In this paper, we present some common fixed point theorems for a pair of self-mappings in fuzzy cone metric spaces under the generalized fuzzy cone contraction conditions. We extend and improve some recent results given in the literature.

1. Introduction

In 1965, Zadeh [1] came up with a fabulous idea. He introduced the theory of fuzzy set, which is the generalization of crisp sets. A mapping G is from X to [0,1]; then G is known as a fuzzy set. Later on, the fuzzy metric space concept was given by Kramosil and Michalek [2], which is performing the probabilistic metric space and would approach the fuzzy set. In [3], George and Veeramani were given the stronger form of the fuzz metric. Some more set-valued mapping results for fixed point on fuzzy metric spaces can be seen, for example, in [4–6] and the references therein.

In 2007, Som [7] proved some continuous self-mapping results for common fixed point in fuzzy metric spaces. He generalized the results of Pant [8], Som [9], and Vasuki [10]. Some other common fixed point results in the fuzzy metric space can be found in [11–16] and the references therein.

Huang and Zhang [17] introduced the concept of cone metric space. They proved the convergent sequences, Cauchy sequences, and some fixed point theorems for contractive-type mappings in cone metric spaces. Later on, Abbas and Jungck [18] proved some noncommuting mapping results in cone metric spaces. After that, a series of authors proved some fixed point and common fixed point results for different contractive-type mappings in cone metric spaces (see, e.g., [19–25]).

Recently, the concept of fuzzy cone metric space was introduced by Oner et al. [26]. They proved some basic properties and a Banach contraction theorem for fixed point with the assumption of Cauchy sequences. Rehman and Li [27] generalized the result of Oner et. al. [26] and proved some fixed point theorems in fuzzy cone metric spaces without the assumption of Cauchy sequences. Some more fixed point and common fixed point results in fuzzy cone metric spaces can be found in [27–31].

In the demonstration of this research work, we generalize the results of Oner [26] and Rehman [27] for a pair of self-mappings in fuzzy cone metric spaces and prove some unique common fixed theorems with illustrative examples.

A 3-tuple (X,M,*) is known as a fuzzy cone metric space, if ∗ is a continuous t-norm, X is an arbitrary set, P is a cone of E, and M is a fuzzy set on X×X×int(P) if the following hold:

A mapping G:X→X is known as fuzzy cone contractive in a fuzzy cone metric space (X,M,*), if ∃c∈(0,1) such that(4)1MGw,Gx,s-1≤c1Mw,x,s-1,for all w,x∈X, s≫ϑ, and c is known as a contraction constant of G.

A self-mapping in a complete fuzzy cone metric space, in which the fuzzy cone contractive sequences are Cauchy, has a unique fixed point.

Further, in this paper, we shall study some common fixed point results in (X,M,*). Let F,G:X→X be two self-mappings satisfying the following more generalized fuzzy cone contraction condition:(5)1MFx,Gw,s-1≤c11Mx,w,s-1+c21Mx,Fx,s-1+c31Mw,Gw,s-1+c41Mw,Fx,s-1+c51Mx,Gw,s-1,where s≫ϑ and the constants c1,c2,c3,c4,c5∈[0,+∞). It is noted that (5) is the same as (4) if F=G, c2=c3=c4=c5=0, and c1∈(0,1). On the other hand, the mappings F and G may not hold the fuzzy cone contraction condition if (5) is satisfied, which is shown in Example 14. Thus, in this research work, we generalize some recent results given in the literature (see Remark 13 and Example 14).

3. Main ResultTheorem 10.

Let F, G:X→X be two self mappings and M is triangular in a complete fuzzy cone metric space (X,M,*) which satisfies (5) with (c1+c2+c3+2max{c4,c5})<1. Then F and G have a unique common fixed point in X.

Proof.

