Common Fixed Points of a Pair of Hardy Rogers Type Mappings on a Closed Ball in Ordered Dislocated Metric Spaces

Abstract

Common fixed point results for mappings satisfying locally contractive conditions on a closed ball in an ordered complete dislocated metric space have been established. The notion of dominated mappings is applied to approximate the unique solution of nonlinear functional equations. Our results improve several well-known conventional results.

1. Introduction and Preliminaries

Let 𝑇∶𝑋→𝑋 be a mapping. A point 𝑥∈𝑋 is called a fixed point of 𝑇if 𝑥=𝑇𝑥. Fixed points results of mappings satisfying certain contractive conditions on the entire domain have been at the centre of rigorous research activity, (e.g., see [1, 2]), and it has a wide range of applications in different areas such as nonlinear and adaptive control systems, parameterize estimation problems, computing magnetostatic fields in a nonlinear medium, and convergence of recurrent networks (see [3–5]).

Recently, many results appeared related to fixed point theorems in complete metric spaces endowed with a partial ordering. Ran and Reurings [6] proved an analogue of Banach's fixed point theorem in metric space endowed with a partial order and gave applications to matrix equations. In this way, they weakened the usual contractive condition. Subsequently, Nieto and Rodríguez-López [7] extended this result in [6] for nondecreasing mappings and applied it to obtain a unique solution for a 1st-order ordinary differential equation with periodic boundary conditions. Thereafter, many works related to fixed point problems have also been considered in partially ordered metric spaces (see [8–10]).

Hitzler and Seda [11] introduce the concept of a dislocated topology and its corresponding generalized metric named as dislocated metric and have established a fixed point theorem in complete dislocated metric spaces which generalizes the Banach contraction principle. The notion of dislocated topology has useful applications in the context of logic programming semantics (see [12]). Further useful results can be seen in [13–16].

From the application point of view the situation is not yet completely satisfactory because it frequently happens that a mapping 𝑇 is a contraction not on the entire space 𝑋 but merely on a subset 𝑌 of 𝑋. However, if 𝑌 is closed and a sequence {𝑥_𝑛} in 𝑋 converges to some 𝑥 in 𝑋, then by imposing a subtle restriction on the choice of 𝑥₀, one may force the sequence to stay eventually in 𝑌. In this case, one can establish the existence of a fixed point of 𝑇. Arshad et al. [17] proved a significant result concerning the existence of fixed points of a mapping satisfying contractive conditions on closed ball in a complete dislocated metric space. Other results on closed ball can be seen in [18–20]. The dominated mapping [21]which satisfies the condition 𝑓𝑥⪯𝑥 occurs very naturally in several practical problems. For example, if 𝑥 denotes the total quantity of food produced over a certain period of time and 𝑓(𝑥) gives the quantity of food consumed over the same period in a certain town, then we must have 𝑓𝑥⪯𝑥. In this paper, we will exploit this concept. We have obtained fixed point theorems for contractive dominated self-mappings in an ordered complete dislocated metric space on a closed ball to generalize, extend, and improve a classical fixed point result in [22]. We have used weaker contractive conditions and weaker restrictions to obtain unique fixed point. Our results do not exist even yet in metric spaces. We have given an example which shows how this result can be used when the corresponding results cannot.

Consistent with [12, 21, 22], the following definitions and results will be needed in the sequel.

Definition 1. Let 𝑋 be a nonempty set and let 𝑑_𝑙∶𝑋×𝑋→[0,+∞) be a function, called a dislocated metric (or simply 𝑑_𝑙-metric) if the following conditions hold for any 𝑥,𝑦,𝑧∈𝑋: (i)if 𝑑_𝑙(𝑥,𝑦)=0, then 𝑥=𝑦,(ii)𝑑_𝑙(𝑥,𝑦)=𝑑_𝑙(𝑦,𝑥), (iii)𝑑_𝑙(𝑥,𝑦)≤𝑑_𝑙(𝑥,𝑧)+𝑑_𝑙(𝑧,𝑦)−𝑑_𝑙(𝑧,𝑧). The pair (𝑋,𝑑_𝑙) is then called a dislocated metric space.

It is clear that if 𝑑_𝑙(𝑥,𝑦)=0, then from (i), 𝑥=𝑦. But if 𝑥=𝑦, 𝑑_𝑙(𝑥,𝑦) may not be 0.

Remark 2. From (iii) of Definition 1, we deduce 𝑑_𝑙𝑥,𝑦≤𝑑_𝑙𝑥,𝑦+𝑑_𝑙(𝑧,𝑧)≤𝑑_𝑙(𝑥,𝑧)+𝑑_𝑙𝑧,𝑦 (1) for all 𝑥,𝑦,𝑧∈𝑋.

Recently Sarma and Kumari [15] proved results that establish existence of a topology induced by a dislocated metric and that this topology is metrizable. This topology has as a base the family of sets {𝐵(𝑥,𝜀)∪{𝑥}∶𝑥∈𝑋,𝜀>0}, where 𝐵(𝑥,𝜀) is an open ball and 𝐵(𝑥,𝜀)={𝑦∈𝑋∶𝑑_𝑙(𝑥,𝑦)<𝜀} for some 𝑥∈𝑋 and 𝜀>0. Examples of the 𝑑_𝑙-metric spaces are given below.

Example 3. If 𝑋=[0,+∞), then 𝑑_𝑙(𝑥,𝑦)=𝑥+𝑦 defines a dislocated metric 𝑑_𝑙 on 𝑋.

Example 4. If 𝑋=[0,+∞), then 𝑑_𝑙(𝑥,𝑦)=max{𝑥,𝑦} defines a dislocated metric 𝑑_𝑙 on 𝑋.

Definition 5 (see [13]). A sequence {𝑥_𝑛} in a 𝑑_𝑙-metric space (𝑋,𝑑_𝑙) is called a Cauchy sequence if given 𝜀>0, there exists 𝑛₀∈𝑁 such that for all 𝑛,𝑚≥𝑛₀, we have 𝑑_𝑙(𝑥_𝑚,𝑥_𝑛)<𝜀.

Definition 6 (see [13]). A sequence {𝑥_𝑛} in a 𝑑_𝑙-metric space converges with respect to 𝑑_𝑙 if there exists 𝑥∈𝑋 such that 𝑑_𝑙(𝑥_𝑛,𝑥)→0 as 𝑛→+∞. In this case, 𝑥 is called limit of {𝑥_𝑛}, and we write 𝑥_𝑛→𝑥.

Definition 7. A 𝑑_𝑙-metric space (𝑋,𝑑_𝑙) is called complete if every Cauchy sequence in 𝑋 converges to a point in 𝑋.

Definition 8. Let 𝑋 be a nonempty set. Then (𝑋,⪯,𝑑_𝑙) is called an ordered dislocated metric space if (i)𝑑_𝑙 is a dislocated metric on 𝑋 and (ii) ⪯ is a partial order on 𝑋.

Definition 9. Let (𝑋,⪯) be a partial ordered set. Then 𝑥,𝑦∈𝑋 are called comparable if 𝑥⪯𝑦 or 𝑦⪯𝑥 holds.

