# New Generalizations of Set Valued Interpolative Hardy-Rogers Type Contractions in b-Metric Spaces

Authors: Muhammad Usman Ali; Hassen Aydi; Monairah Alansari
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Debnath and De La Sen introduced the notion of set valued interpolative Hardy-Rogers type contraction mappings on b-metric spaces and proved that on a complete b-metric space, whose all closed and bounded subsets are compact, the set valued interpolative Hardy-Rogers type contraction mapping has a fixed point. This article presents generalizations of above results by omitting the assumption that all closed and bounded subsets are compact.


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## Body

## 1. Introduction

There are numerous studies on interpolation inequalities in literature. In 1999, Chua [1] gave some weighted Sobolev interpolation inequalities on product spaces. Badr and Russ [2] proved some Littlewood-Paley inequalities and interpolation results for Sobolev spaces. Interpolation is considered as one of the central concepts in pure logic. Various interpolation properties find their applications in computer science and have many deep purely logical consequences (see [3, 4]). Gogatishvili and Koskela [5] presented variant interpolation properties of Besov spaces defined on metric spaces. Going in the same direction in the setting of metric spaces via contraction mappings, Karapinar [6] presented the concept of an interpolative Kannan contraction mapping and proved that this mapping admits a fixed point on complete metric spaces. Later on, this notation has been extended into several directions (see [7–18]).In [6], Karapinar presented the interpolative Kannan contraction as follows: a mapping K:W,dW→W,dW is an interpolative Kannan contraction if
(1)dWKwa,Kwb≤δdWwa,Kwaι1dWwb,Kwb1−ι1for all wa,wb∈W with wa≠Kwa, where δ∈0,1 and ι1∈0,1. This inequality was further refined by Karapinar et al. [7] by
(2)dWKwa,Kwb≤δdWwa,Kwaι1dWwb,Kwb1−ι1for all wa,wb∈W\fixK, where δ∈0,1, ι1∈0,1, and fixK=wa∈W:Kwa=wa.Gaba and Karapinar [9] further modified the interpolative Kannan contraction concept in the following way: a mapping K:W,dW→W,dW is a δ,ι1,ι2-interpolative Kannan contraction, if
(3)dWKwa,Kwb≤δdWwa,Kwaι1dWwb,Kwbι2for all wa,wb∈W\fixK, where δ∈0,1, ι1,ι2∈0,1 with ι1+ι2<1. Karapinar et al. [10] gave the interpolative Hardy-Rogers type contraction as follows: a mapping K:W,dW→W,dW is called an interpolative Hardy-Rogers type contraction if
(4)dWKwa,Kwb≤δdWwa,wbι1dWwa,Kwaι2dWwb,Kwbι3×12ρdWwa,Kwb+dWKwa,wb1−ι1−ι2−ι3for each wa,wb∈W\fixK, where δ∈0,1 and ι1,ι2,ι3∈0,1 with ι1+ι2+ι3<1.Later on, Debnath and De La Sen [12] extended the above definition to set valued interpolative Hardy-Rogers type contraction mappings on b-metric spaces and proved that on complete b-metric spaces, whose all closed and bounded subsets are compact, the set valued interpolative Hardy-Rogers type contraction mapping has a fixed point.On the other hand, Bakhtin [19] and Czerwik [20] introduced the notion of b-metric spaces.Definition 1 (see [19, 20]).
LetW be a nonempty set and dW:W×W→0,∞ be a function so that for all i,j,ℓ∈X and some ρ≥1,
(5)dWi,j=0⇔i=j,dWi,j=dWj,i,dWi,j≤ρdWi,ℓ+dWℓ,j.
Then,dW is a b-metric on W, and W,dW,ρ is called a b-metric space with a coefficient ρ≥1.
For related works in this setting, see [21–23]. From now on, W,dW,ρ is a b-metric space with a coefficient ρ≥1. In the whole paper, ρ≥1 is the coefficient of the b-metric space.Definition 2 (see [20]).
We have the following:(a)
A sequenceηn in W is said to be Cauchy if limn,m→∞dWηn,ηm=0(b)
A sequenceηn in W is said to be convergent to η if limn,m→∞dWηn,η=0(c)
W,dW,ρ is said to be complete if every Cauchy sequence ηn in W is convergent
Denote byCBW the set of nonempty closed bounded subsets of W. For A,B∈CBX, consider
(6)ΔWA,B=supdWω,B;ω∈A,where dWω,B=infdWω,μ,μ∈B. The functional HW:CBW×CBW→0,∞ defined by
(7)HWA,B=maxΔWA,B,ΔWB,Ais known as the Pompieu-Hausdorff b-metric on CBW. We state the following known lemma.Lemma 3 (see [24]).