Fix x0∈X and we define the iterative sequences in X as(6)x2j+1=Fx2jandx2j+2=Gx2j+1,j≥0. By view of (5), for s≫ϑ, (7)1Mx2j+1,x2j+2,s-1=1MFx2j,Gx2j+1,s-1≤c11Mx2j,x2j+1,s-1+c21Mx2j,Fx2j,s-1+c31Mx2j+1,Gx2j+1,s-1+c41Mx2j+1,Fx2j,s-1+c51Mx2j,Gx2j+1,s-1≤c11Mx2j,x2j+1,s-1+c21Mx2j,x2j+1,s-1+c31Mx2j+1,x2j+2,s-1+c51Mx2j,x2j+1,s-1+1Mx2j+1,x2j+2,s-1. Then(8)1Mx2j+1,x2j+2,s-1≤α1Mx2j,x2j+1,s-1,where α=(c1+c2+c5)/(1-c3-c5)<1, since (c1+c2+c3+2max{c4,c5})<1.

Let us denote 1/M(xj,xj+1,s)-1 by Mj; then, from (8), we have(9)M2j+1≤αM2j.Similarly, (10)1Mx2j+2,x2j+3,s-1=1MFx2j+2,Gx2j+1,s-1≤c11Mx2j+1,x2j+2,s-1+c21Mx2j+2,Fx2j+2,s-1+c31Mx2j+1,Gx2j+1,s-1+c41Mx2j+1,Fx2j+2,s-1+c51Mx2j+2,Gx2j+1,s-1≤c11Mx2j+1,x2j+2,s-1+c21Mx2j+2,x2j+3,s-1+c31Mx2j+1,x2j+2,s-1+c41Mx2j+1,x2j+2,s-1+1Mx2j+2,x2j+3,s-1.Then(11)1Mx2j+2,x2j+3,s-1≤β1Mx2j+1,x2j+2,s-1,where β=(c1+c3+c4)/(1-c2-c4)<1, since (c1+c2+c3+2max{c4,c5})<1. Then (11) can be written as(12)M2j+2≤βM2j+1.Now, from (9) and (12), we can get the following inequalities: (13)M2j≤βM2j-1≤βαM2j-2≤⋯≤αβjM0,M2j+1≤αM2j≤αβM2j-1≤α2βM2j-2≤⋯≤ααβjM0,M2j+2≤βM2j+1≤βαM2j≤β2αM2j-1≤⋯≤αβj+1M0,M2j+3≤αM2j+2≤αβM2j+1≤α2βM2j≤⋯≤ααβj+1M0. Thus, we have (14)M2j+M2j+1≤αβj1+αM0,M2j+1+M2j+2≤ααβj1+βM0,M2j+2+M2j+3≤αβj+11+αM0,M2j+3+M2j+4≤ααβj+11+βM0. Hence, from the above, we conclude that a sequence (xj) is fuzzy cone contractive in X; that is,(15)limj→∞Mxj,xj+1,s=1,fors≫ϑ.Let j,k∈N and let (xj) be the above sequence; we assume that k>j. Then, two cases arise.