Definition 10 (see [21]). Let (𝑋,⪯) be a partially ordered set. A self-mapping 𝑓 on 𝑋 is called dominated if 𝑓𝑥⪯𝑥 for each 𝑥 in 𝑋.

Example 11. Let 𝑋=[0,1] be endowed with the usual ordering and 𝑓∶𝑋→𝑋 defined by 𝑓𝑥=𝑥^𝑛 for some 𝑛∈ℕ. Since 𝑓𝑥=𝑥^𝑛≤𝑥 for all 𝑥∈𝑋, therefore 𝑓 is a dominated mapping.

Definition 12. Let 𝑋 be a nonempty set and 𝑇,𝑓∶𝑋→𝑋. A point 𝑦∈𝑋 is called point of coincidence of 𝑇 and 𝑓 if there exists a point 𝑥∈𝑋 such that 𝑦=𝑇𝑥=𝑓𝑥. The mappings 𝑇,𝑓 are said to be weakly compatible if they commute at their coincidence point (i.e., 𝑇𝑓𝑥=𝑓𝑇𝑥 whenever 𝑇𝑥=𝑓𝑥). If ⪯ is a partial order on 𝑋, then the mapping 𝑇 is called 𝑓-dominated if 𝑇𝑥⪯𝑓𝑥 for all 𝑥∈𝑋; see [23].

We require the following lemmas for subsequent use.

Lemma 13 (see [24]). Let 𝑋 be a nonempty set and 𝑓∶𝑋→𝑋 a function. Then there exists a subset 𝐸⊂𝑋 such that 𝑓𝐸=𝑓𝑋 and 𝑓∶𝐸→𝑋 is one-to-one.

Lemma 14 (see [25]). Let 𝑋 be a nonempty set and the mappings 𝑆,𝑇,𝑓∶𝑋→𝑋 have a unique point of coincidence 𝑣 in 𝑋. If (𝑆,𝑓) and (𝑇,𝑓) are weakly compatible, then 𝑆,𝑇,𝑓 have a unique common fixed point.

Theorem 15 (see [22]). Let (𝑋,𝑑) be a complete metric space, 𝑆∶𝑋→𝑋 a mapping, 𝑟>0, and 𝑥₀ an arbitrary point in 𝑋. Suppose that there exists 𝑘∈[0,1) with 𝑑𝑆𝑥,𝑆𝑦≤𝑘𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑌=𝐵𝑥₀,𝑟 (2) and 𝑑(𝑥₀,𝑆𝑥₀)<(1−𝑘)𝑟. Then there exists a unique point 𝑥^∗ in 𝐵(𝑥₀,𝑟) such that 𝑥^∗=𝑆𝑥^∗.

2. Fixed Points of Hardy-Rogers Mappings

The Banach contraction principle in the setting of dislocated metric spaces was established in [26] (see also [11]).

Theorem 16. Let (𝑋,𝑑_𝑙) be a complete dislocated metric space, and let 𝑇∶𝑋→𝑋 be a contraction. Then 𝑇 has a unique fixed point.

The following example shows that there exists a contraction in a dislocated metric space that is not a contraction in a metric space.

Example 17 (see [27, Example 1]). Let 𝑋=[0,1] and 𝑑_𝑙(𝑥,𝑦)=max{𝑥,𝑦}. Fix 𝑟∈[0,1), and define 𝑇∶𝑋→𝑋 by 𝑟𝑇𝑥=0if𝑥=02𝑛−𝑟2𝑛−1()12𝑛2𝑛𝑥−1if12𝑛≤𝑥≤𝑟2𝑛−12𝑛+𝑟2𝑛+1()12𝑛2𝑛𝑥−1if12𝑛+1≤𝑥≤.2𝑛 (3)
For all 𝑥,𝑦∈𝑋, we have 𝑑_𝑙𝑇𝑥,𝑇𝑦=max𝑇𝑥,𝑇𝑦≤max𝑟𝑥,𝑟𝑦=𝑟max𝑥,𝑦=𝑟𝑑_𝑙𝑥,𝑦, (4) which ensures that 𝑇 is a contraction, and so, by Theorem 16, 𝑇 has a unique fixed point in 𝑋.
Now, we show that 𝑇 is not a contraction in the complete metric space (𝑋,𝑑), where 𝑑(𝑥,𝑦)=|𝑥−𝑦| for all 𝑥,𝑦∈𝑋. In fact, if for odd 𝑛>1 we choose 𝑥=1/(2𝑛−1) and 𝑦=1/(𝑛−1), we have 𝑑=𝑟𝑇𝑥,𝑇𝑦=𝑛𝑛−1≤𝑑𝑥,𝑦()≤3𝑛−1)(2𝑛−15(𝑛−1) (5) which is not satisfied if 𝑟>3/5. Consequently, 𝑇 is not a contraction if 𝑟>3/5.

Thus, the above example gives a strong motivation for studying fixed point results in the setting of dislocated metric spaces.

For some results of fixed point in the setting of complete dislocated metric spaces for mappings that satisfy a contractive condition of Hardy and Rogers see [28]. The following theorem is one of our main results.

Theorem 18. Let (𝑋,⪯,𝑑_𝑙) be an ordered complete dislocated metric space, 𝑥₀∈𝑋, 𝑟>0, and 𝑆,𝑇∶𝑋→𝑋 two dominated mappings. Suppose that there exists 𝑎,𝑏,𝑐∈[0,1) with 𝑎+2𝑏+2𝑐<1 such that 𝑑_𝑙𝑆𝑥,𝑇𝑦≤𝑎𝑑_𝑙𝑥,𝑦+𝑏𝑑_𝑙(𝑥,𝑆𝑥)+𝑑_𝑙𝑑𝑦,𝑇𝑦+𝑐_𝑙𝑦,𝑆𝑥+𝑑_𝑙𝑥,𝑇𝑦 (6) for all comparable elements 𝑥,𝑦 in 𝐵(𝑥₀,𝑟) and 𝑑_𝑙𝑥₀,𝑆𝑥₀≤(1−𝜆)𝑟, (7) where 𝜆=(𝑎+𝑏+𝑐)/(1−𝑏−𝑐). If for a nonincreasing sequence {𝑥_𝑛} in 𝐵(𝑥₀,𝑟), 𝑥_𝑛→𝑢 implies that 𝑢⪯𝑥_𝑛, then there exists a point 𝑥^∗ in 𝐵(𝑥₀,𝑟) such that 𝑑_𝑙(𝑥^∗,𝑥^∗)=0 and 𝑥^∗=𝑆𝑥^∗=𝑇𝑥^∗.