LetW,dW,ρ be a b-metric space ρ≥1. Let A,B∈CBW and a∈A. We have the two following statements:
(i) For eachε>0, there is b∈B so that
(8)dWa,b≤HWA,B+ε
(ii) For eachh>1, there is ν∈B so that
(9)dWa,ν≤hHWA,B
This article presents two new generalizations of set valued interpolative Hardy-Rogers type contraction mappings. Namely, we ensure the existence of fixed points of such maps on a complete b-metric space without considering the assumption that all closed and bounded subsets must be compact. Two examples are also presented.


## 2. Main Results

First, we define the notion ofξ-interpolative Hardy-Rogers type contractions.Definition 4.
Consider a b-metric spaceW,dW,ρ. Also, consider maps K:W→CBW and ξ:W×W→ℝ\0. Such a map K is called an ξ-interpolative Hardy-Rogers type contraction if
(10)HWKwa,Kwbξwa,wb≤δdWwa,wbι1dWwa,Kwaι2dWwb,Kwbι3×12ρdWwa,Kwb+dWKwa,wb1−ι1−ι2−ι3for each wa,wb∈W with
(11)mindWwa,wb,dWwa,Kwa,dWwb,Kwb>0,where δ∈0,1/ρ2 and ι1,ι2,ι3∈0,1 with ι1+ι2+ι3<1.The following result ensures the existence of a fixed point ofξ-interpolative Hardy-Rogers type contractions.Theorem 5.
Consider a complete b-metric spaceW,dW,ρ and consider an ξ-interpolative Hardy-Rogers type contraction map K. Also, consider the given assertions.
(I)
There must existw0a∈W and w1a∈Kw0a such that ξw0a,w1a=1(II)
For eachwa,wb∈W with ξwa,wb=1, we have ξwc,wd=1∀wc∈Kwa,wd∈Kwb(III)
For eachwma in W with wma→w and ξwma,wm+1a=1∀m∈ℕ, we have ξwma,w=1∀m∈ℕ
Then,K must have a fixed point in W.Proof.
By assertion (I) there arew0a∈W and w1a∈Kw0a with ξw0a,w1a=1. If
(12)mindWw0a,w1a,dWw0a,Kw0a,dWw1a,Kw1a=0,then K has a fixed point. Suppose that
(13)mindWw0a,w1a,dWw0a,Kw0a,dWw1a,Kw1a>0.
By (10), we obtain
(14)HWKw0a,Kw1a=HWKw0a,Kw1aξw0a,w1a≤δdWw0a,w1aι1dWw0a,Kw0aι2dWw1a,Kw1aι3×12ρdWw0a,Kw1a+dWKw0a,w1a1−ι1−ι2−ι3.
This leads to(15)1δdWw1a,Kw1a≤1δHWKw0a,Kw1a≤δdWw0a,w1aι1dWw0a,Kw0aι2dWw1a,Kw1aι3×12ρdWw0a,Kw1a+dWKw0a,w1a1−ι1−ι2−ι3.
Since1/δ>1, there is w2a∈Kw1a such that
(16)dWw1a,w2a≤1δdWw1a,Kw1a.
Thus, by (15),
(17)dWw1a,w2a≤δdWw0a,w1aι1dWw0a,w1aι2dWw1a,w2aι3×12ρdWw0a,w2a+dWw1a,w1a1−ι1−ι2−ι3.
Note thatdWw0a,w2a≤ρdWw0a,w1a+dWw1a,w2a≤2ρmaxdWw0a,w1a,dWw1a,w2a. Hence, by (17), we get
(18)dWw1a,w2a≤δdWw0a,w1aι1dWw0a,w1aι2dWw1a,w2aι3×maxdWw0a,w1a,dWw1a,w2a1−ι1−ι2−ι3.
Now, we considermaxdWw0a,w1a,dWw1a,w2a=dWw0a,w1a. Then, by (18), we get
(19)dWw1a,w2a≤δdWw0a,w1aι1dWw0a,w1aι2dWw0a,w1aι3×dWw0a,w1a1−ι1−ι2−ι3.
This implies(20)dWw1a,w2a≤δdWw0a,w1a.
