By combining the results of Wardowski's cyclic contraction operators and admissible multi-valued mappings, the motif of η-cyclic (α∗,β∗)-admissible type ϝ-contraction multivalued mappings are presented. Moreover, some novel fixed point theorems for such mappings are proved in the context of Mb-metric spaces. Also, two examples are given to clarify and strengthen our theoretical study. Finally, the existence of a solution of a pair of ordinary differential equations is discussed as an application.
Citation: Mustafa Mudhesh, Hasanen A. Hammad, Eskandar Ameer, Muhammad Arshad, Fahd Jarad. Novel results on fixed-point methodologies for hybrid contraction mappings in Mb-metric spaces with an application[J]. AIMS Mathematics, 2023, 8(1): 1530-1549. doi: 10.3934/math.2023077
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By combining the results of Wardowski's cyclic contraction operators and admissible multi-valued mappings, the motif of η-cyclic (α∗,β∗)-admissible type ϝ-contraction multivalued mappings are presented. Moreover, some novel fixed point theorems for such mappings are proved in the context of Mb-metric spaces. Also, two examples are given to clarify and strengthen our theoretical study. Finally, the existence of a solution of a pair of ordinary differential equations is discussed as an application.
The study of fixed points (FPs) is an interesting topic because it has many applications not only in non-linear analysis but also in many aspects of engineering and physics. FP technique has gained a large number of readers due to its smoothness and ease of approach.
In the long term of studying functional analysis, the metric space (MS) is an important topic in it, which has many generalizations and extensions in different formulas. One of these generalizations is motif of b-metric space [1] which open the wide field for researchers to develop metric fixed point theory. In 1994, Matthews introduced another generalization of (MS) which is called partial metric spaces (PMSs) [2] and studied some properties of this space. In 2014, Ma et al. [3] introduced a new type of MSs which generalize the concepts of MSs and operator-valued MSs, they defined C∗-algebra-valued MSs and gave some FP results. An M-metric space (MMS) was redacted by Asadi et al. [4] in the same year of 2014, as an extension of (PMSs). Accordingly, some topological properties of said space and FP results for contraction mapping have been discussed. Altun et al. [5] presented some FP theorems for multivalued mappings of Feng-Liu type on complete MMSs. They inspected of the topological characteristics of (MMS) and asserted that the sequential topology τs is larger than the topology τm induced by open balls and the closure of a subset A of M-metric Ξ with respect to τs is included the closure of a subset A of M -metric Ξ with respect to (wrt) τm. Sahin et al. [6] generalized Feng-Liu techniques and discussed some new FP results for multivalued F-contraction mappings. Very recently, Patle et al. [7] studied Pompeiu-Hausdorff distance induced by the MMSs. Also, they established the Nadler and Kannan type FP theorems for set-valued mappings in such spaces. Monfared et al. [8,9] applied the notion of control and ultra altering distance functions ψ and ϕ for single valued contraction mappings in an MMS. Meanwhile, Mlaiki { et al.} [10] introduced the concept of Fm-expanding contractive mappings and graphic FP theorems in the mentied spaces. Mlaiki et al. [11] generalized the MMS to Mb-metric space (MbMS) and proved the existence and uniqueness of a FP under suitable contraction conditions. Recently, Hu and Gu [12] derived a new concept of the probabilistic MS, which is called the Menger probabilistic S-metric space, and investigated some topological properties of this space and proved related FP theorems for λ-contraction mapping.
In 1973, Geraghty [13] introduced a fruitful generalization of Banach contraction principle and obtained FP results for a single-valued mapping. In 1989, Mizoguchi and Takahashi [14] relaxed the compactness of value of a mapping Γ to closed and bounded subsets of Ξ and they obtained FP results for multi-valued mappings of Geraghty contraction. Popescu [15] proved interesting result for α-Geraghty contraction mappings in MSs. Arshad et al. [16] extended Popescu's results to introduce the new notion of α∗-Geraghty type F-contraction multivalued mapping in b-metric like space.
On the other hand, the notion of cyclic (α,β) -admissible mapping was discussed by Alizadeh et al. [17] and several FP results under this idea were proved. Ameer et al. [18] investigated FPs of cyclic (α∗,β∗)-type-γ-FG-contractive mappings and established some FP theorems in PbMSs. For more details, see [19,20,21,22,23,24,25,26,27,28].
This manuscript is devoted to introduce the concept of η-cyclic (α∗,β∗)-admissible type F -contraction multivalued mappings. Via this idea, some common FP results are obtained in MbMSs. Finally, as an application, the existence of solution to a pair of ordinary differential equations (ODEs) are given.
In this part, we give some elementary discussions about MMSs.
Definition 2.1. [4] Let Ξ≠∅. If the function m:Ξ×Ξ→R+ fulfills the stipulations below, for all λ,γ,κ∈Ξ:
(M1) m(λ,λ)=m(γ,γ)=m(λ,γ) iff λ=γ;
(M2) mλ,γ≤m(λ,γ);
(M3) m(λ,γ)=m(γ,λ);
(M4) (m(λ,γ)−mλ,γ)≤(m(λ,κ)−mλ,κ)+(m(κ,γ)−mκ,γ).
Then the pair (Ξ,m) is called an MMS.
It should be noted that the notion mλ,γ and Mλ,γ are defined by Asadi et al. [4] as follows:
mλ,γ=min{m(λ,λ),m(γ,γ)}, |
and
Mλ,γ=max{m(λ,λ),m(γ,γ)}. |
Definition 2.2. [11] An MbMS on a non-empty set Ξ is a function mb:Ξ2→R+ that fulfills the assumptions below, for all λ,γ,κ∈Ξ,
(Mb1) mb(λ,λ)=mb(γ,γ)=mb(λ,γ) iff λ=γ;
(Mb2) mbλ,γ≤mb(λ,γ);
(Mb3) mb(λ,γ)=mb(γ,λ);
(Mb4) There is a coefficient s≥1 so that for all λ,γ,κ∈Ξ, we have
mb(λ,γ)−mbλ,γ≤s[(mb(λ,κ)−mbλ,κ)+(mb(κ,γ)−mbκ,γ)]−mb(κ,κ). |
Then the pair (Ξ,mb) is called an MbMS.
