# A Multiple Fixed Point Result forθ,ϕ,ψ-Type Contractions in the Partially Ordered s-Distance Spaces with an Application

Authors: Maliha Rashid; Amna Kalsoom; Abdul Ghaffar; Mustafa Inc; Ndolane Sene
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this manuscript, the aim is to prove a multiple fixed point (FP) result for partially ordereds-distance spaces under θ,ϕ,ψ-type weak contractive condition. The result will generalize some well-known results in literature such as coupled FP (Guo and Lakshmikantham, 1987), triple fixed point (Berinde and Borcut, 2011), and quadruple FP results (Karapinar, 2011). Moreover, to validate the result, an application for the existence of solution of a system of integral equations is also provided.


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## Body

## 1. Introduction

In pure mathematics, the Banach fixed-point theorem [1] (contractive mapping theorem or also known as the contraction mapping theorem) is main result in the study of metric spaces; it assurances the uniqueness and existence of FP of certain self-maps of metric spaces and requires a constructive technique to discover those FP. It can be understood as an abstract formulation of Picard’s method of successive approximations. The theorem is named after Stefan Banach (1892-1945) who first stated it in 1922. It has numerous applications in different fields such as computer science, physics, engineering, and various branches of mathematics itself. FP theory as a whole got an upward flight after this celebrated result.Many authors started working in this field, and soon it became a hot field of research. A number of authors have extended this fundamental result to nonlinear analysis [2]. Following this streak, Guo and Lakshmikantham [3] established the idea of doubled FP. This is considered to be the first of its nature and was extended to triple FP by Berinde and Borcut [4]. Continuing in this direction, Karapinar [5] used four variable to strengthen the idea of quadruple FP in partially ordered metric spaces. In 2012, Berzig and Samet [6] discussed the existence of the fixed point of N-order for m-mixed monotone mappings in complete ordered metric spaces. In the same year, Roldán et al. [7] extended the notion of the FP of N-order to the ϕ-fixed point and obtained some ϕ-fixed point theorems for a mixed monotone mapping in partially ordered complete metric spaces. In [8], Karapinar et al. studied the existence and uniqueness of a FP of the multidimensional operators which satisfy Meir-Keeler type contraction condition. Soon after, a number of articles were published to discussed the concept of a “ multidimensional FP” or “an m-tuples fixed point.” For applications of such results, we refer the reader to [9] and the references cited therein.In 2016, Choban and Berinde [10] generalized metric spaces to distance spaces. They established multidimensional FP results for ordered spaces with distance under certain contractive conditions [4, 11]. They pointed out that the concept allowed them to reduce the multidimensional case of FP and coincidence points to the one dimensional case. Recently, Rashid et al. [12] established some multiple FP findings for the C-distance spaces in the existence of various contractive mapping.The abovementioned ideas serve as motivation of the work in the present paper. We have extended the results of [10] for partially ordered s-distance space, in terms of a significant multiple FP result under θ,ϕ,ψ-type contractive condition [13]. In support of this result, an example is also given.Since, in Section1, introduction and historical background of generalized metric spaces is given. In Section 2, preliminaries and some basic definitions are stated. In Section 3, main result is stated, and in Section 2, some consequences and examples of our main result will be described. In Section 3, an application is stated to support our main result. In the last section, article is concluded.


