Abstract
The aim of this paper is to present some common fixed point results for six selfmappings satisfying generalized weakly (ψ,φ)contractive condition in the setup of partially ordered Gmetric spaces. Our results extend and generalize the comparable results in the work of Abbas from the context of ordered metric spaces to the setup of ordered Gmetric spaces. Also, our results are supported by an example.
Keywords:
Common fixed point; Generalized weakly contraction; Generalized metric space; Partially ordered set; Weak annihilator map; Dominating mapIntroduction and preliminaries
Alber and GuerreDelabrere [1] defined weakly contractive mappings on Hilbert spaces as follows:
Definition 1.1
A mapping f:X→X is said to be a weakly contractive mapping if
where x,y∈X and φ: [0,∞)→ [0,∞) is a continuous and nondecreasing function such that φ(t) = 0 if and only if t = 0.
Theorem 1.2
[2] Let (X,d) be a complete metric space and f:X→X be a weakly contractive mapping. Then f has a unique fixed point.
Recently, Zhang and Song [3] have introduced the concept of a generalized φweak contractive condition and obtained a common fixed point for two maps.
Definition 1.3
Two mappings T,S:X→X are called generalized φweak contractions if there exists a lower semicontinuous function φ: [ 0,∞)→ [ 0,∞) with φ(0)=0 and φ(t)>0 for all t>0 such that
for all x,y∈X, where
Zhang and Song proved the following theorem.
Theorem 1.4
Let (X,d) be a complete metric space and T,S:X→X be generalized φweak contractive mappings where φ: [ 0,∞)→ [ 0,∞) is a lower semicontinuous function with φ(0) = 0 and φ(t)>0 for all t>0. Then, there exists a unique point u∈X such that u = Tu = Su.
Dorić [4], Moradi et al. [5], Abbas and Dorić [6], and Razani et al. [7] obtained some common fixed point theorems which are extensions of the result of Zhang and Song in the framework of complete metric spaces. Also, in these years many authors have focused on different contractive conditions in complete metric spaces with a partially order and have obtained some common fixed point theorems. For more details on fixed point theory, its applications, comparison of different contractive conditions and related results in ordered metric spaces we refer the reader to [815] and the references mentioned therein.
The concept of a generalized metric space, or a Gmetric space, was introduced by Mustafa and Sims [16]. In recent years, many authors have obtained different fixed point theorems for mappings satisfying various contractive conditions on Gmetric spaces (see e.g., [9,1734]).
Definition 1.5
[16] (Gmetric space) Let X be a nonempty set and G:X×X×X→R^{+} be a function satisfying the following properties:
(G1) G(x,y,z)=0 iff x = y = z;
(G2) 0<G(x,x,y), for all x,y∈X with x≠y;
(G3) G(x,x,y)≤G(x,y,z), for all x,y,z∈X with z≠y;
(G4) G(x,y,z)=G(x,z,y)=G(y,z,x)=⋯, (symmetry
in all three variables);
(G5) G(x,y,z)≤G(x,a,a)+G(a,y,z), for all x,y,z,a∈X
(rectangle inequality).
Then the function G is called a Gmetric on X and the pair (X,G) is called a Gmetric space.
Definition 1.6
[16] Let (X,G) be a Gmetric space and let {x_{n}} be a sequence of points in X. A point x∈X is said to be the limit of the sequence {x_{n}} and if and one says that the sequence {x_{n}} is Gconvergent to x. Thus, if x_{n}→x in a Gmetric space (X,G), then for any ε>0, there exists a positive integer N such that G(x,x_{n},x_{m})<ε, for all n,m≥N.
Definition 1.7
[16] Let (X,G) be a Gmetric space. A sequence {x_{n}} is called GCauchy if for every ε>0, there is a positive integer N such that G(x_{n},x_{m},x_{l})<ε, for all n,m,l≥N, that is, if G(x_{n},x_{m},x_{l})→0, as n,m,l→∞.
Lemma 1.8
[16] Let (X,G) be a Gmetric space. Then the following are equivalent:
(1) {x_{n}} is Gconvergent to x.