Note that when we takemaxdWw0a,w1a,dWw1a,w2a=dWw1a,w2a in (18), then we get dWw1a,w2a=0, that is, w1a∈Kw1a; hence, this choice is not possible. As ξw0a,w1a=1 and w1a∈Kw0a and w2a∈Kw1a, then by assertion (II), we get ξw1a,w2a=1. Again, we consider
(21)mindWw1a,w2a,dWw1a,Kw1a,dWw2a,Kw2a>0,then by (10), we get
(22)1δdWw2a,Kw2a≤1δHWKw1a,Kw2a=1δHWKw1a,Kw2aξw1a,w2a≤δdWw1a,w2aι1dWw1a,Kw1aι2dWw2a,Kw2aι3×12ρdWw1a,Kw2a+dWKw1a,w2a1−ι1−ι2−ι3.
Since1/δ>1, there is w3a∈Kw2a such that
(23)dWw2a,w3a≤1δdWw2a,Kw2a.
Thus, by (22), we conclude
(24)dWw2a,w3a≤δdWw1a,w2aι1dWw1a,w2aι2dWw2a,w3aι3×12ρdWw1a,w3a+dWw2a,w2a1−ι1−ι2−ι3.
Note thatdWw1a,w3a≤ρdWw1a,w2a+dWw2a,w3a≤2ρmaxdWw1a,w2a,dWw2a,w3a. Hence, by (24), we get
(25)dWw2a,w3a≤δdWw1a,w2aι1dWw1a,w2aι2dWw2a,w3aι3×maxdWw1a,w2a,dWw2a,w3a1−ι1−ι2−ι3.
Now, we considermaxdWw1a,w2a,dWw2a,w3a=dWw1a,w2a. Then, by (18), we get
(26)dWw2a,w3a≤δdWw1a,w2aι1dWw1a,w2aι2dWw1a,w2aι3×dWw1a,w2a1−ι1−ι2−ι3.
This yields that(27)dWw2a,w3a≤δdWw1a,w2a.
Note that if we takemaxdWw1a,w2a,dWw2a,w3a=dWw2a,w3a in (25), then dWw2a,w3a=0, that is, w2a∈Kw2a, which is not possible. From (27) and (20), we get
(28)dWw2a,w3a≤δ2dWw0a,w1a.
Proceeding in this way, we can obtain a sequencewma in W with wm+1a∈Kwma, ξwma,wm+1a=1 for all m∈W and
(29)dWwma,wm+1a≤δmdWw0a,w1a∀m∈ℕ.
Also, by the construction ofwma, we get
(30)mindWwma,wm+1a,dWwma,Kwma,dWwm+1a,Kwm+1a>0∀m∈ℕ.
By a triangular inequality, we have forn>m,
(31)dWwna,wma≤∑j=mn−1ρjdWwja,wj+1a≤∑j=mn−1ρjδjdWw0a,w1a.
Since the above series is convergent,wma is a Cauchy sequence in W. Completeness of W gives w∗a in W such that wma→w∗a. By considering assertion (III), we get ξwma,w∗a=1∀m∈ℕ. Here, we claim w∗a∈Kw∗a. If the claim is wrong, then mindWwma,w∗a,dWwma,Kwma,dWw∗a,Kw∗a>0 for all m>m0, for some m0∈ℕ. From (10), we get
(32)dWwm+1a,Kw∗a≤HWKwma,Kw∗a=HWKwma,Kw∗aξwma,w∗a≤δdWwma,w∗aι1dWwma,Kwmaι2dWw∗a,Kw∗aι3×12ρdWwma,Kw∗a+dWKwma,w∗a1−ι1−ι2−ι3≤δdWwma,w∗aι1dWwma,Kwmaι2dWw∗a,Kw∗aι3×12ρdWwma,Kw∗a+dWwm+1a,w∗a1−ι1−ι2−ι3∀m>m0.
From the above, we getlimm→∞dWwm+1a,Kw∗a=0. By the triangular inequality, we have
(33)dWw∗a,Kw∗a≤ρdWw∗a,wm+1a+dWwm+1a,Kw∗a∀m∈ℕ.
By taking the limitm→∞, we get dWw∗a,Kw∗a=0, that is, w∗a∈Kw∗a. Therefore, our claim is valid.Example 1.
ConsiderW=ℤ with dWwn,wm=wn−wm2 for all wn,wm∈W. Define K:W→CBW by
(34)Kwn=0,wn∈0,1,2,3,⋯−wn−22,wn∈−1,−2,−3,⋯and ξ:W×W→ℝ\0 by
(35)ξwn,wm=1,wn,wm∈0,1,2,3,⋯−wn+wm+4,otherwise.Note thatCase 1.