Proof. Choose a point 𝑥₁ in 𝑋 such that 𝑥₁=𝑆𝑥₀. As 𝑆𝑥₀⪯𝑥₀ so 𝑥₁⪯𝑥₀, and let 𝑥₂=𝑇𝑥₁. Now 𝑇𝑥₁⪯𝑥₁ gives 𝑥₂⪯𝑥₁, continuing this process and choosing 𝑥_𝑛 in 𝑋 such that 𝑥_2𝑘+1=𝑆𝑥_2𝑘,𝑥_2𝑘+2=𝑇𝑥_2𝑘+1,where𝑘=0,1,2,…. (8) Clearly, 𝑥_2𝑘+1=𝑆𝑥_2𝑘⪯𝑥_2𝑘=𝑇𝑥_2𝑘−1⪯𝑥_2𝑘−1, and this implies that the sequence {𝑥_𝑛} is nonincreasing.
Now, we will prove that 𝑥_𝑛∈𝐵(𝑥₀,𝑟) for all 𝑛∈𝑁, by mathematical induction. By using inequality (7), we have 𝑑_𝑙𝑥₀,𝑥₁≤𝑟, (9) and so 𝑥₁∈𝐵(𝑥₀,𝑟). Assume that 𝑥₂,…,𝑥_𝑗∈𝐵(𝑥₀,𝑟) for some 𝑗∈𝑁. Now, if 2𝑘+1≤𝑗, by using inequality (6) and Remark 2, we obtain 𝑑_𝑙𝑥_2𝑘+1,𝑥_2𝑘+2=𝑑_𝑙𝑆𝑥_2𝑘,𝑇𝑥_2𝑘+1≤𝑎𝑑_𝑙𝑥_2𝑘,𝑥_2𝑘+1+𝑏𝑑_𝑙𝑥_2𝑘,𝑆𝑥_2𝑘+𝑑_𝑙𝑥_2𝑘+1,𝑇𝑥_2𝑘+1+𝑐𝑑_𝑙𝑥_2𝑘,𝑇𝑥_2𝑘+1+𝑑_𝑙𝑥_2𝑘+1,𝑆𝑥_2𝑘≤𝑎𝑑_𝑙𝑥_2𝑘,𝑥_2𝑘+1+𝑏𝑑_𝑙𝑥_2𝑘,𝑥_2𝑘+1+𝑑_𝑙𝑥_2𝑘+1,𝑥_2𝑘+2+𝑐𝑑_𝑙𝑥_2𝑘,𝑥_2𝑘+1+𝑑_𝑙𝑥_2𝑘+1,𝑥_2𝑘+2, (10) and hence 𝑑_𝑙𝑥_2𝑘+1,𝑥_2𝑘+2≤𝜆𝑑_𝑙𝑥_2𝑘,𝑥_2𝑘+1. (11) Similarly, if 2𝑘≤𝑗, we deduce 𝑑_𝑙𝑥_2𝑘+1,𝑥_2𝑘≤𝜆𝑑_𝑙𝑥_2𝑘,𝑥_2𝑘−1. (12) By the previous inequalities, we get 𝑑_𝑙𝑥_2𝑘+1,𝑥_2𝑘+2≤𝜆𝑑_𝑙𝑥_2𝑘,𝑥_2𝑘+1≤⋯≤𝜆^2𝑘+1𝑑_𝑙𝑥₀,𝑥₁,𝑑_𝑙𝑥_2𝑘+1,𝑥_2𝑘≤𝜆𝑑_𝑙𝑥_2𝑘,𝑥_2𝑘−1≤⋯≤𝜆^2𝑘𝑑_𝑙𝑥₀,𝑥₁. (13) Thus, from inequality (13), we have 𝑑_𝑙𝑥_𝑗,𝑥_𝑗+1≤𝜆^𝑗𝑑_𝑙𝑥₀,𝑥₁forsome𝑗∈𝑁. (14) Now, using (14) and (7), we get 𝑑_𝑙𝑥₀,𝑥_𝑗+1≤𝑑_𝑙𝑥₀,𝑥₁+⋯+𝑑_𝑙𝑥_𝑗,𝑥_𝑗+1−𝑑_𝑙𝑥₁,𝑥₁+⋯+𝑑_𝑙𝑥_𝑗,𝑥_𝑗≤𝑑_𝑙𝑥₀,𝑥₁1+⋯+𝜆^𝑗−1+𝜆^𝑗≤(1−𝜆)𝑟1−𝜆^𝑗+11−𝜆<𝑟. (15) Thus 𝑥_𝑗+1∈𝐵(𝑥₀,𝑟); that is, 𝑥_𝑛∈𝐵(𝑥₀,𝑟) for all 𝑛∈𝑁. Since 𝑥_𝑛+1⪯𝑥_𝑛 for all 𝑛∈𝑁, then it follows that 𝑑_𝑙𝑥_𝑛,𝑥_𝑛+1≤𝜆^𝑛𝑑_𝑙𝑥₀,𝑥₁∀𝑛∈𝑁. (16) So we have 𝑑_𝑙𝑥_𝑛+𝑖,𝑥_𝑛≤𝑑_𝑙𝑥_𝑛+𝑖,𝑥_{𝑛+𝑖−1}+⋯+𝑑_𝑙𝑥_𝑛+1,𝑥_𝑛≤𝜆^{𝑛+𝑖−1}𝑑_𝑙𝑥₀,𝑥₁+⋯+𝜆^𝑛𝑑_𝑙𝑥₀,𝑥₁≤𝜆^𝑛𝑑_𝑙𝑥₀,𝑥₁1−𝜆^𝑖1−𝜆⟶0as𝑛⟶+∞. (17) Hence the sequence {𝑥_𝑛} is a Cauchy sequence in (𝐵(𝑥₀,𝑟),𝑑_𝑙). Now, 𝐵(𝑥₀,𝑟) is complete since it is closed in 𝑋. Therefore there exists a point 𝑥^∗∈𝐵(𝑥₀,𝑟) with lim_𝑛→+∞𝑑_𝑙𝑥_𝑛,𝑥^∗=0. (18) By assumptions 𝑥^∗⪯𝑥_𝑛 as 𝑥_𝑛→𝑥^∗, we have 𝑑_𝑙𝑆𝑥^∗,𝑥^∗≤𝑑_𝑙𝑆𝑥^∗,𝑇𝑥_2𝑛+1+𝑑_𝑙𝑥_2𝑛+2,𝑥^∗−𝑑_𝑙𝑥_2𝑛+2,𝑥_2𝑛+2≤𝑑_𝑙𝑥^∗,𝑥_2𝑛+2+𝑎𝑑_𝑙𝑥^∗,𝑥_2𝑛+1+𝑏𝑑_𝑙𝑥^∗,𝑆𝑥^∗+𝑑_𝑙𝑥_2𝑛+1,𝑇𝑥_2𝑛+1+𝑐𝑑_𝑙𝑥^∗,𝑇𝑥_2𝑛+1+𝑑_𝑙𝑥_2𝑛+1,𝑆𝑥^∗≤𝑑_𝑙𝑥^∗,𝑥_2𝑛+2+𝑎𝑑_𝑙𝑥^∗,𝑥_2𝑛+1+𝑏𝑑_𝑙𝑥^∗,𝑆𝑥^∗+𝑑_𝑙𝑥_2𝑛+1,𝑥_2𝑛+2+𝑐𝑑_𝑙𝑥^∗,𝑥_2𝑛+2+𝑑_𝑙𝑥_2𝑛+1,𝑥^∗+𝑑_𝑙𝑥^∗,𝑆𝑥^∗. (19) On taking limit as 𝑛→+∞ in the previous inequality, by using inequality (16) and (18), we get (1−𝑏−𝑐)𝑑_𝑙𝑆𝑥^∗,𝑥^∗≤0, (20) which implies that 𝑥^∗=𝑆𝑥^∗. Similarly, from 𝑑_𝑙𝑥^∗,𝑇𝑥^∗≤𝑑_𝑙𝑥^∗,𝑥_2𝑛+1+𝑑_𝑙𝑥_2𝑛+1,𝑇𝑥^∗−𝑑_𝑙𝑥_2𝑛+1,𝑥_2𝑛+1, (21) we can obtain 𝑥^∗=𝑇𝑥^∗. Hence 𝑆 and 𝑇 have a common fixed point in 𝐵(𝑥₀,𝑟). Now, 𝑑_𝑙𝑥^∗,𝑥^∗=𝑑_𝑙𝑆𝑥^∗,𝑇𝑥^∗≤𝑎𝑑_𝑙𝑥^∗,𝑥^∗+𝑏𝑑_𝑙𝑥^∗,𝑆𝑥^∗+𝑑_𝑙𝑥^∗,𝑇𝑥^∗+𝑐𝑑_𝑙𝑥^∗,𝑆𝑥^∗+𝑑_𝑙𝑥^∗,𝑇𝑥^∗=(𝑎+2𝑏+2𝑐)𝑑_𝑙𝑥^∗,𝑥^∗<𝑑_𝑙𝑥^∗,𝑥^∗, (22) and this implies that 𝑑_𝑙𝑥^∗,𝑥^∗=0. (23)