Case (i). If j is an even number,(16)1Mxj,xk,s-1≤1Mxj,xj+1,s-1+1Mxj+1,xj+2,s-1+1Mxj+2,xj+3,s-1+1Mxj+3,xj+4,s-1+⋯+1Mxk-1,xk,s-1=Mj+Mj+1+Mj+2+Mj+3+⋯+Mk-2+Mk-1≤αβj/2+αβj/2+1+⋯+αβk/2-11+αM0≤αβj/21-αβ1+αM0.Case (ii). If j is an odd number,(17)1Mxj,xk,s-1≤1Mxj,xj+1,s-1+1Mxj+1,xj+2,s-1+1Mxj+2,xj+3,s-1+1Mxj+3,xj+4,s-1+⋯+1Mxk-1,xk,s-1=Mj+Mj+1+Mj+2+Mj+3+⋯+Mk-2+Mk-1≤αβj-1/2+αβj+1/2+⋯+αβk-3/2α1+βM0≤αβj-1/2α1+β1-αβM0. Thus, the right-hand sides of (16) and (17) converge to zero as j→∞, which yields that (xj) is a Cauchy sequence. Since X is complete, ∃z∈X such that(18)limj→∞Mz,xj,s=1,fors≫ϑ.Since M is triangular, (19)1Mz,Fz,s-1≤1Mz,x2j+2,s-1+1Mx2j+2,Fz,s-1,fors≫ϑ.By using (5), (15), and (18), for s≫ϑ, (20)1Mx2j+2,Fz,s-1=1MGx2j+1,Fz,s-1≤c11Mz,x2j+1,s-1+c21Mz,Fz,s-1+c31Mx2j+1,Gx2j+1,s-1+c41Mx2j+1,Fz,s-1+c51Mz,Gx2j+1,s-1≤c11Mz,x2j+1,s-1+c21Mz,Fz,s-1+c31Mx2j+1,x2j+2,s-1+c41Mx2j+1,z,s-1+1Mz,Fz,s-1+c51Mz,x2j+2,s-1→c2+c41Mz,Fz,s-1,asi→∞. Then(21)limsupj→∞1Mx2j+2,Fz,s-1≤c2+c41Mz,Fz,s-1,fors≫ϑ.The above (21) together with (18) and (19) implies that(22)1Mz,Fz,s-1≤c2+c41Mz,Fz,s-1,fors≫ϑ.(c2+c4)<1, since (c1+c2+c3+2max{c4,c5})<1; then M(z,Fz,s)=1; that is, Fz=z. Similarly, by M triangular,(23)1Mz,Gz,s-1≤1Mz,x2j+1,s-1+1Mx2j+1,Gz,s-1,fors≫ϑ.Again, by using (5), (15), and (18), similar to the above, after simplification, we can get(24)limsupj→∞1Mx2j+1,Gz,s-1≤c3+c51Mz,Gz,s-1,fors≫ϑ.The above (24) together with (18) and (23) implies that(25)1Mz,Gz,s-1≤c3+c51Mz,Gz,s-1,fors≫ϑ.(c3+c5)<1, since (c1+c2+c3+2max{c4,c5})<1; then M(z,Gz,s)=1; that is, Gz=z. Hence, the fact that z is the common fixed point of F and G in X is proven.

Uniqueness: let z∗∈X be the other common fixed point of F and G in X. Then, again by view of (5), for s≫ϑ, (26)1Mz∗,z,s-1=1MFz∗,Gz,s-1≤c11Mz∗,z,s-1+c21Mz∗,Fz∗,s-1+c31Mz,Gz,s-1+c41Mz,Fz∗,s-1+c51Mz∗,Gz,s-1=c1+c4+c51Mz∗,z,s-1. We note that (c1+c4+c5)<1, where (c1+c2+c3+2max{c4,c5})<1. Therefore M(z∗,z,s)=1, implying that z=z∗. Hence the fact that the common fixed point of F and G is unique is proven.

Corollary 11.

Let F, G:X→X be two self-mappings and M is triangular in the complete fuzzy cone metric space (X,M,*) which satisfies(27)1MFx,Gw,s-1≤c11Mx,w,s-1+c21Mx,Fx,s-1+c31Mw,Gw,s-1,for all w,x∈X, s≫ϑ, and c1,c2,c3∈[0,∞) such that (c1+c2+c3)<1. Then F and G have a unique common fixed point in X.

Corollary 12.

Let F, G:X→X be two self-mappings and M is triangular in the complete fuzzy cone metric space (X,M,*) which satisfies(28)1MFx,Gw,s-1≤c11Mx,w,s-1+c41Mw,Fx,s-1+c51Mx,Gw,s-1,for all w,x∈X, s≫ϑ, and c1,c4,c5∈[0,∞) such that (c1+2max{c4,c5})<1. Then F and G have a unique common fixed point in X.

Remark 13.

(i) In special case, Theorem 10, Corollaries 11 and 12, and [26, Theorem 3.3] (i.e., Theorem 9) all have the same results. In fact, if G=F, c1∈(0,1) and c2=c3=c4=c5=0 in (5).