Note.Symbols mbλ,γ and Mbλ,γ defined in [11] as follows:
mbλ,γ=min{mb(λ,λ),mb(γ,γ)}, |
and
Mbλ,γ=max{mb(λ,λ),mb(γ,γ)}. |
Example 2.3. [11] Let Ξ=[0,∞) and p>1 be a constant. Define mb:Ξ2⟶[0,∞) by
mb(λ,γ)=(max{λ,γ})p+|λ−γ|p,∀λ,γ∈Ξ. |
Then (Ξ,mb) is an MbMS (with coefficient s=2p) and not MMS.
Example 2.4. [29] Let Ξ=[0,1] and mb:Ξ×Ξ⟶[0,∞) be defined by
mb(λ,γ)=(λ+γ2)2,∀λ,γ∈Ξ. |
Then (Ξ,mb) is an MbMS (with coefficient s=2) which is not an MMS.
Definition 2.5. [11] Let (Ξ,mb) be an MbMS. Then
● A sequence {λn} in Ξ converges to a point λ if and only if
limn→∞(mb(λn,λ)−mbλn,λ)=0. |
● A sequence {λn} in Ξ is called mb-Cauchy sequence iff
limn,m→∞(mb(λn,λm)−mbλn,λm)andlimn,m→∞(Mbλn,λm−mbλn,λm) |
exist and finite.
● An MbMS is called mb-complete if every mb-Cauchy sequence {λn} converges to a point λ so that
limn→∞(mb(λn,λ)−mbλn,λ)=0andlimn→∞(Mbλn,λ−mbλn,λ)=0. |
The first result concerning with the existence of FPs in the MbMS presented by Mlaiki et al. [11] as follows:
Theorem 2.6. Let (Ξ,mb) be an MbMS with coefficient s≥1 and Γ be a self-mapping on Ξ. If there is k∈[0,1) so that
mb(Γλ,Γγ)≤kmb(λ,γ),∀λ,γ∈Ξ. |
Then Γ has a unique FP ς in Ξ.
The concepts of cyclic (α,β)-admissible and cyclic (α∗,β∗)-admissible mappings are showed in the work of [17,18] as follows:
Definition 2.7. Let Ξ≠∅, α,β:Ξ→[0,∞) be two functions. A mapping Γ:Ξ→Ξ is called cyclic (α,β)-admissible if for some λ∈Ξ,
α(λ)≥1⇒β(Γλ)≥1, |
and
β(λ)≥1⇒α(Γλ)≥1. |
Definition 2.8. Let Ξ≠∅, α,β:Ξ→[0,∞) be mappings and A,B be subsets of Ξ. A mapping Γ:Ξ→CB(Ξ) is called cyclic (α∗,β∗)-admissible if for some λ∈Ξ,
α(λ)≥1⇒β∗(Γλ)≥1, |
and
β(λ)≥1⇒α∗(Γλ)≥1, |
where β∗(A)=infa∈Aβ(a) and α∗(B)=infb∈Bα(b).
Theorem 2.9. [13] Let Ξ be a complete metric space and Γ:Ξ→Ξ. If there is φ∈ξ so that
d(Γλ,Γγ)≤φ(d(λ,γ))d(λ,γ),∀λ,γ∈Ξ, |
holds, where ξ is the set of all functions φ:[0,∞)→[0,1) satisfying limn→∞tn=0 whenever limn→∞φ(tn)=1. Then Γ has a unique FP λ∗∈Ξ and for each λ∈Ξ, the sequence {Tnλ} converges to λ∗.
In 2012, Wardowski [30] made a great contribution to the study of new theories related to fixed points in the context of ordinary metric spaces. This contribution is called ϝ-contraction mappings.
Definition 2.10. [30] Let ϝ:R+→R be a mapping fulfilling the stipulations below:
(ϝ1) ϝ is strictly increasing, i.e., if α<β, then ϝ(α)<ϝ(β), ∀α,β∈R+;
(ϝ2) for any sequence {αn}∞n=1 of positive real numbers, limn→∞αn=0 iff limn→∞ϝ(αn)=−∞;
(ϝ3) there is k∈(0,1) so that limn→∞αkϝ(αn)=0.
Felhi [31] generalized the Definition 2.10 by adding the condition below to the stipulations (ϝ1)−(ϝ3):
(ϝ4) for any sequence {αn}∞n=1 of positive real numbers so that
τ+ϝ(sαn)≤ϝ(αn−1),s≥1, |
for all n∈N and some τ>0, then
τ+ϝ(snαn)≤ϝ(sn−1αn−1),∀n∈N. |
Here, ϝw and ϝs denote the sets of all functions ϝ fulfilling (ϝ1)−(ϝ3) and (ϝ1)−(ϝ4), respectively.
Remark 2.11. [32]If ϝ is right continuous and satisfies (ϝ1), then
ϝ(infA)=infϝ(A)∀ϝ⊂(0,∞)withinf(ϝ)>0. |
Assume that (Ξ,mb) is MbMS andCBmb(Ξ) is the family of all non-empty, bounded and closed subsets of Ξ. For ℵ,Q∈CBmb(Ξ), define
Hmb(ℵ,Q)=max{δmb(ℵ,Q),δmb(Q,ℵ)}, |
where δmb(ℵ,Q)=sup{mb(p,Q):p∈ℵ} and mb(p,Q)=inf{mb(p,q):q∈Q}.
The following results are very useful in our study. These results are taken from [4,7].
Lemma 2.12. Let ℵ be a non-empty set in an MbMS (Ξ,mb), then p∈ _ℵ iff
mb(p,ℵ)=supλ∈ℵmbp,λ, |
where _ℵ denotes the closure of ℵ wrt mb.
Lemma 2.13. Let ℵ,Q,ℜ∈CBmb(Ξ), then
(a) δmb(ℵ,ℵ)=supp∈ℵ{supq∈ℵmbpq},
(b) for s≥1, we have
(δmb(ℵ,Q)−supp∈ℵsupq∈Qmbpq)≤s[(δmb(ℵ,ℜ)−infp∈ℵinfr∈ℜmbpr)+(δmb(ℜ,Q)−infr∈ℜinfq∈Qmbrq)]−infr∈ℜmb(r,r). |
Lemma 2.14. Let ℵ,Q,ℜ∈CBmb(Ξ), then
(1)
Hmb(ℵ,ℵ)=δmb(ℵ,ℵ)=supp∈ℵ{supq∈Qmbp,q}, |
(2) Hmb(ℵ,Q)=Hmb(Q,ℵ),
(3) for s≥1, we get
(Hmb(ℵ,Q)−supp∈ℵsupq∈Qmbpq)≤s[(Hmb(ℵ,ℜ)−infp∈ℵinfr∈ℜmbp,r)+(Hmb(ℜ,Q)−infr∈ℜinfq∈Qmbr,q)]−infr∈ℜmb(r,r). |
Lemma 2.15. Let ℵ,Q∈CBmb(Ξ) and h>1, then for all p∈ℵ, there is q∈Q so that
(i) mb(p,q)≤hHmb(ℵ,Q),
(ii) mb(p,q)≤Hmb(ℵ,Q)+h.