(2) G(x_{n},x_{n},x)→0, as n→∞.
(3) G(x_{n},x,x)→0, as n→∞.
Lemma 1.9
[35] If (X,G) is a Gmetric space, then {x_{n}} is a GCauchy sequence if and only if for every ε>0, there exists a positive integer N such that G(x_{n},x_{m},x_{m})<ε, for all m>n≥N.
Definition 1.10
[16] A Gmetric space (X,G) is said to be Gcomplete if every GCauchy sequence in (X,G) is convergent in X.
Definition 1.11
[16] Let (X,G) and (X^{′},G^{′}) be two Gmetric spaces. A function f:X→X^{′} is Gcontinuous at a point x∈X if and only if it is Gsequentially continuous at x, that is, whenever {x_{n}} is Gconvergent to x, {f(x_{n})} is G^{′}convergent to f(x).
Definition 1.12
A Gmetric on X is said to be symmetric if G(x,y,y)=G(y,x,x), for all x,y∈X.
The concept of an altering distance function was introduced by Khan et al. [36] as follows.
Definition 1.13
The function ψ : [ 0,∞)→ [ 0,∞) is called an altering distance function if the following conditions hold:
1. ψ is continuous and nondecreasing.
2. ψ(t)=0 if and only if t = 0.
Definition 1.14
[8] Let (X,≼) be a partially ordered set. A mapping f is called a dominating map on X if x≼fx, for each x in X.
Example 1.15
[8] Let X = [ 0,1] be endowed with the usual ordering. Let f:X→X be defined by Then, , for all x∈X. Thus, f is a dominating map.
Example 1.16
[8] Let X = [ 0,∞) be endowed with the usual ordering. Let f:X→X be defined by for x∈ [ 0,1) and fx = x^{n} for x∈ [ 1,∞), for any positive integer n. Then for all x∈X, x≤fx; that is, f is a dominating map.
A subset W of a partially ordered set X is said to be well ordered if every two elements of W be comparable [8].
Definition 1.17
[8] Let (X,≼) be a partially ordered set. A mapping f is called a weak annihilator of g if fgx≼x for all x∈X.
Jungck in [37] introduced the following definition.
Definition 1.18
[37] Let (X,d) be a metric space and f,g:X→X be two mappings. The pair (f,g) is said to be compatible if and only if , whenever {x_{n}} is a sequence in X such that , for some t∈X.
Definition 1.19
[38,39] Let (X,G) be a Gmetric space and f,g:X→X be two mappings. The pair (f,g) is said to be compatible if and only if , whenever {x_{n}} is a sequence in X such that , for some t∈X.
Definition 1.20
[40] Let f and g be two self mappings of a metric space (X,d). The f and g are said to be weakly compatible if for all x∈X, the equality fx = gx implies fgx = gfx.
Let X be a nonempty set and f:X→X be a given mapping. For every x∈X, let f^{−1}(x)={u∈X:fu = x}.
Definition 1.21
Let (X,≼) be a partially ordered set and f,g,h:X→X be mappings such that fX⊆hX and gX⊆hX. The ordered pair (f,g) is said to be partially weakly increasing with respect to h if for all x∈X, fx≼gy, for all y∈h^{−1}(fx) [41].
Since we are motivated by the work in [8] in this paper, we prove some common fixed point theorems for nonlinear generalized (ψ,φ)weakly contractive mappings in partially ordered Gmetric spaces.
Main results
Abbas et al. [8] proved the following theorem.
Theorem 2.1
Let (X,≼,d) be an ordered complete metric space. Let f, g, S and T be selfmaps on X, (T,f) and (S,g) be partially weakly increasing with f(X)⊆T(X) and g(X)⊆S(X), dominating maps f and g be weak annihilators of T and S, respectively. Suppose that there exist altering distance functions ψ and φ such that for every two comparable elements x,y∈X,
is satisfied where
If for a nondecreasing sequence {x_{n}} with x_{n}≼y_{n} for all n, y_{n}→u implies that x_{n}≼u and either of the following:
(a) (f,S) are compatible, f or S is continuous, and (g,T) are weakly compatible or
(b) (g,T) are compatible, g or T is continuous, and (f,S) are weakly compatible,
then f, g, S, and T have a common fixed point. Moreover, the set of common fixed points of f, g, S and T is well ordered if and only if f, g, S, and T have one and only one common fixed point.