Example 19. Let 𝑋=[0,+∞)∩ℚ be endowed with the order 𝑥⪯𝑦 if 𝑑_𝑙(𝑥,𝑥)≤𝑑_𝑙(𝑦,𝑦), and define 𝑑_𝑙∶𝑋×𝑋→𝑋 as 𝑑_𝑙(𝑥,𝑦)=𝑥+𝑦. Then (𝑋,𝑑_𝑙) is an ordered complete dislocated metric space. Let 𝑆,𝑇∶𝑋→𝑋 be defined by 𝑥𝑆𝑥=7[]1if𝑥∈0,1∩X𝑥−3if𝑥∈(1,+∞)∩X,𝑇𝑥=2𝑥7[]1if𝑥∈0,1∩X𝑥−4if𝑥∈(1,+∞)∩X. (24) Clearly, 𝑆 and 𝑇are dominated mappings. For 𝑥₀=1, 𝑟=2, 𝑎=1/7, and 𝑏=𝑐=1/10, then 𝜆=3/7, 𝐵(𝑥₀,𝑟)=[0,1]∩𝑋, and (1−𝜆)𝑟=8/7=𝑑_𝑙(𝑥₀,𝑆𝑥₀).
Now, if 𝑥,𝑦∈(1,+∞)∩𝑋, then 1𝑥−31+𝑦−4≥17+1𝑥+𝑦1102𝑥−31+2𝑦−4+1110𝑦+𝑥−31+𝑥+𝑦−4, (25) and so 𝑑_𝑙𝑆𝑥,𝑇𝑦≥𝑎𝑑_𝑙𝑥,𝑦+𝑏𝑑_𝑙(𝑥,𝑆𝑥)+𝑑_𝑙𝑑𝑦,𝑇𝑦+𝑐_𝑙𝑦,𝑆𝑥+𝑑_𝑙𝑥,𝑇𝑦. (26) Thus, the contractive condition does not hold on 𝑋. Now if 𝑥,𝑦∈𝐵(𝑥₀,𝑟), then 𝑑_𝑙=𝑥𝑆𝑥,𝑇𝑦7+2𝑦7=17≤1𝑥+2𝑦7+1𝑥+𝑦𝑥10𝑥+7+𝑦+2𝑦7+1𝑥10𝑦+7+𝑥+2𝑦7=𝑎𝑑_𝑙𝑥,𝑦+𝑏𝑑_𝑙(𝑥,𝑆𝑥)+𝑑_𝑙𝑑𝑦,𝑇𝑦+𝑐_𝑙𝑦,𝑆𝑥+𝑑_𝑙𝑥,𝑇𝑦. (27) Therefore, all the conditions of Theorem 18 are satisfied. Moreover, 0 is the common fixed point of 𝑆 and 𝑇 and 𝑑_𝑙(0,0)=0.

If in Theorem 18 we choose 𝑇=𝑆, then we obtain the following corollary.

Corollary 20. Let (𝑋,⪯,𝑑_𝑙) be an ordered complete dislocated metric space, 𝑥₀∈𝑋, 𝑟>0, and 𝑆∶𝑋→𝑋 a dominated mapping. Suppose that there exist 𝑎,𝑏,𝑐∈[0,1) with 𝑎+2𝑏+2𝑐<1 such that 𝑑_𝑙𝑆𝑥,𝑆𝑦≤𝑎𝑑_𝑙𝑥,𝑦+𝑏𝑑_𝑙(𝑥,𝑆𝑥)+𝑑_𝑙𝑑𝑦,𝑆𝑦+𝑐_𝑙𝑦,𝑆𝑥+𝑑_𝑙𝑥,𝑆𝑦, (28) for all comparable elements 𝑥,𝑦 in 𝐵(𝑥₀,𝑟) and 𝑑_𝑙𝑥₀,𝑆𝑥₀≤(1−𝜆)𝑟, (29) where 𝜆=(𝑎+𝑏+𝑐)/(1−𝑏−𝑐). If for a nonincreasing sequence {𝑥_𝑛} in 𝐵(𝑥₀,𝑟),𝑥_𝑛→𝑢 implies that 𝑢⪯𝑥_𝑛, then there exists a point 𝑥^∗ in 𝐵(𝑥₀,𝑟) such that 𝑥^∗=𝑆𝑥^∗ and 𝑑_𝑙(𝑥^∗,𝑥^∗)=0.

In Theorem 18, the condition “for a nonincreasing sequence 𝑥_𝑛→𝑢 implies that 𝑢⪯𝑥_𝑛” is imposed to restrict condition (6) only for comparable elements. However, the following result relaxes this restriction but imposes that condition (6) holds for all elements in 𝐵(𝑥₀,𝑟). In Theorem 18, the common fixed point of 𝑆 and 𝑇 may not be unique, whereas without order we can obtain a unique fixed point of 𝑆 and 𝑇 separately, which is proved in the following theorem.

Theorem 21. Let (𝑋,𝑑_𝑙) be a complete dislocated metric space, 𝑥₀∈𝑋, 𝑟>0, and 𝑆,𝑇∶𝑋→𝑋 two dominated mappings. Suppose that there exist 𝑎,𝑏,𝑐∈[0,1) with 𝑎+2𝑏+2𝑐<1 such that 𝑑_𝑙𝑆𝑥,𝑇𝑦≤𝑎𝑑_𝑙𝑑𝑥,𝑦+𝑏_𝑙(𝑥,𝑆𝑥)+𝑑_𝑙𝑑𝑦,𝑇𝑦+𝑐_𝑙𝑦,𝑆𝑥+𝑑_𝑙𝑥,𝑇𝑦, (30) for all elements 𝑥,𝑦 in 𝐵(𝑥₀,𝑟) and 𝑑_𝑙𝑥₀,𝑆𝑥₀≤(1−𝜆)𝑟, (31) where 𝜆=(𝑎+𝑏+𝑐)/(1−𝑏−𝑐). Then there exists a unique point 𝑥^∗ in 𝐵(𝑥₀,𝑟) such that 𝑥^∗=𝑆𝑥^∗=𝑇𝑥^∗ and 𝑑_𝑙(𝑥^∗,𝑥^∗)=0. Moreover, 𝑆 and 𝑇 have no fixed point other than 𝑥^∗.

