existence of common fixed points of f, g, and h follows from Theorem 2.6. To prove the uniqueness, suppose that if v is another common fixed point of f, g, and h, then
where
U(u, v, v) = 2G(v, u, u), implies that
which gives u = v. And U(u, v, v) = G(u, v, v), gives
U = v. Hence, u is a unique common fixed point of f, g, and h. □
Corollary 2.8. Let f, g, and h be self maps on a complete G-metric space X satisfying
where and
for all x, y, z ∈ X. Then f, g and h have a unique common fixed point in X. Moreover, any fixed point of f is a fixed point g and h and conversely.
Proof. It follows from Theorem 2.6, that f m , g m , and h m have a unique common fixed point p. Now f(p) = f(f m (p)) = f m+1(p) = f m (f(p)), g(p) = g(g m (p)) = g m+1(p) = g m (g(p)) and h(p) = h(h m (p)) = h m+1(p) = h m (h(p)) implies that f(p), g(p) and h(p) are also fixed points for f m , g m , and h m . Now we claim that p = g(p) = h(p), if not then in case when p ≠ g(p) and p ≠ h(p), we obtain
Now if U(p, gp, hp) = G(gp, p, p) + G(hp, p, p), then
a contradiction. Also if U(p, gp, hp) = G(p, gp, gp) + G(hp, gp, gp), then
a contradiction. Finally, if U(p, gp, hp) = G(p, hp, hp) + G(gp, hp, hp), then
a contradiction.
Also similarly when p ≠ g(p) and p = h(p) or when p ≠ h(p) and p = g(p), we arrive at a contradiction following the similar arguments to those given above. Hence, in all cases, we conclude that f(p) = g(p) = h(p) = p □
Example 2.9. Let X = [0, 1] and G(x, y, z) = max{|x - y|, |y - z|, |z - x|} be a G-metric on X. Define f, g, h : X → X by
and
Note that f, g and h are discontinuous maps. Also , , , , and , , which shows that f, g and h does not commute with each other.
Note that for ,
and
Now
For U(x, y, z) = G(x, fx, fx) + G(y, fx, fx) + G(z, fx, fx), we obtain
In case U(x, y, z) = G(x, gy, gy) + G(y, gy, gy) + G(z, gy, gy), then
And for U(x, y, z) = G(x, hz, hz) + G(y, hz, hz) + G(z, hz, hz), we have
Thus, (2.5) is satisfied for .
For
and
Now,
For U(x, y, z) = G(x, fx, fx) + G(y, fx, fx) + G(z, fx, fx), we obtain
In case, U(x, y, z) = G(x, gy, gy) + G(y, gy, gy) + G(z, gy, gy), then
And U(x, y, z) = G(x, hz, hz) + G(y, hz, hz) + G(z, hz, hz) gives that
Hence (2.5) is satisfied for .
Now for , ,
and
Also
Now for U(x, y, z) = G(x, fx, fx) + G(y, fx, fx) + G(z, fx, fx), then
In case U(x, y, z) = G(x, gy, gy) + G(y, gy, gy) + G(z, gy, gy), then
And for U(x, y, z) = G(x, hz, hz) + G(y, hz, hz) + G(z, hz, hz), we have
Thus, (2.5) is satisfied for .
For and
and
Now for U(x, y, z) = G(x, fx, fx) + G(y, fx, fx) + G(z, fx, fx), we obtain
If U(x, y, z) = G(x, gy, gy) + G(y, gy, gy) + G(z, gy, gy), then
For U(x, y, z) = G(x, hz, hz) + G(y, hz, hz) + G(z, hz, hz), we have
Thus, (2.5) is satisfied for . So all the conditions of Theorem 2.6 are satisfied for all x, y, z ∈ X. Moreover, 0 is the unique common fixed point of f, g, and h.
3. Probabilistic G-Metric Spaces
K. Menger introduced the notion of a probabilistic metric space in 1942 and since then the theory of probabilistic metric spaces has developed in many directions [8]. The idea of Menger was to use distribution functions instead of nonnegative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to situations when we do not know exactly the distance between two points, but we know probabilities of possible values of this distance. A probabilistic generalization of metric spaces appears to be interest in the investigation of physical quantities and physiological thresholds. It is also of fundamental importance in probabilistic functional analysis.
Throughout this article, the space of all probability distribution functions (d.f.'s) is denoted by Δ+ = {F : ℝ ∪ {-∞, +∞} → [0, 1]: F is left-continuous and nondecreasing on ℝ, F(0) = 0 and F(+∞) = 1} and the subset D + ⊆ Δ+ is the set D + = {F ∈ Δ+ : l - F(+∞) = 1}. Here, l - f(x) denotes the left limit of the function f at the point x, . The space Δ+ is partially ordered by the usual pointwise ordering of functions, i.e., F ≤ G if and only if F(t) ≤ G(t) for all t in ℝ. The maximal element for Δ+ in this order is the d.f. given by
Definition 3.1. [8] A mapping T : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if T satisfies the following conditions
-
(a)
T is commutative and associative;
-
(b)
T is continuous;
-
(c)
T(a, 1) = a for all a ∈ [0, 1];
-
(d)
T(a, b) ≤ T(c, d) whenever a ≤ c and c ≤ d, and a, b, c, d ∈ [0, 1].