Let (X,≼,G) be an ordered Gmetric space and f,g,h,R,S,T:X→X be six self mappings. In the rest of this paper, unless otherwise stated, for all x,y,z∈X, let
Our first result is the following.
Theorem 2.2
Let (X,≼,G) be a partially ordered complete Gmetric space. Let f,g,h,R,S,T:X→X be the six mappings such that f(X)⊆R(X), g(X)⊆S(X), h(X)⊆T(X) and dominating maps f, g, and h are weak annihilators of R, S, and T, respectively. Suppose that for every three comparable elements x,y,z∈X,
where ψ,φ: [ 0,∞)→ [ 0,∞) are altering distance functions. Then, f, g, h, R, S, and T have a common fixed point in X provided that for a nondecreasing sequence {x_{n}} with x_{n}≼y_{n} for all n, y_{n}→u implies that x_{n}≼u and either of the following:
(i) One of g or R and one of f or T are continuous, the pairs (f,T) and (g,R) are compatible, and the pair (h,S) is weakly compatible or
(ii) One of h or S and one of f or T are continuous, the pairs (f,T) and (h,S) are compatible, and the pair (g,R) is weakly compatible or
(iii) One of g or R and one of h or S are continuous, the pairs (g,R) and (h,S) are compatible, and the pair (f,T) is weakly compatible.
Moreover, the set of common fixed points of f, g, h, R, S, and T is well ordered if and only if f, g, h, R, S, and T have one and only one common fixed point.
Proof 2.3
Let x_{0}∈X be an arbitrary point. Since f(X)⊆R(X), we can choose x_{1}∈X such that fx_{0}=Rx_{1}. Since g(X)⊆S(X), we can choose x_{2}∈X such that gx_{1}=Sx_{2}. Also, as h(X)⊆T(X), we can choose x_{3}∈X such that hx_{2}=Tx_{3}.
Continuing this process, we can construct a sequence {z_{n}} defined by
and
for all n≥0.
Now, since f, g and h are dominating and f, g, and h are weak annihilators of R, S and T, we obtain that
By continuing this process, we get
We will complete the proof in three steps.
Step I. We will prove that
Define G_{k}=G(z_{k},z_{k+1},z_{k+2}). Suppose for some k_{0}. Then, . Consequently, the sequence {z_{k}} is constant, for k≥k_{0}. Indeed, let k_{0}=3n. Then z_{3n}=z_{3n+1}=z_{3n+2}, and we obtain from (1),
where
Now from (2),
and so, φ(G(z_{3n+1},z_{3n+2},z_{3n+3}))=0, that is, z_{3n+1}=z_{3n+2}=z_{3n+3}.
Similarly, if k_{0}=3n+1 or k_{0}=3n+2, one can easily obtain that z_{3n+2}=z_{3n+3}=z_{3n+4} and z_{3n+3}=z_{3n+4}=z_{3n+5}, and so the sequence {z_{k}} is constant (for k≥k_{0}), and is a common fixed point of R,S, T, f,g, and h.