Proof. (i) Suppose that there exists an p∈ℵ such that
mb(p,q)>hHmb(ℵ,Q), |
for all q∈Q. This implies that
inf{mb(p,q):q∈Q}. |
Now
mb(ℵ,Q)≥δmb(ℵ,Q)=sup{mb(p,Q):p∈ℵ}≥mb(p,Q)≥Hmb(ℵ,Q), |
this is a contradiction since Hmb(ℵ,Q)≠0 and h>1. Hence,
mb(p,q)≤hHmb(ℵ,Q). |
(ii) Suppose that there exists p∈ℵ such that mb(p,q)>Hmb(ℵ,Q)+h for all q∈Q, then we have
mb(ℵ,Q)+h≤mb(p,q)≤δmb(ℵ,Q)≤Hmb(ℵ,Q)+h, |
a contradiction again. Since Hmb(ℵ,Q)≠0 and h>1. Thus,
mb(p,q)≤Hmb(ℵ,Q)+h. |
Remark 2.16. For all λ,γ,κ in an MbMS (Ξ,mb), then
(1) Mbλ,γ+mbλ,γ=mb(λ,λ)+mb(γ,γ),
(2) Mbλ,γ−mbλ,γ=|mb(λ,λ)−mb(γ,γ)|,
(3) For s≥1, we have
Mbλ,γ−mbλ,γ≤s[(Mbλ,κ−mbλ,κ)+(Mbκ,γ−mbκ,γ)]. |
Notice that:
If s=1, then we get Remark 1.1 in [4].
In this part, the following are established:
● Definition of η-cyclic (α∗,β∗)-admissible mappings,
● The notion of Geraghty contraction type mappings,
● Some new common FP theorem for a pair of generalized (α∗,β∗)-Geraghty ϝ-contraction multivalued mapping in an MbMS.
Definition 3.1. Let Ξ≠∅, α,β,η:Ξ→[0,∞) be mappings and A,B be subsets of Ξ. A mapping Γ:Ξ→CBmb(Ξ) is called η-cyclic (α∗,β∗)-admissible if for some λ∈Ξ,
α(λ)≥η(λ)⇒β∗(Γλ)≥η∗(Γλ), |
and
β(λ)≥η(λ)⇒α∗(Γλ)≥η∗(Γλ) |
where β∗(A)=infa∈Aβ(a) and α∗(B)=infb∈Bα(b).
Definition 3.2. Let Ξ≠∅, α,β,η:Ξ→[0,∞) be mappings and A,B be subsets of Ξ. Two mappings ℑ,Γ:Ξ→CBmb(Ξ) are called η-cyclic (α∗,β∗) -admissible if for some λ∈Ξ,
α(λ)≥η(λ)⇒β∗(ℑλ)≥η∗(ℑλ), |
and
β(λ)≥η(λ)⇒α∗(Γλ)≥η∗(Γλ). |
Notice that:
● If η=η∗=1 and ℑ=Γ, then we get Definition 2.2 in [18].
● Definition 3.2 reduces to Definition 3.1, if we put ℑ=Γ.
Example 3.3. Let Ξ=[0,∞). Define the mappings ℑ,Γ:Ξ→CBmb(Ξ) and α,β,η:Ξ→[0,∞) by ℑλ={3λ}, Γλ={λ2}, η(λ)=λ, ∀λ∈Ξ,
α(λ)={e3λ2,ifλ>0,1,otherwise,andβ(λ)={52λ,ifλ>0,1,otherwise. |
For all λ>0, we get
α(λ)=e3λ2≥λ=η(λ)⇒β∗(ℑλ)=β∗(3λ)=56λ≥3λ=η∗(ℑλ). |
Similarly,
β(λ)=52λ≥λ=η(λ). |
Otherwise, for λ=0 the conditions of definition are satisfied. Then the pair (ℑ,Γ) is η-cyclic (α∗,β∗)-admissible mappings.
In the setting of the MbMS, we define a generalized (α∗,β∗)-Geraghty ϝ-contraction mappings as follows:
Definition 3.4. Let (Ξ,mb) be an MbMS, α,β,η:Ξ→[0,∞) be functions. Two multivalued mappings ℑ,Γ:Ξ→CBmb(Ξ) is called a pair of generalized (α∗,β∗)-Geraghty ϝ -contraction mappings if there exist φ∈ξ and ϝ∈ϝs so that for all λ,γ∈Ξ, s≥1 and τ∈R+ with Hmb(ℑλ,Γγ)>0,
β∗(ℑλ)α∗(Γγ)≥η∗(ℑλ)η∗(Γγ)⟹τ+ϝ(sHmb(ℑλ,Γγ))≤ϝ(φ(Mmb(λ,γ))Mmb(λ,γ)), | (3.1) |
where
Mmb(λ,γ)=max{mb(λ,γ),mb(λ,ℑγ),mb(γ,Γγ),mb(λ,ℑλ)mb(γ,Γγ)s+mb(λ,γ)}. | (3.2) |
Theorem 3.5. Let (Ξ,mb) be a complete MbMS, α,β,η:Ξ→[0,∞) be a given functions, and ℑ,Γ:Ξ→CBmb(Ξ) be two multivalued mappings satisfy the postulates below:
(1) the pair (ℑ,Γ) is generalized (α∗,β∗)-Geraghty ϝ-contraction;
(2) the pair (ℑ,Γ) is η-cyclic (α∗,β∗)-admissible;
(3) either there is λ0∈Ξ so that α∗(Γλ0)≥η∗(Γλ0) or γ0∈Ξ so that β∗(ℑγ0)≥η∗(ℑγ0).
Then ℑ and Γ have a common FP λ∗∈Ξ.