## 2. Preliminaries

Definition 1 [11].
Consider theM, as a nonempty set and a function σ:M×M⟶ℝ is called s-distance on M if for all ς,ξ,η∈M, σ satisfies the following axioms:
(1)
σς,ξ≥0,∀ς,ξ∈M(2)
σς,ξ+σξ,ς=0 if and only if ς=ξ,∀ς,ξ∈M(3)
For a positive real numbers>0(1)σς,ξ≤sσς,η+ση,ξ,∀ς,ξ,η∈M.
Ans-distance space M,σ is said to be a symmetric s-distance space if σς,ξ=σξ,ς,∀ς,ξ∈M. Now, some remarks and examples as given below.Remark 2.
(1)
Everyb-metric space is an s-distance space but not conversely(2)
Ins-distance space, distance σ is not necessarily a continuous function, i.e., if ςn⟶ς and ξn⟶ξ then σςn,ξn↛σς,ξ(3)
In ans-distance space, the limit of a convergent sequence may not be uniqueExample 3.
LetM=α1,α2,α3, σ:M×M⟶0,∞ and
(2)σα1,α2=14,σα2,α1=12,σα1,α3=σα2,α3=1,σαi,αi=0,σα3,α1=σα3,α213,σαi,αj≥0,σαi,αj+σαj,αi=0⇔αi=αj,for all i,j∈1,2,3.(3)σα1,α2<σα1,α3+σα3,α2,σα1,α3<σα1,α2+σα2,α3,σα2,α3<σα2,α1+σα1,α3,σα2,α1<σα2,α3+σα3,α1,σα3,α1<σα3,α2+σα2,α1,σα2,α3<σα2,α1+σα1,α3.
Hence,M,σ is an s-distance space with s=1. However, it is not a b-metric space.Fixr∈ℕ and Γ=Γ1,⋯,Γr is said to be a collection of mappings such that
(4)Γi´:1,2,3,⋯,r⟶1,2,3,⋯,r;1≤i≤r.LetM,σ be a distance space and also G:Mr⟶M be a mapping. The mapping ΓG:Mr⟶Mr which is a composition of G and Γ is gvien as
(5)ΓGς1,⋯,ςr=ξ1,⋯,ξr,ξi=GςΓi1,⋯,ςΓir,for any point ς1,⋯,ςr∈Mr and i´∈1,2,⋯,r. A point a=a1,⋯,ar∈Mr is considered as a Γ-multiple FP of G if it is a FP of ΓG, i.e., a=ΓGa and
(6)ai´=GaΓi´1,⋯,aΓi´rforanyi´∈1,2,3,⋯,r.LetM,σ be a distance space, r∈ℕ=1,2,⋯. On Mr, consider the distance
(7)σrς1,⋯,ςr,ξ1,⋯,ξr=supσςi´,ξi´:i´≤r.Obviously,Mr,σr is a distance space, too.The following are some basic concepts from [14]:Consider a partially ordered distance spaceM,σ,≼, r be a positive integer and J,K be a partition of Jr=1,2,⋯,r, i.e., J,K≠ϕ, J∪K=Jr, and J∩K=ϕ. Define Mr=M×M×… ×Mrtimes the Cartesian product of the set M. Define a partial order ≼r over Mr as follows:For anyω=ς1,ς2,⋯,ςr,ν=ξ1,ξ2,⋯,ξr∈Mr, ω≼rν if and only if ςi´≼i´ξi´ for all i´∈Jr, where
(8)ς≼i′ξiffς≼ξ,ifi´∈J,ς≽ξ,ifi´∈K.The functionσr:Mr×Mr⟶0,+∞ given by
(9)σrς,ξ=sup1≤i´≤rσςi´,ξi´,defines a distance on Mr, where ς=ς1,ς2,⋯,ςr and ξ=ξ1,ξ2,⋯,ξr. Obviously, Mr,σr,≼r is a partially ordered distance space, so that it inherits the properties of M,σ,≼ and σrςϑ,ς⟶0 if and only if σςi´ϑ,ςi´⟶0 as ϑ⟶∞ for all i´∈Jr.Definition 4. [15].
Letg be a self mapping on M. A mapping G has the mixed g-monotone property with respect to the partition J,K, if G is g-monotone nondecreasing in arguments of J and g-monotone nonincreasing in arguments of K, i.e., ∀ς1,ς2,⋯,ςr, ξ,η∈M, i´∈Jr, and
(10)gξ≼gη⇒Gς1,⋯,ςi´−1,ξ,ςi´+1,⋯,ςr≼i´Gς1,⋯,ςi´−1,η,ςi´+1,⋯,ςr.Ifg is the identity mapping on M, then the mapping G has the mixed monotone property with respect to the partition J,K. Define a set of mappings by
(11)ΩJ,K=Γi´:Jr⟶Jr:Γi´J⊆J,Γi´K⊆K,ΩJ,K′=Γi´:Jr⟶Jr:Γi´J⊆K,Γi´K⊆J,such that Γi´∈ΩJ,K if i´∈J and Γi´∈ΩJ,K′ if i´∈K.Definition 5.
If a functionψ:0,∞⟶0,∞ is continuous, nondecreasing, and ψ0=0, then it is called an altering distance function.Definition 6.
The metric spaceM,σ,≼ is called regular if it satisfies the following properties:
(1)
Ifςr is a nondecreasing sequence and ςr⟶ς then ςr≼ς for all r≥1(2)
Ifςr is a nonincreasing sequence and ςr⟶ς then ςr±ς for all r≥1