Suppose
for all k. We prove that for each k = 1,2,3,⋯
Let k = 3n. Since x_{k−1}≼x_{k}, using (1) we obtain that
where
Since ψ is a nondecreasing function from (5), we get
If for an n≥0, G(z_{3n+1},z_{3n+2},z_{3n+3})>G(z_{3n},z_{3n+1}, z_{3n+2})>0, then
Therefore, (5) implies that
which is only possible when G(z_{3n+1},z_{3n+2},z_{3n+3})=0. This is a contradiction to (3). Hence, G(z_{3n+1},z_{3n+2},z_{3n+3}) ≤G(z_{3n},z_{3n+1},z_{3n+2}) and
Therefore, (4) is proved for k = 3n. Similarly, it can be shown that
and
Hence, we conclude that {G(z_{k},z_{k+1},z_{k+2})} is a nondecreasing sequence of nonnegative real numbers. Thus, there is an r≥0 such that
Since
letting k→∞ in (10), we get
Letting n→∞ in (5) and using (9) and (11) and the continuity of ψ and φ, we get ψ(r)≤ψ(r)−φ(r)≤ψ(r) and hence φ(r)=0. This gives us
from our assumptions about φ. Also, from Definition 1.5, part (G3), we have
Step II. We will show that {z_{n}} is a GCauchy sequence in X. Therefore, we will show that for every ε>0, there exists a positive integer k such that for all m,n≥k, G(z_{m},z_{n},z_{n})<ε. Suppose the above statement is false. Then there exists ε>0 for which we can find subsequences {z_{m(k)}} and {z_{n(k)}} of {z_{n}} such that n(k)>m(k)≥k and
(a) m(k)=3t and n(k)=3t^{′}+1, where t and t^{′} are nonnegative integers.
(b)
(c) n(k) is the smallest number such that the condition (b) holds; i.e.,
From rectangle inequality and (15), we have
Making k→∞ in (16) from (12) and (15), we conclude that
Again, from rectangle inequality,
and
Hence, in (18) and (19), if k→∞, using (12), (14), and (17), we have
On the other hand,
and
Hence, in (21) and (22), if k→∞ is from (13), (17), and (20), we have
In a similar way, we have
and
and therefore, from (24) and (25) by taking limit when k→∞, using (13) and (20), we get that
Also,
and
Hence in (27) and (28), if k→∞ from (13), (23), and (25), we have
Also,
and
So from (13), (26), (29), and (30), we have
Finally,
and
Hence in (33) and (34), if k→∞ and by using (12) and (32), we have
Since x_{m(k)}≼x_{n(k)}≼x_{n(k)+1}, putting x = x_{m(k)}, y = x_{n(k)}, and z = x_{n(k)+1} in (1) for all k≥0, we have
where
Now, from (13), (19), (26), and (35), if k→∞ in (36), we have
Hence, ε = 0, which is a contradiction. Consequently, {z_{n}} is a GCauchy sequence.
Step III. We will show that f, g, h, R, S, and T have a common fixed point.
Since {z_{n}} is a GCauchy sequence in the complete Gmetric space X, there exists z∈X such that
and
Let (i) holds. Assume that R and T are continuous and let the pairs (f,T) and (g,R) are compatible. This implies that
and
Since
by using (1) we obtain that
where
as n→∞.
On taking the limit as n→∞ in (43), we obtain that
and hence, Tz = Rz = z.
Since x_{3n+1}≼x_{3n+2}≼hx_{3n+2} and hx_{3n+2}→z, as n→∞, we have x_{3n+1}≼x_{3n+2}≼z. Therefore, from (1),
where
as n→∞.
If in (45) n→∞, we obtain that
hence fz = z.
Since x_{3n+2}≼hx_{3n+2} and hx_{3n+2}→z, as n→∞, we have x_{3n+2}≼z. Hence from(1),
where
as n→∞.
Making n→∞ in (47), we obtain that
which implies that gz = z.
Since g(X)⊆S(X), there exists a point w∈X such that z = gz = Sw. Suppose that hw≠Sw. Since z≼gz = Sw≼gSw≼w, we have z≼w. Hence, from (1), we obtain that
where
as n→∞.
On taking the limit as n→∞ in (49), we obtain that
which yields that hw = z.
Now, Since h and S are weakly compatible, we have hz = hSw = Shw = Sz. Thus, z is a coincidence point of h and S.
Now, we are ready to show that hz = z.
Since x_{3n}≼fx_{3n} and fx_{3n}→z, as n→∞, we have x_{3n}≼z. Hence, from (1),
where
as n→∞.
Letting n→∞ in (51), we obtain that
hence hz = z. Therefore, fz = gz = hz = Rz = Sz = Tz = z.
Similarly, the result follows when (ii) or (iii) hold.