Proof. By following similar arguments, as we used to prove Theorem 18, we can obtain a point 𝑥^∗ in 𝐵(𝑥₀,𝑟) such that 𝑥^∗=𝑆𝑥^∗=𝑇𝑥^∗ and 𝑑_𝑙(𝑥^∗,𝑥^∗)=0. Let 𝑦=𝑇𝑦. Then, 𝑦 is a fixed point of 𝑇, and it may not be the fixed point of 𝑆. Then, 𝑑_𝑙𝑥^∗,𝑦=𝑑_𝑙𝑆𝑥^∗,𝑇𝑦≤𝑎𝑑_𝑙𝑥^∗𝑑,𝑦+𝑏_𝑙𝑥^∗,𝑆𝑥^∗+𝑑_𝑙𝑑𝑦,𝑇𝑦+𝑐_𝑙𝑦,𝑆𝑥^∗+𝑑_𝑙𝑥^∗,𝑇𝑦≤𝑎𝑑_𝑙𝑥^∗,𝑦+𝑏𝑑_𝑙𝑦,𝑥^∗+𝑑_𝑙𝑥^∗,𝑦+2𝑐𝑑_𝑙𝑥^∗≤()𝑑,𝑦𝑎+2𝑏+2𝑐_𝑙𝑥^∗,𝑦<𝑑_𝑙𝑥^∗,𝑦. (32) This shows that 𝑥^∗=𝑦. Thus 𝑇 has no fixed point other than 𝑥^∗. Similarly 𝑆 has no fixed point other than 𝑥^∗.

In Theorem 18, condition (7) is imposed to restrict condition (6) only for 𝑥,𝑦 in 𝐵(𝑥₀,𝑟), and Example 19 explains the utility of this restriction. However, the following result relaxes condition (7) but imposes condition (6) for all comparable elements in the whole space 𝑋. Moreover, we introduce a weaker restriction to obtain a unique common fixed point.

Theorem 22. Let (𝑋,⪯,𝑑_𝑙) be an ordered complete dislocated metric space, 𝑥₀∈𝑋, and 𝑆,𝑇∶𝑋→𝑋 two dominated mappings. Suppose that there exist 𝑎,𝑏,𝑐∈[0,1) with 𝑎+2𝑏+2𝑐<1 such that 𝑑_𝑙𝑆𝑥,𝑇𝑦≤𝑎𝑑_𝑙𝑥,𝑦+𝑏𝑑_𝑙(𝑥,𝑆𝑥)+𝑑_𝑙𝑑𝑦,𝑇𝑦+𝑐_𝑙𝑦,𝑆𝑥+𝑑_𝑙𝑥,𝑇𝑦, (33) for all comparable elements 𝑥,𝑦 in 𝑋. If for a nonincreasing sequence {𝑥_𝑛} in 𝑋,𝑥_𝑛→𝑢 implies that 𝑢⪯𝑥_𝑛, then there exists a point 𝑥^∗ in 𝑋 such that 𝑥^∗=𝑆𝑥^∗=𝑇𝑥^∗ and 𝑑_𝑙(𝑥^∗,𝑥^∗)=0. Moreover, the point 𝑥^∗ is unique if for any two points 𝑥,𝑦 in 𝑋 there exists a point 𝑧₀∈𝑋 such that 𝑧₀⪯𝑥 and 𝑧₀⪯𝑦.

Proof. By following similar arguments of Theorem 18, we can obtain a point 𝑥^∗ in 𝑋 such that 𝑥^∗=𝑆𝑥^∗=𝑇𝑥^∗. By Theorem 21, 𝑥^∗ is unique common fixed point for all comparable elements. Now if 𝑥^∗ and 𝑦 are not comparable and 𝑦=𝑆𝑦=𝑇𝑦, then there exists a point 𝑧₀∈𝑋 such that 𝑧₀⪯𝑥^∗ and 𝑧₀⪯𝑦. Choose a point 𝑧₁ in 𝑋 such that 𝑧₁=𝑇𝑧₀. As 𝑇𝑧₀⪯𝑧₀ so 𝑧₁⪯𝑧₀, and let 𝑧₂=𝑆𝑧₁. Now 𝑆𝑧₁⪯𝑧₁ gives 𝑧₂⪯𝑧₁, continuing this process and having chosen 𝑧_𝑛 in 𝑋 such that 𝑧_2𝑖+1=𝑇𝑧_2𝑖,𝑧_2𝑖+2=𝑆𝑧_2𝑖+1,𝑧_2𝑖+1=𝑇𝑧_2𝑖⪯𝑧_2𝑖=𝑆𝑧_2𝑖−1⪯𝑧_2𝑖−1,where𝑖=0,1,2,…. (34) It follows that 𝑧_𝑛+1⪯𝑧_𝑛⪯⋯⪯𝑧₀⪯𝑥^∗. Following similar arguments as we have used to prove inequality (16), we have 𝑑_𝑙𝑧_𝑛,𝑧_𝑛+1≤𝜆^𝑛𝑑_𝑙𝑧₀,𝑧₁∀𝑛∈𝑁. (35) As 𝑧₀⪯𝑥^∗ and 𝑧₀⪯𝑦, it follows that 𝑧_𝑛⪯𝑇𝑥^∗ and 𝑧_𝑛⪯𝑇𝑦 for all 𝑛∈𝑁. Then, for 𝑖∈𝑁, 𝑑_𝑙𝑆𝑧_2𝑖−1,𝑇𝑥^∗≤𝑎𝑑_𝑙𝑧_2𝑖−1,𝑥^∗+𝑏𝑑_𝑙𝑧_2𝑖−1,𝑆𝑧_2𝑖−1+𝑑_𝑙𝑥^∗,𝑇𝑥^∗+𝑐𝑑_𝑙𝑧_2𝑖−1,𝑇𝑥^∗+𝑑_𝑙𝑥^∗,𝑆𝑧_2𝑖−1, (36) and so 𝑑_𝑙𝑧_2𝑖,𝑥^∗=𝑑_𝑙𝑆𝑧_2𝑖−1,𝑥^∗≤𝛿𝑑_𝑙𝑧_2𝑖−1,𝑥^∗+𝜇𝑑_𝑙𝑧_2𝑖−1,𝑧_2𝑖, (37) where 𝛿=(𝑎+𝑐)/(1−𝑐) and 𝜇=𝑏/(1−𝑐).
Similarly, we deduce 𝑑_𝑙𝑥^∗,𝑧_2𝑖−1=𝑑_𝑙𝑆𝑥^∗,𝑇𝑧_2𝑖−2≤𝛿𝑑_𝑙𝑥^∗,𝑧_2𝑖−2+𝜇𝑑_𝑙𝑧_2𝑖−2,𝑧_2𝑖−1. (38) Since 𝛿,𝜇≤𝜆, we get 𝑑_𝑙𝑥^∗,𝑧_𝑛≤𝛿𝑑_𝑙𝑥^∗,𝑧_𝑛−1+𝜇𝑑_𝑙𝑧_𝑛−1,𝑧_𝑛≤𝜆𝑑_𝑙𝑥^∗,𝑧_𝑛−1+𝜆^𝑛𝑑_𝑙𝑧₀,𝑧₁ (39) and hence 𝑑_𝑙𝑥^∗,𝑧_𝑛≤𝜆^𝑛𝑑_𝑙𝑥^∗,𝑧₀+𝑛𝜆^𝑛𝑑_𝑙𝑧₀,𝑧₁. (40)
On taking limit as 𝑛→+∞ in (40), we have 𝑑_𝑙𝑥^∗,𝑧_𝑛=0. (41) Similarly, 𝑑_𝑙𝑧_𝑛,𝑦⟶0as𝑛⟶+∞. (42) Now by using inequality (41) and (42), we have 𝑑_𝑙𝑥^∗,𝑦≤𝑑_𝑙𝑥^∗,𝑧_𝑛+𝑑_𝑙𝑧_𝑛,𝑦−𝑑_𝑙𝑧_𝑛,𝑧_𝑛⟶0as𝑛⟶+∞. (43) So, 𝑥^∗=𝑦.