## 3. Main Result

Berinde and Borcut [4] extended the concept of multidimensional FP to ordered distance spaces by utilizing the properties of contractive type mappings. Keeping ourselves in touch with all these concepts, we are extending these results to symmetric s-distance spaces by using a combination of altering distance functions. This result will generalize the main theorems of [14], in which the space considered is a metric space. It is also valid for b-metric spaces.Theorem 7.
Consider a complete partially ordered symmetrics-distance space.
M,σ,≼ and Γ=Γ1,Γ2,⋯,Γr be collection of mappings verifying Γi´∈ΩJ,K if i´∈J and Γi´∈ΩJ,K′ if i´∈K. If the mapping G:Mr⟶M satisfies the following conditions:
(a)
Forμ>0
ψμ−θμ−φμ>0,(∗)
whereψ is an altering distance function and θ:0,∞⟶0,∞ and φ:0,∞⟶0,∞ are upper semicontinuous and increasing functions such that θ0=φ0=0 satisfying
(12)ψσGς,Gξ≤θσrς,ξs+1+φσrς,ξs+1,for all ς,ξ∈Mr with ς≼rξ.(b)
There isς0=ς10,ς20,⋯,ςr0 such that ςi´0≼i´GςΓi´10,⋯,ςΓi´r0 for all i´∈Jr(c)
G has mixed monotone property with respect to J,K(d)
G is continuous, or M,σ,≼ is regular; then, G has Γ-multiple FP(e)
Moreover, if forς and ξ in Mr, there is η∈Mr such that ς≼rη and ξ≼rη; then, G has a unique Γ-multiple FPProof.
Step 1. Letςϑ=ΓGϑς0 be the nth Picard iterate of ς0 under ΓG, i.e., ςϑ=ΓGϑς0=ς1ϑ,ς2ϑ,⋯,ςrϑ, where
(13)ς1ϑ=GςΓ11ϑ−1,ςΓ12ϑ−1,⋯ςΓ1rϑ−1,ς2ϑ=GςΓ21ϑ−1,ςΓ22ϑ−1,⋯ςΓ2rϑ−1,⋮ςrϑ=GςΓr1ϑ−1,ςΓr2ϑ−1,⋯ςΓrrϑ−1.
By conditionb and definition of ΓG, it follows that ς0≼rς1. Since, G has mixed monotone property so ΓG is monotone nondecreasing [15]; therefore
(14)ςϑ−1≼rςϑ∀ϑ≥1.
Step 2. We need to show thatlimϑ⟶∞σrςϑ−1,ςϑ=0. Set Di´ϑ=σςi´ϑ−1,ςi´ϑ and Dϑ=supi´∈JrDi´ϑ=σrςϑ−1,ςϑ. If Dϑ=0 for some ϑ≥1 then ςϑ−1=ΓGςϑ−1 which means G has Γ-multiple FP which completes the proof. Assume Dϑ>0 for all ϑ≥1. Since ςϑ−1≼rςϑ and Γi´Jr⊆Jr, it follows that
(15)ςΓi´1ϑ−1,ςΓi´2ϑ−1,⋯,ςΓi´rϑ−1≼rςΓi´1ϑ,ςΓi´2ϑ,⋯ςΓi´rϑ,for any i´∈Jr and ϑ≥1. Now, using condition (12)
(16)ψDi′ϑ=ψσGςΓi´1ϑ−2,ςΓi´2ϑ−2,⋯,ςΓi´rϑ−2,GςΓi´1ϑ−1,ςΓi´2ϑ−1,⋯,ςΓi´rϑ−1,≤θsupj∈JrσςΓi´jϑ−2,ςΓi´jϑ−1s+1+φsupj∈JrσςΓi´jϑ−2,ςΓi´jϑ−1s+1.