Two typical examples of continuous t-norm are T P (a, b) = ab and T M (a, b) = Min(a, b).
Now t-norms are recursively defined by T 1 = T and
for n ≥ 2 and x i ∈ [0, 1], for all i ∈ {1, 2, ..., n + 1}.
We say that a t-norm T is of Hadžić type if the family {T n } n∈ℕis equicontinuous at x = 1, that is,
T M is a trivial example of a t-norm of Hadžić type, but T P is not of Hadžić type (see [9–11]).
Definition 3.2. A Menger Probabilistic Metric space (briefly, Menger PM-space) is a triple , where X is a nonempty set, T is a continuous t-norm, and is a mapping from X × X into D + such that, if F x, y denotes the value of at the pair (x, y), the following conditions hold: for all x, y, z in X,
(PM1) F x, y (t) = 1 for all t > 0 if and only if x = y;
(PM2) F x, y (t) = F y, x (t);
(PM3) F x, z (t + s) ≥ T(F x, y (t), F y, z (s)) for all x, y, z ∈ X and t, s ≥ 0.
Using PM-space we define probabilistic G-metric spaces.
Definition 3.3. A Menger Probabilistic G-Metric space (briefly, Menger PGM-space) is a triple , where X is a nonempty set, T is a continuous t-norm, and is a mapping from X × X × X into D + such that, if G x, y, z denotes the value of at the triple (x, y, z), the following conditions hold: for all x, y, z in X,
(PGM1) G x, y, z (t) = 1 for all t > 0 if and only if x = y = z;
(PGM2) G x, y, z (t) < 1 for all t > 0 if and only if x ≠ y;
(PGM3) G x, y, z (t) = G y, x, z (t) = G y, z, x (t) = ⋯;
(PGM4) G x, y, z (t + s) ≥ T(G x, a, a (t), G a, y, z (s)) for all x, y, z, a ∈ X and t, s ≥ 0.
Definition 3.4. A probabilistic G-metric is said to be symmetric if G x, y, y (t) = G y, x, x (t) for all x, y ∈ X.
Example 3.5. Let be a PM-space. Define
Then, is a PGM-space.
Now, we generalize the definition of G- Cauchy and G- convergent (see Definition 1.3) to Menger PGM-spaces.
Definition 3.6. Let be a Menger PGM-space.
-
(1)
A sequence {x n } n in X is said to be PG-convergent to x in X if, for every ε > 0 and λ > 0, there exists positive integer N such that whenever m, n ≥ N.
-
(2)
A sequence {x n } n in X is called PG-Cauchy sequence if, for every ε > 0 and λ > 0, there exists positive integer N such that whenever n, m, l ≥ N.
-
(3)
A Menger PM-space is said to be complete if and only if every PG-Cauchy sequence in X is PG-convergent to a point in X.
Definition 3.7. Let be a Menger PGM space. For each p in X and λ > 0, the strong λ-neighborhood of p is the set
and the strong neighborhood system for X is the union where .
4. Fixed Point Theorems in PGM-Spaces
Lemma 4.1. Let be a Menger PGM-space with T of Hadžić-type and {x n } be a sequence in X such that, for some k ∈ (0, 1),
Then, {x n } is a PG-Cauchy sequence.
Proof. Let T be Hadžić-type, then
Since is a Menger PGM-space, we have then there exists a t 0 > 0 such that , then
Let t > 0. Since the series is convergent, there exists n 1 ∈ ℕ such that for n ≥ n 0 we have . Then, for all n ≥ n 1 and m, l ∈ ℕ (put m + l - 1 = N), we have
Hence, the sequence {x n } is PG-Cauchy. □
It is not difficult to see that more general fixed point results in probabilistic G-metric spaces can be proved in this manner. For example, we also have the following generalization of Theorem 2.1.
Theorem 4.2. Let f, g, and h be self maps on a complete PGM-space satisfying
where and
for all x, y, z ∈ X. Then f, g, and h have a unique common fixed point in X. Moreover, any fixed point of f is a fixed point g and h and conversely.
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The authors are thankful to the anonymous referees for their critical remarks which helped greatly to improve the presentation of this article.
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Abbas, M., Nazir, T. & Saadati, R. Common fixed point results for three maps in generalized metric space. Adv Differ Equ 2011, 49 (2011). https://doi.org/10.1186/1687-1847-2011-49
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DOI: https://doi.org/10.1186/1687-1847-2011-49