Proof. Let λ0∈Ξ so that α(λ0)≥η(λ0), by axiom (2) ∃λ1∈ℑλ0 and λ2∈Γλ1 so that
α(λ0)≥η(λ0)⇒β(λ1)≥β∗(ℑλ0)≥η∗(ℑλ0) |
and
α(λ2)≥α∗(Γλ1)≥η∗(Γλ1). |
Therefore
α∗(Γλ1)β∗(ℑλ0)≥η∗(Γλ1)η∗(ℑλ0), |
Since F is right continuous, then from Remark 2.11, we have
ϝ(smb(λ1,Γλ1))=infγ∈Γλ1ϝ(smb(λ1,γ)). |
Thus, there is γ=λ2∈Γλ1, so that
ϝ(smb(λ1,λ2))≤ϝ(sHmb(ℑλ0,Γλ1))≤ϝ(φ(Mmb(λ0,λ1))Mmb(λ0,λ1))−τ, | (3.3) |
where
Mmb(λ0,λ1)=max{mb(λ0,λ1),mb(λ0,ℑλ0),mb(λ1,Γλ1),mb(λ0,ℑλ0)mb(λ1,Γλ1)s+mb(λ0,λ1)}=max{mb(λ0,λ1),mb(λ1,λ2),mb(λ0,λ1)mb(λ1,λ2)s+mb(λ0,λ1)}≤max{mb(λ0,λ1),mb(λ1,λ2),mb(λ0,λ1)mb(λ1,λ2)mb(λ0,λ1)}=max{mb(λ0,λ1),mb(λ1,λ2)}. |
If Mmb(λ0,λ1)≤mb(λ1,λ2), then from (3.3), we can write
ϝ(smb(λ1,λ2))≤ϝ(φ(mb(λ1,λ2))mb(λ1,λ2))−τ<ϝ(mb(λ1,λ2)). |
Applying (ϝ1), we have
smb(λ1,λ2)<mb(λ1,λ2), |
a contradiction. If Mmb(λ0,λ1)≤mb(λ0,λ1), then by (3.3), we get
ϝ(smb(λ1,λ2))≤ϝ(φ(mb(λ0,λ1))mb(λ0,λ1))−τ<ϝ(mb(λ0,λ1)). |
Again, from (ϝ1), we obtain
smb(λ1,λ2)<mb(λ0,λ1), |
Analogous to (3.3), there is λ3∈ℑλ2 so that
ϝ(smb(λ2,λ3))≤ϝ(φ(Mmb(λ1,λ2)))Mmb(λ1,λ2))−τ≤ϝ(φ(mb(λ1,λ2)))mb(λ1,λ2))−τ. |
Continuing with the same scenario, we construct a sequence {λn} in Ξ so that λ2n+1∈ℑλ2n and λ2n+2∈Γλ2n+1. Since the pair (ℑ,Γ) is η-cyclic (α∗,β∗) -admissible, we have
β∗(ℑλ2n)α∗(Γλ2n+1)≥η∗(ℑλ2n)η∗(Γλ2n+1),∀n≥0. |
Subsequently, by (3.1), we get
τ+ϝ(smb(λ2n+1,λ2n+2))≤τ+ϝ(sHmb(ℑλ2n,Γλ2n+1))≤ϝ(φ(Mmb(λ2n,λ2n+1))Mmb(λ2n,λ2n+1))≤ϝ(φ(mb(λ2n,λ2n+1))mb(λ2n,λ2n+1))≤ϝ(mb(λ2n,λ2n+1), | (3.4) |
therefore (3.4) implies that
τ+ϝ(smb(λ2n+1,λ2n+2))≤ϝ(mb(λ2n,λ2n+1). |
Set ρ2n+1=mb(λ2n+1,λ2n+2) and μ2n+1=s2n+1ρ2n+1, ∀n≥0, then, we can write
τ+ϝ(sρ2n+1)≤ϝ(ρ2n),∀n≥0. |
By (ϝ4), one can obtain
τ+ϝ(μ2n+1)≤ϝ(μ2n),∀n≥0. | (3.5) |
Repeating the inequality (3.5), we obtain
ϝ(μ2n+1)≤ϝ(μ2n)−τ≤...≤ϝ(μ0)−(2n+1)τ,∀n≥0. | (3.6) |
Letting n→∞ in (3.6), we have
limn→∞ϝ(μ2n+1)=−∞. |
It follows from (ϝ2) that
limn→∞μ2n+1=0. |
By (ϝ3), there is k∈(0,1) so that
limn→∞μk2n+1ϝ(μ2n+1)=0. |
From (3.6), we get
μk2n+1ϝ(μ2n+1)−μk2n+1ϝ(μ0)≤(μk2n+1(ϝ(μ0)−(2n+1)τ)−μk2n+1ϝ(μ0))=−μk2n+1(2n+1)τ≤0,∀n≥0. |
Taking the limit as n→∞ and since τ>0, we obtain
limn→∞μk2n+1(2n+1)=0. |
Thus, there is n1∈N so that
μk2n+1(2n+1)≤1⇒μ2n+1≤1(2n+1)1k∀n≥n1. |
This leads to the series ∑nμ2n+1 is convergent.