Since,θ and φ both are increasing and s>0 so that
(17)ψDi´ϑ≤θsupj∈JrσςΓi´jϑ−2,ςΓi´jϑ−1+φsupj∈JrσςΓi´jϑ−2,ςΓi´jϑ−1,for any i´∈Jr, since Jr is a finite set, there is an index i´ϑ∈Jr such that supi´∈JrDi´ϑ=Di´ϑϑ. From above inequality, it follow that
(18)ψDi´ϑϑ=ψDi´ϑϑ,=ψσGςΓi´ϑ1ϑ−2,ςΓi´ϑ2ϑ−2,⋯,ςΓi´ϑrϑ−2,GςΓi´ϑ1ϑ−1,ςΓi´ϑ2ϑ−1,⋯,ςΓi´ϑrϑ−1,≤θsupj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1s+1+φsupj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1s+1,≤θsupj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1+φsupj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1.
Since(19)0<supj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1≤Dϑ−1,for all ϑ≥1. Therefore, from the inequality ∗, we get
(20)θsupj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1+φsupj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1<ψsupj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1≤ψsupj∈Jrσςjϑ−2,ςjϑ−1.
Combining (19) and (20), we have
(21)ψDϑ<ψsupj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1≤ψDϑ−1,for all ϑ≥1. Since ψ is an altering distance function, it follows that
(22)Dϑ<supj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1≤Dϑ−1.
Hence, the sequenceDϑ and supj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1 are monotone decreasing and bounded below. So, we have τ≥0 such that
(23)limϑ⟶∞Dϑ=limϑ⟶∞supj∈JrσςΓi´ϑjϑ−2,ςΓi´ϑjϑ−1=τ.
We need to show thatτ=0. Suppose τ>0, then by applying limit as ϑ⟶∞ in (19) and utilizing the properties of ψ,θ and φ, we have ψτ−θτ−φτ≤0, which contradicts ∗. Hence
(24)limϑ⟶∞Dϑ=limϑ⟶∞σrςϑ−1,ςϑ=0.
Step 3. Now to prove that the sequenceςϑϑ∈ℕ is Cauchy. Suppose on contrary that it is not Cauchy, then there exists ε>0 for which there are subsequences ςϑζ and ςrζ of ςϑ with ϑζ>rζ>ζ such that
(25)σrςrς,ςϑς≥ε.
Letϑζbe the smallest integer satisfying above, then
(26)σrςrς,ςϑς−1<ε.
Assume thats<1, then consider
(27)σrςrζ,ςϑζ≤sσrςrζ,ςϑζ−1+σrςϑζ−1,ςϑζ<ε+σrςϑζ−1,ςϑζ.
On lettingζ⟶∞ and using condition (24), we get
(28)ε≤limsupζ⟶∞σrςrς,ςϑζ<ε,which leads to a contradiction. Now consider s≥1, then
(29)σrςrζ,ςϑζ≤sσrςrζ,ςϑζ−1+σrςϑζ−1,ςϑζ<sε+σrςϑζ−1,ςϑζ.