Remark 23. In Theorem 18, the common fixed point of 𝑆 and 𝑇 may not be unique. However, fixed point is unique in Theorem 18, if for every pair of elements 𝑥,𝑦 in 𝐵(𝑥₀,𝑟)there exists a point 𝑧₀∈𝐵(𝑥₀,𝑟) such that 𝑧₀⪯𝑥 and 𝑧₀⪯𝑦 and 𝑧_𝑛∈𝐵(𝑥₀,𝑟), where 𝑧₁=𝑇𝑧₀, 𝑧₂=𝑆𝑧₁,….

Metric version of Theorem 18 is given below.

Theorem 24. Let (𝑋,⪯,𝑑) be an ordered complete metric space, 𝑥₀∈𝑋, 𝑟>0, and 𝑆,𝑇∶𝑋→𝑋 two dominated mappings. Suppose that there exist 𝑎,𝑏,𝑐∈[0,1) with 𝑎+2𝑏+2𝑐<1 such that 𝑑𝑑()𝑑,𝑆𝑥,𝑇𝑦≤𝑎𝑑𝑥,𝑦+𝑏𝑥,𝑆𝑥+𝑑𝑦,𝑇𝑦+𝑐𝑦,𝑆𝑥+𝑑𝑥,𝑇𝑦 (44) for all comparable elements 𝑥,𝑦 in 𝐵(𝑥₀,𝑟) and 𝑑𝑥₀,𝑆𝑥₀≤(1−𝜆)𝑟, (45) where 𝜆=(𝑎+𝑏+𝑐)/(1−𝑏−𝑐). If for a nonincreasing sequence {𝑥_𝑛} in 𝐵(𝑥₀,𝑟), 𝑥_𝑛→𝑢 implies that 𝑢⪯𝑥_𝑛, then there exists a point 𝑥^∗ in 𝐵(𝑥₀,𝑟) such that 𝑥^∗=𝑆𝑥^∗=𝑇𝑥^∗.

Now we apply our Theorem 22 to obtain a unique common fixed point of three mappings in ordered complete dislocated metric space. To this aim the innovative technique introduced in [29].

Theorem 25. Let (𝑋,⪯,𝑑_𝑙) be an ordered dislocated metric space and 𝑆,𝑇,𝑓∶𝑋→𝑋 such that 𝑆,𝑇 are 𝑓-dominated mappings and 𝑆𝑋∪𝑇𝑋⊂𝑓𝑋. Assume that the following condition holds for 𝑎,𝑏,𝑐∈[0,1) such that 𝑎+2𝑏+2𝑐<1: 𝑑_𝑙𝑆𝑥,𝑇𝑦≤𝑎𝑑_𝑙𝑓𝑥,𝑓𝑦+𝑏𝑑_𝑙𝑓𝑥,𝑆𝑥+𝑑_𝑙𝑑𝑓𝑦,𝑇𝑦+𝑐_𝑙𝑓𝑦,𝑆𝑥+𝑑_𝑙𝑓𝑥,𝑇𝑦, (46) for all comparable elements 𝑓𝑥,𝑓𝑦∈𝑓𝑋.
Suppose that for a nonincreasing sequence {𝑥_𝑛} in 𝑓𝑋, 𝑥_𝑛→𝑢 implies that 𝑢⪯𝑥_𝑛. Also for any two points 𝑧 and 𝑥 in 𝑓𝑋 there exists a point 𝑦∈𝑓𝑋 such that 𝑦⪯𝑧, 𝑦⪯𝑥. If fX is a complete subspace of 𝑋and (𝑆,𝑓) and (𝑇,𝑓) are weakly compatible, then 𝑆, 𝑇 and 𝑓 have a unique common fixed point 𝑓𝑧 in 𝑓𝑋. Moreover 𝑑_𝑙(𝑓𝑧,𝑓𝑧)=0.

Proof. By Lemma 13, there exists 𝐸⊂𝑋 such that 𝑓𝐸=𝑓𝑋 and 𝑓∶𝐸→𝑋 is one-to-one. Now since 𝑆𝑋∪𝑇𝑋⊂𝑓𝑋, we define two mappings 𝑔,ℎ∶𝑓𝐸→𝑓𝐸 by 𝑔(𝑓𝑥)=𝑆𝑥 and ℎ(𝑓𝑥)=𝑇𝑥, respectively. Since 𝑓 is one-to-one on 𝐸, then 𝑔,ℎ are well defined. As 𝑆𝑥⪯𝑓𝑥 implies that 𝑔(𝑓𝑥)⪯𝑓𝑥 and 𝑇𝑥⪯𝑓𝑥 implies that ℎ(𝑓𝑥)⪯𝑓𝑥, therefore 𝑔 and ℎ are dominated mappings. Let 𝑦₀=𝑓𝑥₀; choose a point 𝑦₁ in 𝑓𝑋 such that 𝑦₁=ℎ(𝑦₀).As ℎ(𝑦₀)⪯𝑦₀, so 𝑦₁⪯𝑦₀, and let 𝑦₂=𝑔(𝑦₁). Now 𝑔(𝑦₁)⪯𝑦₁ gives 𝑦₂⪯𝑦₁. Continuing this process and having chosen 𝑦_𝑛 in 𝑓𝑋 such that 𝑦_2𝑖+1=ℎ𝑦_2𝑖,𝑦_2𝑖+2=𝑔𝑦_2𝑖+1,where𝑖=0,1,2,…, (47) then 𝑦_𝑛+1⪯𝑦_𝑛 for all 𝑛∈𝑁. Note that for 𝑓𝑥,𝑓𝑦∈𝑓𝑋, where 𝑓𝑥 and 𝑓𝑦 are comparable and 𝑎,𝑏,𝑐∈[0,1) such that 𝑎+2𝑏+2𝑐<1, by using inequality (42), we have 𝑑_𝑙𝑔𝑓𝑥,ℎ𝑓𝑦≤𝑎𝑑_𝑙𝑑𝑓𝑥,𝑓𝑦+𝑏_𝑙𝑓𝑥,𝑔𝑓𝑥+𝑑_𝑙𝑑𝑓𝑦,ℎ𝑓𝑦+𝑐_𝑙𝑓𝑦,𝑔𝑓𝑥+𝑑_𝑙𝑓𝑥,ℎ𝑓𝑦. (48)
As 𝑓𝑋 is a complete space, so that all conditions of Theorem 22 are satisfied, we deduce that there exists a unique common fixed point 𝑓𝑧∈𝑓𝑋 of 𝑔 and ℎ. Also 𝑑_𝑙(𝑓𝑧,𝑓𝑧)=0. Now 𝑓𝑧=𝑔(𝑓𝑧)=ℎ(𝑓𝑧) or 𝑓𝑧=𝑆𝑧=𝑇𝑧=𝑓𝑧. Thus 𝑓𝑧 is a point of coincidence of 𝑆, 𝑇 and 𝑓. Let 𝑣∈𝑓𝑋 be another point of coincidence of 𝑓, 𝑆 and 𝑇; then there exists 𝑢∈𝑋 such that 𝑣=𝑓𝑢=𝑆𝑢=𝑇𝑢, which implies that 𝑓𝑢=𝑔(𝑓𝑢)=ℎ(𝑓𝑢). A contradiction as 𝑓𝑧∈𝑓𝑋 is a unique common fixed point of 𝑔 and ℎ. Hence 𝑣=𝑓𝑧. Thus 𝑆, 𝑇 and 𝑓 have a unique point of coincidence 𝑓𝑧∈𝑓𝑋. Now since (𝑆,𝑓) and (𝑇,𝑓) are weakly compatible, by Lemma 14𝑓𝑧 is a unique common fixed point of 𝑆, 𝑇 and 𝑓.