Now, we prove that {λn} is an mb-Cauchy sequence in Ξ. Using (Mb4), we get
mb(λ2n+1,λ2n+3)−mbλ2n+1,λ2n+3≤s[mb(λ2n+1,λ2n+2)−mbλ2n+1,λ2n+2+mb(λ2n+2,λ2n+3)−mbλ2n+2,λ2n+3]−mb(λ2n+2,λ2n+2)≤s[mb(λ2n+1,λ2n+2)−mbλ2n+1,λ2n+2+mb(λ2n+2,λ2n+3)−mbλ2n+2,λ2n+3]≤smb(λ2n+1,λ2n+2)+s2mb(λ2n+2,λ2n+3). |
Similarly
mb(λ2n+1,λ2n+4)−mbλ2n+1,λ2n+4≤s[mb(λ2n+1,λ2n+2)−mbλ2n+1,λ2n+2+mb(λ2n+2,λ2n+4)−mbλ2n+2,λ2n+4]−mb(λ2n+2,λ2n+2)≤smb(λ2n+1,λ2n+2)+s2mb(λ2n+2,λ2n+3)+s3mb(λ2n+3,λ2n+4). |
In general, for all q>p>n1 with p=2n+1, we obtain
mb(λp,λq)−mbλp,λq≤q−1∑i=psi−p+1mb(λi,λi+1)≤q−1∑i=psimb(λi,λi+1)≤∞∑i=pμi. |
The convergence of the series ∞∑i=pμi leads to
limp,q→∞(mb(λp,λq)−mbλp,λq)=0. |
By the same way and from Remark 2.16, we obtain
Mbλ2n+1,λ2n+4−mbλ2n+1,λ2n+4≤s(Mbλ2n+1,λ2n+2−mbλ2n+1,λ2n+2)+s2(Mbλ2n+2,λ2n+3−mbλ2n+2,λ2n+3)+s3(Mbλ2n+3,λ2n+4−mbλ2n+3,λ2n+4). |
In general, for all q>p>n1 with p=2n+1, we obtain
Mbλp,λq−mbλp,λq≤q−1∑i=psi−p+1(Mbλi,λi+1−mbλi,λi+1)≤q−1∑i=psi−p+1Mbλi,λi+1≤q−1∑i=psi−p+1mb(λi,λi)≤q−1∑i=psi−p+1mb(λi,λi+1)≤q−1∑i=psimb(λi,λi+1)≤∞∑i=pμi. |
The convergence of the series ∞∑i=pμi leads to
limp,q→∞(Mbλp,λq−mbλp,λq)=0. |
Therefore, {λn} is an mb-Cauchy sequence in Ξ. Since Ξ is mb-complete, there exists λ∗∈Ξ so λn⟶λ∗ as n⟶∞, implies λ2n+1→λ∗ and λ2n+2→λ∗ as n→∞. Thus, we have
limn→∞(mb(λ2n+1,λ∗)−mbλ2n+1,λ∗)=0. | (3.7) |
Since limn→∞mb(λ2n+1,λ2n+1)=0, then by (3.7), we get
limn→∞mb(λ2n+1,λ∗)=0. | (3.8) |
It follows from (3.1), (3.4) and (3.8) that
limn→∞Hmb(ℑλ2n,Γλ∗)=0. | (3.9) |
Since λ2n+1∈ℑλ2n and
mb(λ2n+1,Γλ∗)≤Hmb(ℑλ2n,Γλ∗). |
Then after taking the limit as n→∞, we obtain that
limn→∞mb(λ2n+1,Γλ∗)=0. | (3.10) |
By (Mb2), one can write
mbλ2n+1,Γλ∗≤mb(λ2n+1,Γλ∗), |
that is
limn→∞mbλ2n+1,Γλ∗=0. | (3.11) |
Now, utilizing (Mb4), we have
mb(λ∗,Γλ∗)−supγ∈Γλ∗mbλ∗,γ≤mb(λ∗,Γλ∗)−mbλ∗,Γλ∗≤s[mb(λ∗,λ2n+1)−mbλ∗,λ2n+1+mb(λ2n+1,Γλ∗)−mbλ2n+1,Γλ∗]−mb(λ2n+1,λ2n+1). | (3.12) |
Letting n→∞ in (3.12) and from (3.8), (3.10) and (3.11), we conclude that
mb(λ∗,Γλ∗)≤supγ∈Γλ∗mbλ∗,γ. | (3.13) |
Using (Mb2), for all γ∈Γλ∗, we get
mbλ∗,γ≤mb(λ∗,γ), |
yields
mbλ∗,γ−mb(λ∗,γ)≤0. |
Thus
sup{mbλ∗,γ−mb(λ∗,γ):γ∈Γλ∗}≤0, |
this implies that
supγ∈Γλ∗mbλ∗,γ−supγ∈Γλ∗mb(λ∗,γ)≤0. |
Therefore
supγ∈Γλ∗mbλ∗,γ≤mb(λ∗,Γλ∗). | (3.14) |
From (3.13) and (3.14), we obtain
mb(λ∗,Γλ∗)=supγ∈Γλ∗mbλ∗,γ. |
Hence by Lemma 2.12, we get λ∗∈¯Γλ∗=Γλ∗. Similarly, we can easily conclude that λ∗∈ℑλ∗. Therefore λ∗ is a common FP of ℑ and Γ.
Remark 3.6. Theorem 3.5 still valid if we consider the following:
● If we put s=1 in Definition 3.4, then generalized (α∗,β∗)-Geraghty ϝ-contraction mappings take the form: Hm(ℑλ,Γγ)>0,
α∗(ℑλ)β∗(Γγ)≥η∗(ℑλ)η∗(Γγ)⟹τ+ϝ(Hm(ℑλ,Γγ))≤ϝ(φ(M(λ,γ))M(λ,γ)) |
where φ∈ξ, ϝ∈ϝs, τ∈R+ and
M(λ,γ)=max{m(λ,γ),m(λ,ℑλ),m(γ,Γγ),m(λ,ℑλ)m(γ,Γγ)1+m(λ,γ)}. |
Moreover, under the same conditions (1)–(3) of Theorem 3.5, ℑ and Γ have a common FP in a complete MMS (Ξ,m).
● If we take Mmb(λ,γ)=mb(λ,γ) in Definition 3.4, then we have a common FP of ℑ and Γ in complete MbMS, provided that the stipulations (1)–(3) of Theorem 3.5 hold.
● If we consider ℑ=Γ in Definition 3.4, then the result is given quickly in the same manner as the proof of Theorem 3.5.
The example below supports Theorem 3.5.
Example 3.7. Let Ξ=[0,∞) and mb:Ξ×Ξ⟶[0,∞) defined by
mb(λ,γ)=max{λ,γ}p+|λ−γ|p,∀λ,γ∈Ξ. |
Clearly, (Ξ,mb) is an MbMS with p>1 and s=2p.