Again lettingζ⟶∞ and using condition (24), we get
(30)ε≤limsupζ⟶∞σrςrζ,ςϑζ<sε.
Now(31)σrςrζ−1,ςϑζ−1≤sσrςrζ−1,ςrζ+σrςrζ,ςϑζ−1,limsupζ⟶∞σrςrζ−1,ςϑζ−1<sε.
Fromςϑ−1≼rςϑ, it follows that
(32)ςrζ≼rςrζ+1≼r⋯≼rςϑζ−1≼rςϑζ≼rςϑζ+1.
Sinceςrζ−1≼rςϑζ−1 and Γi´Jr⊆Jr, we have
(33)ςri1rζ−1,ςri2rζ−1,⋯,ςrirrζ−1≼rςri1ϑζ−1,ςri2ϑζ−1,⋯,ςΓi´rϑζ−1,for any i´∈Jr and ζ≥1. By condition a, it follows that
(34)ψσςi´rζ,ςi´ϑζ≤ψσGςΓi´1rζ−1,ςΓi´2rζ−1,⋯ςΓi´rrζ−1,GςΓi´1rζ−1,ςΓi´2rζ−1,⋯ςΓi´rrζ−1,≤θsupj∈JrσςΓi´jrζ−1,ςΓi´jϑζ−1s+1+φsupj∈JrσςΓi´jrζ−1,ςΓi´jϑζ−1s+1,for any i´∈Jr. Since Jr is finite, there will be an index i´ζ in Jr, so
(35)supi´∈Jrσςi´rζ,ςi´ϑζ=σςi´ζrζ,ςi´ζϑζ.
From inequality (29)
(36)ψε≤ψσrςrζ,ςϑζ,≤ψσGςΓi´ζ1rζ−1,ςΓi´ζ2rζ−1,⋯ςΓi´ζrrζ−1,GςΓi´ζ1rζ−1,ςΓi´ζ2rζ−1,⋯ςΓi´ζrrζ−1,≤θsupj∈JrσςΓi´ζjrζ−1,ςΓi´ζjϑζ−1s+1+φsupj∈JrσςΓi´ζjrζ−1,ςΓi´ζjϑζ−1s+1.
Sinceθ and φ are increasing functions and Γi′Jr⊆Jr, so
(37)ψε≤θsupj∈JrσςΓi´ζjrζ−1,ςΓi´ζjϑζ−1s+φsupj∈JrσςΓi´ζjrζ−1,ςΓi´ζjϑζ−1s,≤θsupj∈Jrσςjrζ−1,ςjϑζ−1s+φsupj∈Jrσςjrζ−1,ςjϑζ−1s,≤θσrςrζ−1,ςϑζ−1s+φσrςrζ−1,ςϑζ−1s.
Applyinglimsupζ⟶∞ over above inequality and then using (29), we get
(38)ψε<θsεε+φsεε,<θε+φε,which is a contradiction to the condition ∗. Hence, ςϑϑ∈ℕ is a Cauchy sequence.
Step 4. Since the spaceM,σ is complete so Mr,σr is complete. Therefore, we have ν∈Mr such thatςϑϑ∈ℕ⟶ν, i.e.,
(39)limϑ⟶∞ς1ϑ=GςΓ11ϑ−1,ςΓ12ϑ−1,⋯,ςΓ1rϑ−1=ν1,limϑ⟶∞ς2ϑ=GςΓ21ϑ−1,ςΓ22ϑ−1,⋯,ςΓ2rϑ−1=ν2,⋮limϑ⟶∞ςrϑ=GςΓr1ϑ−1,ςΓr2ϑ−1,⋯,ςΓrrϑ−1=νr.
Next, we show thatν=ν1,ν2,⋯,νr∈Mr is a Γ-multiple FP of operator G. If condition c holds and G is continuous, then
(40)limϑ⟶∞GςΓ11ϑ−1,ςΓ12ϑ−1,⋯,ςΓ1rϑ−1=GνΓ11,νΓ12,⋯,νΓ1r,limϑ⟶∞GςΓ21ϑ−1,ςΓ22ϑ−1,⋯,ςΓ2rϑ−1=GνΓ21,νΓ22,⋯,νΓ2r,⋮limϑ⟶∞GςΓr1ϑ−1,ςΓr2ϑ−1,⋯,ςΓrrϑ−1=GνΓi1,νΓi2,⋯,νΓir.
Above shows that(41)νi´=GνΓi1,νΓi2,⋯,νΓir,foralli´∈Jr,which means the point ν=ν1,ν2,⋯,νr is a Γ-FP of G. Next, suppose M,σ≼ is regular. The above relation implies ςϑ=ς1ϑ,ς2ϑ,⋯,ςrϑ≼rν=ν1,ν2,⋯,νr. On the other hand
(42)ςΓi1ϑ,ςΓi2ϑ,⋯,ςΓi´rϑ≼rνΓi1,νΓi2,⋯,νΓir.
Since,Γi´Jr⊆Jr for any i´∈Jr. By using a(43)ψσGνΓi´1,νΓi´2,⋯,νΓi´r,GςΓi1ϑ,ςΓi´2ϑ,⋯,ςΓi´rϑ,≤θsupj∈JrσνΓi´j,ςΓi´jϑs+1+φmaxj∈JrσνΓi´j,ςΓi´jϑs+1,≤θsupj∈JrσνΓi´j,ςΓi´jϑ+φsupj∈JrσνΓi´j,ςΓi´jϑ.