Suppose that the set of common fixed points of f, g, h, R, S, and T is well ordered. We claim that common fixed point of f, g, h, R, S, and T is unique. Assume on contrary that fu = gu = hu = Ru = Su = Tu = u, fv = gv = hv = Rv = Sv = Tv = v, and u≠v. By using (1), we obtain
where
On the other hand, as v and u are comparable,
where
From (53) and (54),
Therefore, φ(max{G(u,v,v),G(v,u,u)})=0 which yields that u = v is a contradiction. Conversely, if f, g, h, R, S, and T have only one common fixed point then, clearly, the set of common fixed points of f, g, h, R, S, and T is well ordered.
We assume that
Taking f = g = h in Theorem 2.2, we obtain the following common fixed point result in corollary.
Corollary 2.4
Let (X,≼,G) be a partially ordered complete Gmetric space. Let f,R,S,T:X→X be four mappings such that f(X)⊆R(X)∪S(X)∪T(X) and dominating map f is a weak annihilator of R, S, and T. Suppose that for every three comparable elements x,y,z∈X,
where ψ,φ: [ 0,∞)→ [ 0,∞) are altering distance functions. Then, f, R, S, and T have a common fixed point in X provided that for a nondecreasing sequence {x_{n}} with x_{n}≼y_{n} for all n, y_{n}→u implies that x_{n}≼u and either of the following:
(i) One of f or R and one of f or T are continuous, the pairs (f,T), and (f,R) are compatible, and the pair (f,S) is weakly compatible or
(ii) One of f or S and one of f or T are continuous, the pairs (f,T), and (f,S) are compatible, and the pair (f,R) is weakly compatible or
(iii) One of f or R and one of f or S are continuous, the pairs (f,R), and (f,S) are compatible, and the pair (f,T) is weakly compatible.
Moreover, the set of common fixed points of f, R, S, and T is well ordered if and only if f, R, S, and T have one and only one common fixed point.
Let
Taking T = R = S in Theorem 2.2, we obtain the following common fixed point result.
Corollary 2.5
Let (X,≼,G) be a partially ordered complete Gmetric space. Let f,g,h,T:X→X be four mappings such that f(X)∪g(X)∪h(X)⊆T(X) and dominating maps f, g, and h are weak annihilators of T. Suppose that for every three comparable elements x,y,z∈X,
where ψ,φ: [ 0,∞)→ [ 0,∞) are altering distance functions. Then, f, g, h, and T have a common fixed point in X provided that for a nondecreasing sequence {x_{n}}, with x_{n}≼y_{n} for all n, y_{n}→u implies that x_{n}≼u and either of the following:
(i) One of f or T and one of g or T are continuous, the pairs (f,T) and (g,T) are compatible, and the pair (h,T) is weakly compatible or
(ii) One of f or T and one of h or T are continuous, the pairs (f,T) and (h,T) are compatible, and the pair (g,T) is weakly compatible or
(iii) One of g or T and one of h or T are continuous, the pairs (g,T) and (h,T) are compatible, and the pair (f,T) is weakly compatible.
Moreover, the set of common fixed points of f, g, h, and T is well ordered if and only if f, g, h, and T have one and only one common fixed point.
Let
Taking S = T and g = h in Theorem 2.2, we obtain the following common fixed point result.
Corollary 2.6
Let (X,≼,G) be a partially ordered complete Gmetric space. Let f,g,R,S:X→X be four mappings such that f(X)⊆R(X) and g(X)⊆S(X) and dominating maps f and g are weak annihilators of R and S, respectively. Suppose that for every three comparable elements x,y,z∈X,
where ψ,φ: [ 0,∞)→ [ 0,∞) are altering distance functions. Then, f, g, R, and S have a common fixed point in X provided that for a nondecreasing sequence {x_{n}} with x_{n}≼y_{n} for all n, y_{n}→u implies that x_{n}≼u and either of the following:
(i) One of g or R and one of f or S are continuous, the pairs (f,S) and (g,R) are compatible, and the pair (g,S) is weakly compatible or
(ii) One of g or S and one of f or S are continuous, the pairs (f,S) and (g,S) are compatible, and the pair (g,R) is weakly compatible or
(iii) One of g or R and one of g or S are continuous, the pairs (g,R) and (g,S) are compatible, and the pair (f,S) is weakly compatible.