If we take λ=5, γ=1 and κ=4, we obtain that
mb(λ,γ)−mbλ,γ>mb(λ,κ)−mbλ,κ−mb(κ,γ)−mbκ,γ. |
This means (Ξ,mb) is not MMS. Define ℑ,Γ:Ξ→CBmb(Ξ) by
ℑλ={{λ64},ifλ∈(0,1],{0,116},otherwise,andΓλ={{0,λ48},ifλ∈(0,1],0,otherwise. |
Describe the functions α,β,η:Ξ→[0,∞) as η(λ)=λ+1,
α(λ)={3e2λ2,ifλ>0,3,λ=0,,β(λ)={5λ+1,λ>0,1,λ=0. |
for all λ∈Ξ. Now, for λ∈(0,1], we have α(λ)≥η(λ) implies
β∗(ℑλ)=β∗(λ64)=5λ64+1≥λ64+1=η∗({λ64})=η∗(ℑλ). |
When Γλ=0, then β(λ)≥η(λ) implies
α∗(Γλ)=α∗(0)=3>1=η∗(0)=η∗(Γλ), |
if Γλ=λ48, then, we get β(λ)≥η(λ) implies
α∗(Γλ)=α∗(λ48)=3e2(λ48)2≥λ48+1=η∗(λ48)=η∗(Γλ). |
Hence the pair (ℑ,Γ) is η-cyclic (α∗,β∗)-admissible mappings. Consider φ(t)=16p, so for λ,γ∈(0,1], one can write
ϝ(sHmb(ℑλ,Γγ))=ϝ(2p(max{supa∈ℑλmb(a,Γγ),supb∈Γγmb(ℑλ,b)}))=ϝ(2pmax(mb(λ64,{0,γ48}),mb(λ64,γ48)))=ϝ(2pmax(mb(λ64,0),mb(λ64,γ48)))=ϝ(2pmb(λ64,γ48))=ϝ(2p16pmb(λ4,γ3))=ϝ((18)pmb(λ4,γ3))=ϝ(123pmb(λ4,γ3))≤ϝ((168×12)pmb(λ,γ))=ϝ(16pmb(λ,γ)). |
By taking ϝ(λ)=lnλ, we have
ln(sHmb(ℑλ,Γγ))≤ln(16pmb(λ,γ))=−pln(6)+ln(mb(λ,γ)), |
which implies that
ln(sHmb(ℑλ,Γγ))≤ln(mb(λ,γ))−τ. |
Since mb(λ,γ)≤Mb(λ,γ), and using the definition of φ, then we obtain
ϝ(sHmb(ℑλ,Γγ))≤ϝ(φ(Mb(λ,γ)))−τ≤ϝ(φ(Mmb(λ,γ))Mmb(λ,γ))−τ. |
Otherwise, the inequality below holds
ϝ(sHmb(ℑλ,Γγ)≤ϝ(φ(Mmb(λ,γ))Mmb(λ,γ))−τ. |
Analogously, for each λ,γ∈Ξ, we can find some τ>0 satisfy the above inequality. Hence, all hypotheses of Theorem 3.5 are fulfilled with τ=pln(6) and λ∗=0 is a common FP of ℑ and Γ.
This part is a reduction of the previous part by taking ℑ and Γ are single-valued mappings.
Definition 4.1. Let Ξ≠∅ and α,β,η:Ξ→[0,∞) be given functions. The mapping Γ:Ξ→Ξ is called η-cyclic (α,β)-admissible if for some λ∈Ξ,
α(λ)≥η(λ)⇒β(Γλ)≥η(Γλ), |
and
β(λ)≥η(λ)⇒α(Γλ)≥η(Γλ). |
Definition 4.2. Let Ξ≠∅ and α,β,η:Ξ→[0,∞) be given functions. The mappings ℑ,Γ:Ξ→Ξ are called η-cyclic (α,β)-admissible if for some λ∈Ξ,
α(λ)≥η(λ)⇒β(ℑλ)≥η(ℑλ), |
and
β(λ)≥η(λ)⇒α(Γλ)≥η(Γλ). |
Now, we present some results related to the existence of FPs which can be proven in a similar way to Theorem 3.5.
Corollary 4.3. Let (Ξ,mb) be a complete MbMS and α,β,η:Ξ→[0,∞) be given functions. Assume that the mapping Γ:Ξ→Ξ satisfies the following condition: There are φ∈ξ and ϝ∈ϝs so that for all λ,γ∈Ξ, s≥1 and τ>1,
α(λ)β(γ)≥η(λ)η(γ)⟹τ+ϝ(smb(Γλ,Γγ))≤ϝ(φ(Mmb(λ,γ))Mmb(λ,γ)), |
where Mmb(λ,γ) is defined in (3.2). Assume also that the following hypotheses are satisfied:
(i) Γ is an η-cyclic (α,β) -admissible;
(ii) there is λ0∈Ξ so that α(λ0)≥η(λ0)or β(λ0)≥η(λ0).
Then Γ has a FP λ∗∈ Ξ.
Corollary 4.4. Let (Ξ,mb) be a complete MbMS and α,β,η:Ξ→[0,∞) be given functions. Consider the mappings ℑ,Γ:Ξ→Ξ satisfy the assumption below: There are φ∈ξ and ϝ∈ϝs so that for all λ,γ∈Ξ, s≥1 and τ>1,
α(λ)β(γ)≥η(λ)η(γ)⟹τ+ϝ(smb(ℑλ,Γγ))≤ϝ(φ(Mmb(λ,γ))Mmb(λ,γ)), |
where Mmb(λ,γ) is defined in (3.2). Suppose also the following two conditions hold:
(i) (ℑ,Γ) is a pair of η-cyclic (α,β)-admissible;
(ii) there is λ0∈Ξ so that α(λ0)≥η(λ0) or γ0∈Ξ so that β(γ0)≥η(γ0).
Then ℑ and Γ have a common FP λ∗∈Ξ.
If we set α(λ)=β(γ)=η(λ)=η(γ)=1 in Corollary 4.4, we have the following result.
Corollary 4.5. Let (Ξ,mb) be a complete MbMS, ℑ and Γ be self-mappings defined on Ξ. If there are φ∈ξ and ϝ∈ϝs so that for all λ,γ∈Ξ, s≥1 and τ>1,
τ+ϝ(smb(ℑλ,Γγ))≤ϝ(φ(Mmb(λ,γ))Mmb(λ,γ)), | (4.1) |
where Mmb(λ,γ) is described as (3.2). Then ℑ and Γ have a common FP λ∗∈Ξ.
Note. The pair (ℑ,Γ) that satisfy (4.1) is called generalized Geraghty ϝ-contraction mappings.