Taking into account above and lettingϑ⟶∞ in the last inequality, it implies
(44)ψσGνΓi´1,νΓi´2,⋯,νΓi´r,νi´=0.
Hence(45)GνΓi´1,νΓi´2,⋯,νΓi´r=νi´foralli´∈Jr.
This completes the existence ofΓ-multiple FP.
Step 5. In this step, we will show that the FP ofG is unique.
Supposew∈Mr be another FP of G. By condition e, there exists η=η1,η2,⋯,ηr∈Mr such that ν≼rη and w≼rη.
Putη0=η and define
(46)η1ϑ=GηΓ11ϑ−1,ηΓ12ϑ−1,⋯,ηΓ1rϑ−1,η2ϑ=GηΓ21ϑ−1,ηΓ22ϑ−1,⋯,ηΓ2rϑ−1,⋮ηrϑ=GηΓr1ϑ−1,ηΓr2ϑ−1,⋯,ηΓrrϑ−1.
By the induction method, we have(47)ν≼rηϑ,w≼rηϑ,for all ϑ≥0. By condition e, ν≼rη0. Assume that above condition holds for ϑ−1.
Using the procedure of Step 1, it can be shown that(48)ν=GνΓi´1,νΓi´2,⋯,νΓi´r≼i´GηΓi´1ϑ−1,ηΓi´2ϑ−1,⋯,ηΓi´rϑ−1=ηi´ϑ,for all i´∈Jr; that is, ν≼rηϑ. Similarly, we can prove the second inequality. Further, we prove that
(49)limϑ⟶∞σrν,ηϑ=0.
For this, we first show that ifσrν,ηϑ=0 for some ϑ0 then σrν,ηϑ=0, for all ϑ≥ϑ0. Indeed, from above condition, it follows that
(50)νΓi´1,νΓi´2,⋯,νΓi´r≼rηΓi´1ϑ,ηΓi´2ϑ,⋯,ηΓi´rϑ,for all i´∈Jr and ϑ≥1. From a, it follows that
(51)ψσνi´,ηi´ϑ=ψσGνΓi´1,νΓi´2,⋯,νΓi´r,GηΓi´1ϑ−1,ηΓi´2ϑ−1,⋯,ηΓi´rϑ−1,≤θsupj∈JrσνΓi´j,ηΓi´jϑ−1s+1+φsupj∈JrσνΓi´j,ηΓi´jϑ−1s+1,≤θsupj∈JrσνΓi´j,ηΓi′jϑ−1+φsupj∈JrσνΓi´j,ηΓi´jϑ−1,for all i´∈Jr and ϑ≥1. Recall that Γi´Jr⊆Jr. Hence
(52)supj∈JrσνΓi´j,ηΓi´jϑ−1≤σrν,ηϑ−1,for all ϑ≥1. Taking into account ∗, it follows
(53)θsupj∈JrσνΓi´j,ηΓi´jϑ−1+φsupj∈JrσνΓi´j,ηΓi´jϑ−1,≤ψsupj∈JrσνΓi´j,ηΓi´jϑ−1≤ψσrν,ηϑ,for all i´∈Jr and ϑ≥1. Combining (50) and (52), we get
(54)ψsupj∈Jrσνi´,ηi´ϑ≤ψσrν,ηϑ−1,for ϑ≥1. Since, ψ is an altering distance function
(55)ψσrν,ηϑ≤ψσrν,ηϑ−1.
Now, it is obvious that ifσrν,ηϑ0=0, then σrν,ηϑ=0 for all ϑ≥ϑ0.
Next, assume thatσrν,ηϑ>0 for all ϑ≥1. Using the same manner adopted in Step 1, it can be proved that σrν,ηϑ≤σrν,ηϑ−1. Hence, there exists r≥0 such that limϑ⟶∞σrν,ηϑ=r. In similar way, it can be shown
(56)limϑ⟶∞σrν,ηϑ=0andlimϑ⟶∞σrw,ηϑ=0.
On the other hand,(57)σrν,w≤sσrν,ηϑ+σrηϑ,w.
Applyinglimϑ⟶∞ over the above inequality, it follows
(58)σrν,w=0.
This impliesν=w. Hence, G has a unique multidimensional FP.☐