(ii) Theorem 10 and [27, Theorem 3.1] both have similar proof. If G=F, c1,c2,c3,c4∈[0,∞) and c5=0 in (5).

Example 14.

Let X=[0,∞); ∗ is a continuous t-norm and M:X×X×(0,∞)→[0,1] is defined as (29)Mx,w,s=ss+x-w∀w,x∈X and s>0. Then, one can easily prove that M is triangular and (X,M,*) is a complete fuzzy cone metric space. Now we define F,G:X→X as (30)Fx=76x+3,if0≤x≤1,56x+32,if1<x<∞. And (31)Gw=76w+3,if0≤w≤1,34w+94,if1<w<∞. Then F and G are not fuzzy cone contractive, since (32)1MFx,Gw,s-1=761Mx,w,s-1. In special case, if G=F, then Theorem 9 does not hold. But it can be easily proven that all the conditions of Theorem 10 hold with c1=1/6, c2=c3=1/4, andc4=c5=1/8. Thus, F and G have a unique common fixed point in [0,∞), that is, 9.

Theorem 15.

Let F, G:X→X be two self-mappings and M is triangular in the complete fuzzy cone metric space (X,M,*) which satisfies(33)1MFx,Gw,s-1≤α1minMx,Fx,s,Mw,Gw,s,Mw,Fx,s,Mx,Gw,s-1,for all w,x∈X, s≫ϑ, and α∈(0,1). Then F and G have a unique common fixed point in X.

Proof.

Fix x0∈X and a point x1∈X such that Fx0=x1 and ∃x2∈X such that Gx1=x2. If α=0, then we have that (34)1Mx1,x2,s-1=1MFx0,Gx1,s-1=0,which implies that Fx0=Gx1 if and only if x0=x1. Then the proof is complete. Otherwise, we assume that α>0 and let us take β=1/α>1. Now we define the iterative sequence in X such as(35)x2j+1=Fx2jandx2j+2=Gx2j+1,j≥0.By view of (33), (36)1Mx2j+1,x2j+2,s-1≤β1MFx2j,Gx2j+1,s-1≤α1minMx2j,Fx2j,s,Mx2j+1,Gx2j+1,s,Mx2j+1,Fx2j,s,Mx2j,Gx2j+1,s-1=α1minMx2j,x2j+1,s,Mx2j+1,x2j+2,s,Mx2j,x2j+2,s-1.Now there are three possibilities.

(i) If M(x2j,x2j+1,s) is minimum, then 1/M(x2j,x2j+1,s)-1 will be the maximum in the above (36). Then, we have(37)1Mx2j+1,x2j+2,s-1≤α1Mx2j,x2j+1,s-1.

(ii) If M(x2j+1,x2j+2,s) is minimum, then 1/M(x2j+1,x2j+2,s)-1 will be the maximum in the above (36). Then, we have(38)1Mx2j+1,x2j+2,s-1≤α1Mx2j+1,x2j+2,s-1,which is not possible.