Moreover, the set of common fixed points of f, g, R and S is well ordered if and only if f, g, R and S have one and only one common fixed point.
Let
Taking R = S = T and f = g = h in Theorem 2.2, we obtain the following common fixed point result:
Corollary 2.7
Let (X,≼,G) be a partially ordered complete Gmetric space. Let f,T:X→X be two mappings such that f(X)⊆T(X), dominating map f is a weak annihilator of T. Suppose that for every three comparable elements x,y,z∈X,
where ψ,φ: [ 0,∞)→ [ 0,∞) are altering distance functions. Then, f and T have a common fixed point in X provided that for a nondecreasing sequence {x_{n}}, x_{n}≼y_{n} for all n, and y_{n}≼u implies that x_{n}≼u and one f or T is continuous and the pair (f,T) is compatible.
Moreover, the set of common fixed points of f and T is well ordered if and only if f and T have one and only one common fixed point.
Example 2.8
(see also [42]) Let X = [ 0,∞) and G on X be given by G(x,y,z)=x−y+y−z+x−z, for all x,y,z∈X. We define an ordering ‘ ≼’ on X as follows:
Define selfmaps f, g, h, S, T and R on X by
For each x∈X, we have 1+x≤e^{x}, and Hence, fx = ln(1+x)≤x, , and , which yields that x≼fx, x≼gx, and x≼hx, so f, g, and h are dominating.
Also, for each x∈X, we have fRx = ln(1+Rx)=3x≥x,
and since t^{6}−3t+2≥0 for each t≥1, we have
Hence, fRx≼x, gSx≼x and hTx≼x. Thus f, g, and h are weak annihilators of S, T, and R, respectively.
Furthermore, fX = TX = gX = SX = hX = RX = [ 0,∞) and the pairs (f,T), (g,R), and (h,S) are compatible. For example, we will show that the pair (f,T) is compatible. Let {x_{n}} is a sequence in X such that for some t∈X, and Therefore, we have
Since f and T are continuous, we have
On the other hand, ln(1+t)−e^{6t}+1=0⇔t = 0.
Define control functions ψ,φ: [ 0,∞)→ [ 0,∞) with ψ(t)=bt and φ(t)=(b−1)t for all t∈ [ 0,∞), where 1<b≤6.
Now, we will show that f, g, h, R, S and T satisfy (1). Using the mean value theorem, we have
Thus, (1) is satisfied for all x,y,z∈X. Therefore, all the conditions of the Theorem 2.2 are satisfied. Moreover, 0 is the unique common fixed point of f, g, h, R, S, and T.
Denoted by Λ, the set of all functions μ: [ 0+∞)→ [ 0,+∞), verifying the following conditions:
(I) μ is a positive Lebesgue integrable mapping on each compact subset of [ 0,+∞).
Other consequences of the main theorem are the following results for mappings satisfying contractive conditions of integral type.
Corollary 2.9
We replaced the contractive condition (1) of Theorem 2.2 by the following condition: There exists a μ∈Λ such that
Then, f, g, h, R, S, and T have a coincidence point, if the other conditions of Theorem 2.2 be satisfied.
Proof 2.10
Consider the function . Then (62) becomes
Taking ψ_{1}=Γoψ and φ_{1}=Γoφ and applying Theorem 2.2, we obtain the proof (it is easy to verify that ψ_{1} and φ_{1} are altering distance functions).
Similar to [43], let N be a fixed positive integer. Let {μ_{i}}_{1≤i≤N} be a family of N functions which belong to Λ. For all t≥0, we define
We have the following result.
Corollary 2.11
We replaced the inequality (1) of Theorem 2.2 by the following condition:
Then, f, g, h, R, S, and T have a coincidence point, if the other conditions of Theorem 2.2 be satisfied.
Proof 2.12
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
VP, AR, and JRR have worked together on each section of the paper such as the literature review, results and examples. All authors read and approved the final manuscript.
Acknowledgments
The authors thank the referees for the extremely careful reading that contributed to the improvement of the manuscript.
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