## 4. Some Consequences and Example of Theorem7

In the following section, some important concept regarding the consequences of Theorem7 are discussed which are in terms of the main results represented in the articles [4, 16]. An illustrative example is also added in this section which will be helpful for the readers to understand the structure of multidimensional FP under the weak contractions for partially ordered s-distance spaces.Corollary 8.
LetM,σ,≼ is known as complete partially ordered s-distance space and Γ=Γ1,Γ2,⋯,Γr be a collection of mappings verifying Γi´∈ΩJ,K if i´∈J and Γi´∈ΩJ,K′ if i′∈K. Assume that the mapping G:Mr⟶M satisfies the following conditions:
(1)
If there existsγ∈0,1 such that(59)σGς1,ς2,⋯,ςr,Gξ1,ξ2,⋯,ξr≤γσrς,ξ,for all ς=ς1,ς2,⋯,ςr∈Mr and ξ=ξ1,ξ2,⋯,ξr∈Mr with ς≼rξ satisfies conditions.
(2)
Toe of the above theorem then G has a unique multidimensional FPProof.
We can prove this corollary easily by takingψς=ς, θς=s+1γς, and φς=0 in the above theorem.☐Remark 9.
Theorems 2.1 and 2.2 of [16], in s-distance spaces, follow from Corollary 8. In [16], contractive condition is
(60)σFς1,ς2,Fξ1,ξ2≤δ2σς1,ξ1+σς2,ξ2,and for ς,ξ∈M2 such that ς≼2ξ, it implies
(61)σFς1,ς2,Fξ1,ξ2≤δ2σς1,ξ1+σς2,ξ2≤δσ2ς,ξ.
Applying Corollary8, we get the desired result in s-distance space.Remark 10.
Corollary8 also generalizes the main triple FP result of [4], in which the space under consideration is a metric space. We generalize it for b-metric space and for an s-distance space. In [4], Γ=Γ1,Γ2,Γ3, A3 is chosen as A=1,3,B=2, collection of mappings is defined as
(62)Γ11Γ12Γ13Γ21Γ22Γ23Γ31Γ32Γ33=123213312,the contractive condition in [4] is
(63)σFς1,ς2,ς3,Fξ1,ξ2,ξ3≤δ1σς1,ξ1+δ2σς2,ξ2+δ3σς3,ξ3,for any ς=ς1,ς2,ς3,ξ=ξ1,ξ2,ξ3 with ς≼iξ and δ1,δ2,δ3≥0 and δ1+δ2+δ3<1. Obviously
(64)σFς1,ς2,ς3,Fξ1,ξ2,ξ3≤δ1+δ2+δ3σ3ς,ξ.
Applying Corollary8, we get the desired result in s-distance space.Corollary 11.
ConsiderM,σ,≼ as a complete partially ordered b-metric space and Γ=Γ1,Γ2,⋯,Γr be collection of mappings verifying Γi´∈ΩJ,K if i´∈J and Γi´∈ΩJ,K′ if i´∈K. Assume that the mapping G:Mr⟶M satisfies conditions a to e, then G has a unique multidimensional FP.Example 12.
LetM=ℝ, Mr=ℝr. Define σrω,ν=supi´∈Jrσςi´,ξi´ and σς,ξ=ς−ξ, for all ω=ς1,ς2,⋯,ςr,ν=ξ1,ξ2,⋯,ξr∈Mr. Now, define G:Mr⟶M, as Gς1,ς2,⋯,ςr=ς1/2.
Since “≤” is a partial order on M, therefore
(65)ω⋞rvifandonlyifς1⋞iξi,ς⋞iξiffς≤ξi∈J=1,2,⋯,kς≤ξi∈K=k+1,k+2,⋯,r,where J and K partitions Jr=1,2,⋯,r such that J∪K=Jr and J∩K=ϕ.
A functionψ:0,∞⟶0,∞ defined as ψt=1/3t is increasing and continuous, θ:0,∞⟶0,∞ defined as θt=1/9t, θ is increasing and upper semicontinuous function, and φ:0,∞⟶0,∞ defined as φt=2/9t, which is increasing and lower semicontinuous function, for all t∈0,∞.(66)σGω,Gν=ς1−ξ12,ψσGω,Gν=ς1−ξ16,θσrω,νs+1=σrω,ν9s+1,φσrω,νs+1=2σrω,ν9s+1.
Consider(67)θσrω,νs+1+φσrω,νs+1=σrω,ν9s+1+2σrω,ν9s+1=σrω,ν2s+1,≤σrω,ν2=supi´∈Jrςi´−ξi´2.
Thus, we have(68)ς1−ξ16≤supi´∈Jrςi´−ξi´2,which follows
(69)ψσGω,Gν≤θσrω,νs+1+φσrω,νs+1.
Now to showG has mixed monotone property, if ξ≼η⇒ξ≤η, consider
(70)Gς1,ς2,⋯,ςi−1,ξ,⋯,ςr=ς12,Gς1,ς2,⋯,ςi−1,η,⋯,ςr=ς12.
So(71)Gς1,ς2,⋯,ςi´−1,ξ,⋯,ςr≼i´Gς1,ς2,⋯,ςi´−1,η,⋯,ςr,which implies G has mixed monotone property with respect to J,K. Set of mappings is defined as
(72)Γ11Γ12⋯Γ1rΓ21Γ22⋯Γ2r⋮⋮⋮Γr1Γr2⋯Γrr=123⋯r234⋯1345⋯2⋮⋮⋮⋮r12⋯r−1.
Letς10,ς20,⋯,ςr0=ς1,ς2,⋯,ςr∈Mr and
(73)ς1ϑ=Gς1ϑ−1,ς2ϑ−1,⋯,ςrϑ−1=ς1ϑ−12,ς2ϑ=Gς2ϑ−1,ς3ϑ−1,⋯,ςrϑ−1,ς1ϑ−1=ς2ϑ−12,⋮ςrϑ=Gςrϑ−1,ς1ϑ−1,⋯,ςr−2ϑ−1,ςr−1ϑ−1=ςrϑ−12,for ϑ=1(74)ς11=ς102=ς12,ς21=ς202=ς22,⋮ςr1=ςr02=ςr2,for ϑ=2(75)ς12=ς112=ς122,ς22=ς212=ς222,⋮ςr2=ςr12=ςr22.
Continuing this process, we have(76)limϑ⟶∞ς1ϑ,ς2ϑ,⋯,ςrϑ=limϑ⟶∞ς12ϑ,ς22ϑ,⋯,ςr2ϑ=0,0,⋯,0=O1,⋯,Or.
Thus(77)Oi′=GOΓi´1,OΓi´2,⋯,OΓi´r,
Hence,Oi′ is a unique multidimensional FP of G.