(iii) If M(x2j,x2j+2,s) is minimum, then 1/M(x2j,x2j+2,s)-1 will be the maximum in the above (36). Then, we have (39)1Mx2j+1,x2j+2,s-1≤α1Mx2j,x2j+2,s-1≤α1M(x2j,x2j+1,s-1+1Mx2j+1,x2j+2,s-1, which implies that(40)1Mx2j+1,x2j+2,s-1≤γ1Mx2j,x2j+1,s-1,where γ=α/(1-α)<1, since α∈(0,1). Thus, δ=max{γ,α}<1. Now, from (i), (ii), and (iii), for all j≥0 and s≫ϑ, (41)1Mx2j+1,x2j+2,s-1≤δ1Mx2j,x2j+1,s-1≤⋯≤δ2j+11Mx0,x1,s-1, which shows that a sequence (xj) is fuzzy cone contractive. Thus,(42)limj→∞Mx2j+1,x2j+2,s=1,fors≫ϑ.Since M is triangular, for all k>j≥j0, we have (43)1Mxj,xk,s-1≤1Mxj,xj+1,s-1+1Mxj+1,xj+2,s-1+⋯+1Mxk-1,xk,s-1≤δj+δj+1+⋯+δk-11Mx0,x1,s-1≤δj1-δ1Mx0,x1,s-1→0,asj→∞,which shows that (xj) is a Cauchy sequence. Since X is complete and ∃z∈X, we have(44)limj→∞Mz,xj,s=1,fors≫ϑ.Now we shall show that Fz=z. By the triangular property of M, we have(45)1Mz,Fz,s-1≤1Mz,x2j+2,s-1+1Mx2j+2,Fz,s-1,fors≫ϑ.Now, by using (33), (42), and (44), for s≫ϑ, we have (46)1Mx2j+2,Fz,s-1≤β1MFz,Gx2j+1,s-1≤α1minMz,Fz,s,Mx2j+1,Gx2j+1,s,Mx2j+1,Fz,s,Mz,Gx2j+1,s-1≤α1minMz,Fz,s,Mx2j+1,x2j+2,s,Mx2j+1,Fz,s,Mz,x2j+2,s-1→α1Mz,Fz,s-1,asj→∞.Thus,(47)limsupj→∞1Mx2j+2,Fz,s-1≤α1Mz,Fz,s-1,fors≫ϑ.The above (47) together with (44) and (45) implies that(48)1-α1Mz,Fz,s-1≤0,fors≫ϑ, and (1-α)<1, since α∈(0,1). This implies that M(z,Fz,s)=1; that is, Fz=z. Similarly, we can prove that Gz=z. Thus, Fz=Gz=z.

Uniqueness: let z∗∈X such that Fz∗=Gz∗=z∗. Then, by using (33), for every s≫ϑ, we have (49)1Mz,z∗,s-1≤β1MFz,Gz∗,s-1≤α1minMz,Fz,s,Mz∗,Gz∗,s,Mz∗,Fz,s,Mz,Gz∗,s-1=α1minMz,z,s,Mz∗,z∗,s,Mz∗,z,s,Mz,z∗,s-1=α1Mz,z∗,s-1. This implies that (50)1-α1Mz,z∗,s-1≤0.1-α≠0, since α∈(0,1). This implies that M(z,z∗,s)=1; that is, z∗=z. Hence the fact that F and G have a unique common fixed point is proven. That is, Fz=Gz=z∈X.

Corollary 16.

Let F, G:X→X be two self-mappings and M is triangular in the complete fuzzy cone metric space (X,M,*) which satisfies(51)1MFx,Gw,s-1≤α1minMx,Fx,s,Mw,Gw,s-1,for all w,x∈X, s≫ϑ, and α∈(0,1). Then F and G have a unique common fixed point in X.

Example 17.

From Example 14, we define F,G:X→X as (52)Fx=Gx=37x-17,if0≤x≤1,12x+2,if1<x<∞. Then, F and G are fuzzy cone contractive, since (53)1MFx,Gw,s-1=371Mx,w,s-1≤371minMx,Fx,s,Mw,Gw,s,Mw,Fx,s,Mx,Gw,s-1, for all x,w∈X. Then, all the conditions of Theorem 15 easily hold with α=3/7∈(0,1), as well as Theorem 9, if G=F. Thus, F and G have a unique common fixed point in [0,∞), that is, 4.

4. Conclusion

We gave the concept of common fixed point for a pair of self-mappings in fuzzy cone metric spaces and proved some unique common fixed point results in fuzzy cone metric spaces. We also proved that a pair of self-mappings may not be a fuzzy cone contraction if it satisfies (5), which is shown in Example 14. According to this concept, one can study some more common fixed point results for two or more self-mappings in fuzzy cone metric spaces for different contractive-type mappings.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors share equal contributions to the final manuscript.

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