Ifwn,wm≥0 with wn≠wm, we get HWKwn,Kwmξwn,wm=0.Case 2.
Ifwn,wm<0 with wn≠wm, we get
(36)HWKwn,Kwmξwn,wm=1−wn−22+wm−222wn+wm+4.Case 3.
Ifwn<0 and wm≥0, we get HWKwn,Kwmξwn,wm=1/−wn−222wn+wm+4.
After calculating the values, it is easy to see that
For Case1: if wn,wm>0 with wn≠wm, we get
(37)HWKwn,Kwmξwn,wm=0<151·1·1·14≤15dWwn,wmι1dWwn,Kwnι2dWwm,Kwmι3×12ρdWwn,Kwm+dWKwn,wm1−ι1−ι2−ι3for each ι1,ι2,ι3∈0,1 with ι1+ι2+ι3<1.
For Case2: if wn,wm<0 with wn≠wm, we get
(38)HWKwn,Kwmξwn,wm=1−wn−22+wm−222wn+wm+4≤1497<151·1·1·14≤15dWwn,wmι1dWwn,Kwnι2dWwm,Kwmι3×12ρdWwn,Kwm+dWKwn,wm1−ι1−ι2−ι3for each ι1,ι2,ι3∈0,1 with ι1+ι2+ι3<1.
For Case3: if wn<0 and wm>0, we get
(39)HWKwn,Kwmξwn,wm=1−wn−222wn+wm+4≤1816<151·1·1·14≤15dWwn,wmι1dWwn,Kwnι2dWwm,Kwmι3×12ρdWwn,Kwm+dWKwn,wm1−ι1−ι2−ι3for each ι1,ι2,ι3∈0,1 with ι1+ι2+ι3<1. By keeping these calculations in mind, one can check that all the hypotheses of Theorem 5 are valid. Hence, K must have a fixed point.The following definition presents a multiplicativeξ-interpolative Hardy-Rogers type contraction.Definition 6.
Consider a b-metric space W,dW,ρ. Also, consider the maps K:W→CBW and ξ:W×W→0,∞. Such K is called a multiplicative ξ-interpolative Hardy-Rogers type contraction if
(40)ξwa,wbHWKwa,Kwb≤δdWwa,wbι1dWwa,Kwaι2dWwb,Kwbι3×12ρdWwa,Kwb+dWKwa,wb1−ι1−ι2−ι3for each wa,wb∈W with
(41)mindWwa,wb,dWwa,Kwa,dWwb,Kwb>0,where δ∈0,1/ρ2 and ι1,ι2,ι3∈0,1 with ι1+ι2+ι3<1.The following result concerns the existence of fixed points for the above-defined mapping.Theorem 7.
Consider a complete b-metric space W,dW,ρ and consider a multiplicative ξ-interpolative Hardy-Rogers type contraction map K. Also, consider the given assertions:
(i)
There must existw0a∈W and w1a∈Kw0a such that ξw0a,w1a≥1(ii)
For eachwa,wb∈W with ξwa,wb≥1, we have ξwc,wd≥1∀wc∈Kwa,wd∈Kwb(iii)
For eachwma in W with wma→w and ξwma,wm+1a≥1∀m∈ℕ, we have ξwma,w≥1∀m∈ℕ
ThenK possesses a fixed point in W.Proof.
Assertion (i) implies the existence ofw0a∈W and w1a∈Kw0a with ξw0a,w1a≥1. We consider
(42)mindWw0a,w1a,dWw0a,Kw0a,dWw1a,Kw1a>0.
Otherwise,K has a fixed point. Then, by (40), we obtain
(43)HWKw0a,Kw1a≤ξw0a,w1aHWKw0a,Kw1a≤δdWw0a,w1aι1dWw0a,Kw0aι2dWw1a,Kw1aι3×12ρdWw0a,Kw1a+dWKw0a,w1a1−ι1−ι2−ι3.
This yields that(44)1δdWw1a,Kw1a≤1δHWKw0a,Kw1a≤δdWw0a,w1aι1dWw0a,Kw0aι2dWw1a,Kw1aι3×12ρdWw0a,Kw1a+dWKw0a,w1a1−ι1−ι2−ι3.
Since1/δ>1, there is w2a∈Kw1a satisfying
(45)dWw1a,w2a≤1δdWw1a,Kw1a.