## 5. An Application

To give the broad impact of FP results to the different areas of study like physics, biological sciences, engineering, game theory, and economics has always been an interest of various mathematicians. The applications can be found in various directions (like [17, 18]). In the following section, we have also included an application of the main result to find the solution of a system of integral equations.Consider the pointa,b∈ℝ with a<b and let I=a,b. Let the space M=CI of all continuous real valued functions defined on I, which define a partial order ≼ on M by
(78)λ≼μifandonlyifλt≤μt,forallt∈a,b,and the distance
(79)σλ,μ=maxt∈Iλt−μtpforallλ,μ∈M,p≥1,then M,σ is a complete partially ordered s-distance space with s=2p−1.Consider the following system of equations(80)ς1t=κ+∫atKwLς1w,ς2w,⋯,ςrwdw,(81)ςit=κ+∫atKwLςiw,ςi+1w,⋯,ςrw,ς1w,⋯,ςi−1wdw,for i=1,2,⋯,r, where L:ℝr⟶ℝ be a mapping verifying
(i)
L is continuous(ii)
For alle1,e2,⋯,er,h1,h2,⋯,hr∈ℝr(82)Le1,e2,⋯,er−Lh1,h2,⋯,hr≤sup1≤i≤rei−hip1/pb−as+12/p,and K:I⟶ℝ be a continuous mapping such that Kt≥0 and ∫abKtdt≤b−a.Define a mappingG:Mr⟶M for all ς=ς1,ς2,⋯,ςr in Mr and κ∈ℝ such that
(83)Gς1,ς2,⋯,ςrt=κ+∫atKwLς1w,ς2w,⋯,ςrwdw.Clearly,G∈CI. To find solution of the system (79), it is required to show that G has a multiple FP. For this, consider
(84)σGς1,ς2,⋯,ςr,Gν1,ν2,⋯,νr=maxt∈IGς1,ς2,⋯,ςrt−Gν1,ν2,⋯,νrtp,=maxt∈Iκ+∫atKwLς1w,ς2w,⋯,ςrwdw−κ+∫atKwLν1w,ν2w,⋯,νrwdwp,=maxt∈I∫atKwLς1w,ς2w,⋯,ςrw−Lν1w,ν2w,⋯,νrwdwp,≤maxt∈I∫atKwLς1w,ς2w,⋯,ςrw,−Lν1w,ν2w,⋯,νrwdwp,≤maxt∈I∫atKwsup1≤i≤rςiw−νiwp1/pb−as+12/pdwp,≤maxt∈I1b−as+12/p×sup1≤i≤r∫atKwmaxt∈Iςiw−νiwp1/pdwp,=maxt∈I1b−as+12/psup1≤i≤r∫atKwσςi,νi1/pdwp,≤sup1≤i≤rσςi,νis+121b−amaxt∈I∫atKwdwp,σGς1,ς2,⋯,ςr,Gν1,ν2,⋯,νr≤σrς1,ς2,⋯,ςr,ν1,ν2,⋯,νrs+12.Define functionsψ,θ,φ:0,∞⟶0,∞ by
(85)ψt=s+1t,θt=t,φt=0,such that for t>0(86)ψt−θt−φt>0.Then, from above inequality, we get(87)ψσGς,Gν≤θσrς,νs+1+φσrς,νs+1,which by Theorem 7 implies that G has a unique multiple FP, which gives the required solution of system (79).


## 6. Conclusion

The main idea of this paper was to prove a multiple fixed point (FP) result for partially ordereds-distance spaces under θ,ϕ,ψ-type weak contractive condition which is the generalization of some well-known results in literature. Many other related and relevant results can be obtained in the same manner for partially ordered generalized distance spaces such as C-distance space, balanced distance space, and s,q distance space.



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