In this part, we apply Corollary 4.5 to discuss the existence of solution to the pair of ODEs. Consider the following pair of ODEs:
{−d2λdt2=f(t,λ(t)),t∈[0,1]λ(0)=λ(1)=0,and{−d2γdt2=g(t,γ(t)),t∈[0,1],γ(0)=γ(1)=0. | (5.1) |
where f,g:[0,1]×R⟶R are continuous functions. So, the pair of ODEs (5.1) is equivalent to the following integral equations:
λ(t)=∫10G(t,s)f(s,λ(s))dsandγ(t)=∫10G(t,s)g(s,γ(s))ds. | (5.2) |
The Green's function G:[0,1]×[0,1]→R associated with (5.2) is described as
G(t,s)={t(1−s),0≤t≤s≤1,s(1−t),0≤s≤t≤1. |
Let Ξ=C([0,1],R) be the set of all continuous functions defined on [0,1]. Define a function m:Ξ×Ξ→R+ by
mb(λ,γ)=maxt∈I(|λ(t)+γ(t)2|)2,∀λ,γ∈Ξ. |
Obviously, (Ξ,mb) is a complete MbMS with a constant s=2.
The ODEs (5.1) will be considered under the two postulates below:
(1) there is a function ω:R⟶(0,1) so that for all z1,z2∈R, we have
|f(t,z1)|+|g(t,z2)|≤√ω(t)Mmb(z1,z2),∀t∈[0,1], |
where
Mmb(z1,z2)=max{|z1+z22|2,|z1+ℑz12|2,|z2+Γz22|2,|z1+ℑz12|2|z2+Γz22|2s+|z1+z22|2}; |
(2) there is s≥1 so that ∫10G(t,r)dr≤√12e−τ7s, for some τ>0.
Now, we present our main theorem in this part.
Theorem 5.1. Under the postulates (1) and (2), ODEs (5.1) has at least one solution λ∗∈Ξ.
Proof. Describe the operators ℑ,Γ:Ξ⟶Ξ as
ℑλ(t)=∫10G(t,s)f(s,λ(s))dsandΓγ(t)=∫10G(t,s)g(s,γ(s))ds, |
for all t∈[0,1]. Clearly, the solution of the integral equations (5.2) is equivalent to find a common FP of the operators ℑ and Γ. Let λ,γ∈Ξ, by our assumption, for all t∈[0,1], we get
[|ℑλ(t)|+|Γγ(t)|]2=[|∫10G(t,s)f(s,λ(s))ds|+|∫10G(t,s)g(s,γ(s))ds|]2≤[∫10[|G(t,s)f(s,λ(s))|+|G(t,s)g(s,γ(s))|]ds]2≤[∫10G(t,s)(|f(s,λ(s))|+|g(s,γ(s))|)ds]2≤[∫10G(t,s)√ω(t)Mmb(λ,γ)ds]2≤[∫10G(t,s)√ω(t)Mmb(λ,γ)ds]2=[√ω(t)Mmb(λ,γ)]2[∫10G(t,s)ds]2≤[√ω(t)Mmb(λ,γ)]2[√12e−τ7s]2=ω(t)12e−τ7sMmb(λ,γ). |
Consequently, we get
smb(ℑλ,Γγ)≤3ω(t)7e−τMmb(λ,γ)≤e−τφ(Mmb(λ,γ))Mmb(λ,γ), |
which implies that
τ+ln(smb(ℑλ,Γγ))≤ln[φ(Mmb(λ,γ))Mmb(λ,γ)], |
where ϝ(λ)=lnλ∈ϝs and φ(t)=3ω(t)7, for all t∈[0,1]. Thus, all stipulations of Corollary 4.5 are fulfilled. Therefore, the operators ℑ and Γ have a common FP, which is a solution to the ODEs (5.1).
Remark 5.2. It should be noted that under the same conditions, we cannot obtain the solution of the ODEs (5.1) by the classical FP theorem because of the definition of the function m:Ξ×Ξ→R+. It is defined as
mb(λ,γ)=maxt∈I(|λ(t)+γ(t)2|)2,∀λ,γ∈Ξ. |
On a complete metric space, the classical theorem holds true, but the first metric space requirement is not met as follows:
forλ,γ∈Ξ,ifλ=γ,thenmb(λ,λ)=maxt∈I(|λ(t)+λ(t)2|)2=maxt∈I(|λ(t)|)2>0. |
So not equal 0. Hence, (Ξ,mb) is a complete MbMS with a constant s=2 and not a complete metric space.
After the large number of papers published in the field of fixed point, we can assert that this technique is the backbone of non-linear analysis due to its smoothness and pivotality in many life disciplines. Therefore, in our manuscript, a new type of contraction was defined, called η-cyclic (α∗,β∗)-admissible type ϝ -contraction multivalued mappings. Under this contraction, some results concerned with FPs have been proven in the context of MbMSs. Also, our new results generalize and unify many papers in this regard. Moreover, some examples have been discussed to clarify the obtained results. Finally, we applied our main result to study the existence of a solution to a pair of ODEs.
The authors declare that they have no competing interests concerning the publication of this article.