Unique common fixed point result of three mappings in complete dislocated metric space is given below, which can be proved with the help of Theorem 21.

Theorem 26. Let (𝑋,𝑑_𝑙) be a dislocated metric space and𝑆, 𝑇 and 𝑓self-mappings on 𝑋 such that 𝑆𝑋∪𝑇𝑋⊂𝑓𝑋. Assume that the following condition holds: 𝑑_𝑙𝑆𝑥,𝑇𝑦≤𝑎𝑑_𝑙𝑓𝑥,𝑓𝑦+𝑏𝑑_𝑙𝑓𝑥,𝑆𝑥+𝑑_𝑙𝑑𝑓𝑦,𝑇𝑦+𝑐_𝑙𝑓𝑦,𝑆𝑥+𝑑_𝑙𝑓𝑥,𝑇𝑦, (49) for all elements 𝑓𝑥,𝑓𝑦∈𝐵(𝑓𝑥₀,𝑟) where 𝑎,𝑏,𝑐∈[0,1) such that 𝑎+2𝑏+2𝑐<1 and for 𝑟>0, 𝑑_𝑙𝑓𝑥₀,𝑇𝑥₀≤(1−𝜆)𝑟, (50) where𝜆=(𝑎+𝑏+𝑐)/(1−𝑏−𝑐). If 𝑓𝑋, is a complete subspace of 𝑋and (𝑆,𝑓) and (𝑇,𝑓) are weakly compatible, then 𝑆, 𝑇 and 𝑓 have a unique common fixed point 𝑓𝑧 in 𝐵(𝑓𝑥₀,𝑟). Moreover, 𝑑_𝑙(𝑓𝑧,𝑓𝑧)=0.

In the following theorem we use Theorem 21 to establish the existence of a unique common fixed point of four mappings on closed ball in complete dislocated metric space.

Theorem 27. Let (𝑋,𝑑_𝑙) be a dislocated metric space and𝑆, 𝑇, 𝑔, and 𝑓 self-mappings on 𝑋 such that 𝑆𝑋,𝑇𝑋⊂𝑓𝑋=𝑔𝑋. Assume that the following condition holds: 𝑑_𝑙𝑆𝑥,𝑇𝑦≤𝑎𝑑_𝑙𝑓𝑥,𝑔𝑦+𝑏𝑑_𝑙𝑓𝑥,𝑆𝑥+𝑑_𝑙𝑑𝑔𝑦,𝑇𝑦+𝑐_𝑙𝑔𝑦,𝑆𝑥+𝑑_𝑙𝑓𝑥,𝑇𝑦, (51) for all elements 𝑓𝑥,𝑓𝑦∈𝐵(𝑓𝑥₀,𝑟)⊆𝑓𝑋, such that 𝑎,𝑏,𝑐∈[0,1) with 𝑎+2𝑏+2𝑐<1 and for 𝑟>0, 𝑑_𝑙𝑓𝑥₀,𝑆𝑥₀≤(1−𝜆)𝑟, (52) where𝜆=(𝑎+𝑏+𝑐)/(1−𝑏−𝑐). If 𝑓𝑋 is a complete subspace of 𝑋and (𝑆,𝑓) and (𝑇,𝑔) are weakly compatible, then 𝑆, 𝑇, 𝑓, and 𝑔 have a unique common fixed point 𝑓𝑧 in 𝐵(𝑓𝑥₀,𝑟). Moreover, 𝑑_𝑙(𝑓𝑧,𝑓𝑧)=0.