Thus, by (44), we get
(46)dWw1a,w2a≤δdWw0a,w1aι1dWw0a,w1aι2dWw1a,w2aι3×12ρdWw0a,w2a+dWw1a,w1a1−ι1−ι2−ι3.
SincedWw0a,w2a≤ρdWw0a,w1a+dWw1a,w2a≤2ρmaxdWw0a,w1a,dWw1a,w2a, we get using (46),
(47)dWw1a,w2a≤δdWw0a,w1aι1dWw0a,w1aι2dWw1a,w2aι3×maxdWw0a,w1a,dWw1a,w2a1−ι1−ι2−ι3.
ConsidermaxdWw0a,w1a,dWw1a,w2a=dWw0a,w1a. Then, by (47), we get
(48)dWw1a,w2a≤δdWw0a,w1aι1dWw0a,w1aι2dWw0a,w1aι3×dWw0a,w1a1−ι1−ι2−ι3.
This implies that(49)dWw1a,w2a≤δdWw0a,w1a.
If we takemaxdWw0a,w1a,dWw1a,w2a=dWw1a,w2a in (47), then we get dWw1a,w2a=0, that is, w1a∈Kw1a, which is not possible. Since ξw0a,w1a≥1, w1a∈Kw0a, and w2a∈Kw1a, by assertion (ii), we get ξw1a,w2a≥1. Applying (40) and again assertion (ii), we can obtain a sequence wma in W with wm+1a∈Kwma, ξwma,wm+1a≥1 for all m∈W and
(50)dWwma,wm+1a≤δmdWw0a,w1a∀m∈ℕ.
Also, by construction ofwma, we know that
(51)mindWwma,wm+1a,dWwma,Kwma,dWwm+1a,Kwm+1a>0∀m∈ℕ.
By a triangular inequality, we have forn>m,
(52)dWwna,wma≤∑j=mn−1ρjdWwja,wj+1a≤∑j=mn−1ρjδjdWw0a,w1a.
This implies thatwma is a Cauchy sequence in W. Since W is complete, wma→w∗a∈W. By assertion (iii), we get ξwma,w∗a≥1 for all m∈ℕ. Now, we claim that w∗a∈Kw∗a. Assume the claim is wrong, then mindWwma,w∗a,dWwma,Kwma,dWw∗a,Kw∗a>0 for all m>m0 for some m0∈ℕ. Then by (40), we get
(53)dWwm+1a,Kw∗a≤HWKwma,Kw∗a=ξwma,w∗aHWKwma,Kw∗a≤δdWwma,w∗aι1dWwma,Kwmaι2dWw∗a,Kw∗aι3×12ρdWwma,Kw∗a+dWKwma,w∗a1−ι1−ι2−ι3≤δdWwma,w∗aι1dWwma,Kwmaι2dWw∗a,Kw∗aι3×12ρdWwma,Kw∗a+dWwm+1a,w∗a1−ι1−ι2−ι3∀m>m0.
From the above inequality, we getlimm→∞dWwm+1a,Kw∗a=0. By a triangular inequality, we have
(54)dWw∗a,Kw∗a≤ρdWw∗a,wm+1a+dWwm+1a,Kw∗a∀m∈ℕ.
Hence, by taking the limitm⟶∞, we get dWw∗a,Kw∗a=0, that is, w∗a∈Kw∗a.Example 2.
ConsiderW=ℤ with dWwn,wm=∣wn−wm∣ for all wn,wm∈W. Define K:W⟶CBW by
(55)Kwn=0,wn∈0,1,2,3,⋯0,2wn,wn∈−1,−2,−3,⋯and ξ:W×W→ℝ−0 by
(56)ξwn,wm=1,wn,wm∈0,1,2,3,⋯0,otherwise.
One can see that all the hypotheses of Theorem7 are valid. Hence, K must have a fixed point.Remark 8.
Note that ([12], Theorem 2) is not applicable in Example 2. It suffices to take x=−1 and y=−2, then Kx=0,−2 and Ky=0,−4. Thus, we have HKx,Ky=2, dx,y=1, dx,Kx=1, dy,Ky=2, dy,Kx=0, and dx,Ky=1. One then writes
(57)HKx,Ky=2>δ1ι11ι22ι3121−ι1−ι2−ι3for all δ,ι1,ι2,ι3∈0,1. Thus, our main results generalize and improve the result given in [12]. Moreover, when considering the single valued case in Theorem 5 and Theorem 7, that is, for a self-mapping K:W⟶W, we get generalizations of the main results in [9].



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