[1] | S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav., 1 (1993), 5–11. |
[2] | S. G. Matthews, Partial metric topology, Ann. N. Y. Acad. Sci., 728 (1994), 183–197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x |
[3] | Z. Ma, L. Jiang, H. Sun, C∗-algebra-valued metric spaces and related fixed point theorems, Fixed Point Theory A., 206 (2014), 1–11. https://doi.org/10.1186/s13663-015-0471-6 doi: 10.1186/s13663-015-0471-6 |
[4] | M. Asadi, E. Karapınar, P. Salimi, New extension of p-metric spaces with some fixed-point results on M-metric spaces, J. Ineq. Appl., 2014 (2014), 1–9. https://doi.org/10.1186/1029-242X-2014-18 doi: 10.1186/1029-242X-2014-18 |
[5] | I. Altun, H. Sahin, D. Turkoglu, Fixed point results for multivalued mappings of Feng-Liu type on M-metric spaces, J. Nonlin. Funct. Anal., 2018 (2018), 1–8. https://doi.org/10.22436/jnsa.009.06.36 doi: 10.22436/jnsa.009.06.36 |
[6] | H. Sahin, I. Altun, D. Turkoglu, Two fixed point results for multivalued F-contractions on M-metric spaces, RACSAM, 113 (2019), 1839–1849. https://doi.org/10.1007/s13398-018-0585-x doi: 10.1007/s13398-018-0585-x |
[7] | P. R. Patle, D. K. Patel, H. Aydi, D. Gopal, N. Mlaiki, Nadler and Kannan type set valued mappings in M-metric spaces and an application, Mathematics, 7 (2019), 1–14. https://doi.org/10.3390/math7040373 doi: 10.3390/math7040373 |
[8] | H. Monfared, M. Azhini, M. Asadi, Fixed point results on M-metric spaces, J. Math. Anal., 7 (2016), 85–101. |
[9] | H. Monfared, M. Azhini, M. Asadi, C-class and F(ψ,φ)-contractions on M-metric spaces, J. Nonlin. Anal. Appl., 8 (2017), 209–224. |
[10] | N. Mlaiki, Fm-contractive and Fm-expanding mappings in M-metric spaces, J. Math. Comput. Sci., 18 (2018), 262–271. https://doi.org/10.22436/jmcs.018.03.02 doi: 10.22436/jmcs.018.03.02 |
[11] | N. Mlaiki, A. Zarrad, N. Souayah, A. Mukheimer, T. Abdeljawed, Fixed point theorem in Mb-metric spaces, J. Math. Anal., 7 (2016), 1–9. |
[12] | P. Hu, F. Gu, Some fixed point theorems of λ-contractive mappings in Menger PSM-spaces, J. Nonlin. Funct. Anal., 33 (2020), 1–12. https://doi.org/10.23952/jnfa.2020.33 doi: 10.23952/jnfa.2020.33 |
[13] | M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604–608. https://doi.org/10.1090/S0002-9939-1973-0334176-5 doi: 10.1090/S0002-9939-1973-0334176-5 |
[14] | N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl., 141 (1989), 177–188. https://doi.org/10.1016/0022-247X(89)90214-X doi: 10.1016/0022-247X(89)90214-X |
[15] | O. Popescu, Some new fixed point theorems for α-Geraghty contraction type maps in metric spaces, Fixed Point Theory A., 190 (2014), 1–12. https://doi.org/10.1186/1687-1812-2014-190 doi: 10.1186/1687-1812-2014-190 |
[16] | M. Arshad, M. Mudhesh, A. Hussain, E. Ameer, Recent thought of α∗-geraghty F-contraction with application, J. Math. Ext., 16 (2021), 1–28. |
[17] | S. Alizadeh, F. Moradlou, P. Salimi, Some fixed point results for (α,β)-(ψ,ϕ)-contractive mappings, Filomat, 28 (2014), 635–647. https://doi.org/10.1186/1687-1812-2014-190 doi: 10.1186/1687-1812-2014-190 |
[18] | E. Ameer, H. Huang, M. Nazam, M. Arshad, Fixed point theorems for multivalued γ-FG-contractions with (α∗,β∗)-admissible mappings in partial b-metric spaces and application, U.P.B. Sci. Bull., S. A, 81 (2019), 97–108. |
[19] | S. K. Padhan, GVV. J. Rao, A. Al-Rawashdeh, H. K. Nashine, R. P. Agarwal, Existence of fixed point for γ-FG-contractive condition via cyclic (α,β)-admissible mappings in b-metric spaces, J. Nonlinear Sci. Appl., 10 (2017), 5495–5508. https://doi.org/10.22436/jnsa.010.10.31 doi: 10.22436/jnsa.010.10.31 |
[20] | H. Isik, B. Samet, C. Vetro, Cyclic admissible contraction and applications to functional equations in dynamic programming, Fixed Point Theory A., 2015 (2015), 1–19. https://doi.org/10.1186/s13663-015-0410-6 doi: 10.1186/s13663-015-0410-6 |
[21] | M. S. Sezen, Cyclic (α,β)-admissible mappings in modular spaces and applications to integral equations, Universal J. Math. Appl., 2 (2019), 85–93. |
[22] | H. A. Hammad, P. Agarwal, L. G. J. Guirao, Applications to boundary value problems and homotopy theory via tripled fixed point techniques in partially metric spaces, Mathematics, 9 (2021), 2012. https://doi.org/10.3390/math9162012 doi: 10.3390/math9162012 |
[23] | H. A. Hammad, H. Aydi, M. D. la Sen, Analytical solution for differential and nonlinear integral equations via Fϖe-Suzuki contractions in modified ϖe-metric-like spaces, J. Func. Space., 2021 (2021), 6128586. |
[24] | H. A. Hammad, H. Aydi, M. D. la Sen, Solutions of fractional differential type equations by fixed point techniques for multivalued contractions, Complexity, 2021 (2021), 5730853. https://doi.org/10.1155/2021/5730853 doi: 10.1155/2021/5730853 |
[25] | H. A. Hammad, M. D. la Sen, Tripled fixed point techniques for solving system of tripled-fractional differential equations, AIMS Math., 6 (2021), 2330–2343. https://doi.org/10.3934/math.2021141 doi: 10.3934/math.2021141 |
[26] | R. A. Rashwan, H. A. Hammad, M. G. Mahmoud, Common fixed point results for weakly compatible mappings under implicit relations in complex valued g-metric spaces, Inform. Sci. Lett., 8 (2019), 111–119. https://doi.org/10.18576/isl/080305 doi: 10.18576/isl/080305 |
[27] | A. Hammad, M. D. la Sen, Fixed-point results for a generalized almost (s,q)-Jaggi F-contraction-type on b-metric-like spaces, Mathematics, 8 (2020), 63. https://doi.org/10.3390/math8010063 doi: 10.3390/math8010063 |
[28] | S. Anwar, M. Nazam, H. H. Al Sulami, A. Hussain, K. Javed, M. Arshad, Existence fixed-point theorems in the partial b-metric spaces and an application to the boundary value problem, AIMS Math., 7 (2022), 8188–8205. https://doi.org/10.3934/math.2022456 doi: 10.3934/math.2022456 |
[29] | B. Rodjanadid, J. Tanthanuch, Some fixed point results on Mb-metric space via simulation functions, Thai J. Math., 18 (2020), 113–125. |
[30] | D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory A., 2012 (2012), 1–6. https://doi.org/10.1186/1687-1812-2012-94 doi: 10.1186/1687-1812-2012-94 |
[31] | A. Felhi, Some fixed point results for multi-valued contractive mappings in partial b-metric spaces, J. Adv. Math. Stud., 9 (2016), 208–225. |
[32] | I. Altun, G. Minak, H. Daǧ, Multivalued F-contractions on complete metric spaces, J. Nonlin. Convex A., 16 (2015), 659–666. https://doi.org/10.2298/FIL1602441A doi: 10.2298/FIL1602441A |
[33] | M. Delfani, A. Farajzadeh, C. F. Wen, Some fixed point theorems of generalized Ft-contraction mappings in b-metric spaces, J. Nonlin. Var. Anal., 5 (2021), 615–625. |
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