Proof. We use the technique introduced in [29]. By Lemma 13, there exists 𝐸₁,𝐸₂⊂𝑋 such that 𝑓𝐸₁=𝑓𝑋=𝑔𝑋=𝑔𝐸₂, 𝑓∶𝐸₁→𝑋, and 𝑔∶𝐸₂→𝑋 are one-to-one. Now define the mappings 𝐴,𝐵∶𝑓𝐸₁→𝑓𝐸₁ by 𝐴(𝑓𝑥)=𝑆𝑥 and 𝐵(𝑔𝑥)=𝑇𝑥, respectively. Since 𝑓,𝑔 are one-to-one on 𝐸₁ and 𝐸₂, respectively, then the mappings 𝐴,𝐵 are well defined. As 𝑓𝑥₀∈𝐵(𝑓𝑥₀,𝑟)⊆𝑓𝑋, then 𝑓𝑥₀∈𝑓𝑋. Let 𝑦₀=𝑓𝑥₀, choose a point 𝑦₁ in 𝑓𝑋 such that 𝑦₁=𝐴(𝑦₀), and let 𝑦₂=𝐵(𝑦₁); continuing this process and chosen 𝑦_𝑛 in 𝑓𝑋 such that 𝑦_2𝑖+1=𝐴𝑦_2𝑖,𝑦_2𝑖+2=𝐵𝑦_2𝑖+1,where𝑖=0,1,2,…. (53) Following similar arguments of Theorem 18, we deduce that 𝑦_𝑛∈𝐵(𝑓𝑥₀,𝑟). Also by inequality (51), we have 𝑑_𝑙𝑓𝑥₀,𝐴𝑓𝑥₀≤(1−𝜆)𝑟. (54) By using inequality (46), for 𝑓𝑥,𝑔𝑦∈𝐵(𝑓𝑥₀,𝑟) and 𝑎+2𝑏+2𝑐<1 we have 𝑑_𝑙𝐴𝑓𝑥,𝐵𝑔𝑦≤𝑎𝑑_𝑙𝑑𝑓𝑥,𝑔𝑦+𝑏_𝑙𝑓𝑥,𝐴𝑓𝑥+𝑑_𝑙𝑑𝑔𝑦,𝐵𝑔𝑦+𝑐_𝑙𝑔𝑦,𝐴𝑓𝑥+𝑑_𝑙𝑓𝑥,𝐵𝑔𝑦. (55) As 𝑓𝑋 is a complete space, all conditions of Theorem 21 are satisfied; we deduce that there exists a unique common fixed point 𝑓𝑧∈𝐵(𝑓𝑥₀,𝑟) of 𝐴 and 𝐵. Furthermore 𝐴 and 𝐵 have no fixed point other than 𝑓𝑧. Also 𝑑_𝑙(𝑓𝑧,𝑓𝑧)=0. Now 𝑓𝑧=𝐴(𝑓𝑧)=𝐵(𝑓𝑧) or 𝑓𝑧=𝑆𝑧=𝑓𝑧. Thus 𝑓𝑧 is a point of coincidence of 𝑓 and 𝑆. Let 𝑤∈𝐵(𝑓𝑥₀,𝑟) be another point of coincidence of 𝑆 and 𝑓, then there exists 𝑢∈𝐵(𝑓𝑥₀,𝑟) such that 𝑤=𝑓𝑢=𝑆𝑢, which implies that 𝑓𝑢=𝐴(𝑓𝑢). A contradiction as 𝑓𝑧∈𝐵(𝑓𝑥₀,𝑟) is a unique fixed point of 𝐴. Hence 𝑤=𝑓𝑧. Thus 𝑆 and 𝑓 have a unique point of coincidence 𝑓𝑧∈𝐵(𝑓𝑥₀,𝑟). Since (𝑆,𝑓) are weakly compatible, by Lemma 14??𝑓𝑧 is a unique common fixed point of 𝑆 and 𝑓. As 𝑓𝑋=𝑔𝑋 then there exists 𝑣∈𝑋 such that 𝑓𝑧=𝑔𝑣. Now as 𝐴(𝑓𝑧)=𝐵(𝑓𝑧)=𝑓𝑧 implies that 𝐴(𝑔𝑣)=𝐵(𝑔𝑣)=𝑔𝑣⇒𝑇𝑣=𝑔𝑣, thus 𝑔𝑣 is the point of coincidence of 𝑇 and 𝑔. Now if 𝑇𝑥=𝑔𝑥, then we have 𝐵(𝑔𝑥)=𝑔𝑥, a contradiction. This implies that 𝑔𝑣=𝑔𝑥. As (𝑇,𝑔) are weakly compatible, we obtain that 𝑔𝑣 is the unique common fixed point for 𝑇 and 𝑔. But 𝑔𝑣=𝑓𝑧. Thus 𝑆, 𝑇, 𝑔, and 𝑓 have a unique common fixed point 𝑓𝑧∈𝐵(𝑓𝑥₀,𝑟).

Theorem 28. Let (𝑋,⪯,𝑑_𝑙) be an ordered dislocated metric space and 𝑆,𝑇,𝑓∶𝑋→𝑋 such that 𝑆,𝑇 are 𝑓-dominated mappings and 𝑆𝑋∪𝑇𝑋⊂𝑓𝑋. Assume that the following condition holds for 𝑎,𝑏,𝑐∈[0,1) such that 𝑎+2𝑏+2𝑐<1𝑑_𝑙𝑆𝑥,𝑇𝑦≤𝑎𝑑_𝑙𝑓𝑥,𝑓𝑦+𝑏𝑑_𝑙𝑓𝑥,𝑆𝑥+𝑑_𝑙𝑑𝑓𝑦,𝑇𝑦+𝑐_𝑙𝑓𝑦,𝑆𝑥+𝑑_𝑙𝑓𝑥,𝑇𝑦, (56) for all comparable elements 𝑓𝑥,𝑓𝑦∈𝑓𝑋. Suppose that for a nonincreasing sequence {𝑥_𝑛} in 𝑓𝑋, 𝑥_𝑛→𝑢 implies that 𝑢⪯𝑥_𝑛. Also for any two points 𝑧 and 𝑥 in 𝑓𝑋 there exists a point 𝑦∈𝑓𝑋 such that 𝑦⪯𝑧, 𝑦⪯𝑥. If 𝑓𝑋 is a complete subspace of 𝑋, then 𝑆, 𝑇 and 𝑓 have a unique point of coincidence 𝑓𝑧 in 𝑓𝑋. Moreover 𝑑_𝑙(𝑓𝑧,𝑓𝑧)=0.

Theorem 29. Let (𝑋,𝑑_𝑙) be a dislocated metric space and𝑆, 𝑇, 𝑔, and 𝑓 self-mappings on 𝑋 such that 𝑆𝑋,𝑇𝑋⊂𝑓𝑋=𝑔𝑋. Assume that the following condition holds: 𝑑_𝑙𝑆𝑥,𝑇𝑦≤𝑎𝑑_𝑙𝑓𝑥,𝑔𝑦+𝑏𝑑_𝑙𝑓𝑥,𝑆𝑥+𝑑_𝑙𝑑𝑔𝑦,𝑇𝑦+𝑐_𝑙𝑔𝑦,𝑆𝑥+𝑑_𝑙𝑓𝑥,𝑇𝑦, (57) for all elements 𝑓𝑥,𝑓𝑦∈𝐵(𝑓𝑥₀,𝑟)⊆𝑓𝑋, with 𝑎,𝑏,𝑐∈[0,1) such that 𝑎+2𝑏+2𝑐<1 and for 𝑟>0, 𝑑_𝑙𝑓𝑥₀,𝑆𝑥₀≤(1−𝜆)𝑟, (58) where𝜆=(𝑎+𝑏+𝑐)/(1−𝑏−𝑐). If 𝑓𝑋 is a complete subspace of 𝑋, then 𝑆, 𝑇, 𝑓, and 𝑔 have a unique point of coincidence 𝑓𝑧 in 𝐵(𝑓𝑥₀,𝑟). Moreover, 𝑑_𝑙(𝑓𝑧,𝑓𝑧)=0.

Remark 30. We can obtain the metric version of all theorems which are still not present in the literature.

3. Conclusion

Azam et al. [18] very recently exploited the idea of fixed points and proved a significant result concerning the existence of fixed points for fuzzy mappings on closed ball in a complete metric space. We continue their investigations, and in this paper, some common fixed point theorems for mappings under Hardy Rogers contractive conditions on a closed ball in a ordered complete dislocated metric space (𝑋,⪯,𝑑_𝑙) have been discussed. Our analysis is based on the simple observation that fixed point results can be deduced from fixed point theory of mappings on closed balls. Practically speaking, there are many situations in which the mappings are not contractive on the whole space but instead they are contractive on its subsets. We feel that this aspect of finding the fixed points via closed balls was overlooked and our paper will bring a lot of interest into this area. Furthermore, we have applied the concept of dominated mappings in the process of investigating the existence of unique fixed point of contractive mappings on closed balls in the settings of ordered dislocated metric spaces.

Acknowledgment

Pasquale Vetro is supported by Università degli Studi di Palermo (Local University Project ex 60%).