Research article

Novel results on fixed-point methodologies for hybrid contraction mappings in Mb-metric spaces with an application

  • Received: 05 May 2022 Revised: 17 October 2022 Accepted: 19 October 2022 Published: 21 October 2022
  • MSC : 46S40, 47H10, 54H25

  • 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×ΞR+ fulfills the stipulations below, for all λ,γ,κΞ:

    (M1) m(λ,λ)=m(γ,γ)=m(λ,γ) iff λ=γ;

    (M2) mλ,γm(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:Ξ2R+ that fulfills the assumptions below, for all λ,γ,κΞ,

    (Mb1) mb(λ,λ)=mb(γ,γ)=mb(λ,γ) iff λ=γ;

    (Mb2) mbλ,γmb(λ,γ);

    (Mb3) mb(λ,γ)=mb(γ,λ);

    (Mb4) There is a coefficient s1 so that for all λ,γ,κΞ, we have

    mb(λ,γ)mbλ,γs[(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+|λγ|p,λ,γΞ.

    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,γΞ.

    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,λmmbλn,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,λmbλn,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 s1 and Γ be a self-mapping on Ξ. If there is k[0,1) so that

    mb(Γλ,Γγ)kmb(λ,γ),λ,γΞ.

    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:ΞΞ is called cyclic (α,β)-admissible if for some λΞ,

    α(λ)1β(Γλ)1,

    and

    β(λ)1α(Γλ)1.

    Definition 2.8. Let Ξ, α,β:Ξ[0,) be mappings and A,B be subsets of Ξ. A mapping Γ:ΞCB(Ξ) is called cyclic (α,β)-admissible if for some λΞ,

    α(λ)1β(Γλ)1,

    and

    β(λ)1α(Γλ)1,

    where β(A)=infaAβ(a) and α(B)=infbBα(b).

    Theorem 2.9. [13] Let Ξ be a complete metric space and Γ:ΞΞ. If there is φξ so that

    d(Γλ,Γγ)φ(d(λ,γ))d(λ,γ),λ,γΞ,

    holds, where ξ is the set of all functions φ:[0,)[0,1) satisfying limntn=0 whenever limnφ(tn)=1. Then Γ has a unique FP λΞ and for each λΞ, 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,βR+;

    (ϝ2) for any sequence {αn}n=1 of positive real numbers, limnαn=0 iff limnϝ(αn)=;

    (ϝ3) there is k(0,1) so that limnαkϝ(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(αn1),s1,

    for all nN and some τ>0, then

    τ+ϝ(snαn)ϝ(sn1αn1),nN.

    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 ,QCBmb(Ξ), define

    Hmb(,Q)=max{δmb(,Q),δmb(Q,)},

    where δmb(,Q)=sup{mb(p,Q):p} and mb(p,Q)=inf{mb(p,q):qQ}.

    The following results are very useful in our study. These results are taken from [4,7].

    Lemma 2.12. Let be a non-empty set in an MbMS (Ξ,mb), then p _ iff

    mb(p,)=supλmbp,λ,

    where _ denotes the closure of wrt mb.

    Lemma 2.13. Let ,Q,CBmb(Ξ), then

    (a) δmb(,)=supp{supqmbpq},

    (b) for s1, we have

    (δmb(,Q)suppsupqQmbpq)s[(δmb(,)infpinfrmbpr)+(δmb(,Q)infrinfqQmbrq)]infrmb(r,r).

    Lemma 2.14. Let ,Q,CBmb(Ξ), then

    (1)

    Hmb(,)=δmb(,)=supp{supqQmbp,q},

    (2) Hmb(,Q)=Hmb(Q,),

    (3) for s1, we get

    (Hmb(,Q)suppsupqQmbpq)s[(Hmb(,)infpinfrmbp,r)+(Hmb(,Q)infrinfqQmbr,q)]infrmb(r,r).

    Lemma 2.15. Let ,QCBmb(Ξ) and h>1, then for all p, there is qQ so that

    (i) mb(p,q)hHmb(,Q),

    (ii) mb(p,q)Hmb(,Q)+h.

    Proof. (i) Suppose that there exists an p such that

    mb(p,q)>hHmb(,Q),

    for all qQ. This implies that

    inf{mb(p,q):qQ}.

    Now

    mb(,Q)δmb(,Q)=sup{mb(p,Q):p}mb(p,Q)Hmb(,Q),

    this is a contradiction since Hmb(,Q)0 and h>1. Hence,

    mb(p,q)hHmb(,Q).

    (ii) Suppose that there exists p such that mb(p,q)>Hmb(,Q)+h for all qQ, then we have

    mb(,Q)+hmb(p,q)δmb(,Q)Hmb(,Q)+h,

    a contradiction again. Since Hmb(,Q)0 and h>1. Thus,

    mb(p,q)Hmb(,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,γmbλ,γ=|mb(λ,λ)mb(γ,γ)|,

    (3) For s1, we have

    Mbλ,γmbλ,γs[(Mbλ,κmbλ,κ)+(Mbκ,γmbκ,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:ΞCBmb(Ξ) is called η-cyclic (α,β)-admissible if for some λΞ,

    α(λ)η(λ)β(Γλ)η(Γλ),

    and

    β(λ)η(λ)α(Γλ)η(Γλ)

    where β(A)=infaAβ(a) and α(B)=infbBα(b).

    Definition 3.2. Let Ξ, α,β,η:Ξ[0,) be mappings and A,B be subsets of Ξ. Two mappings ,Γ:ΞCBmb(Ξ) are called η-cyclic (α,β) -admissible if for some λΞ,

    α(λ)η(λ)β(λ)η(λ),

    and

    β(λ)η(λ)α(Γλ)η(Γλ).

    Notice that:

    ● If η=η=1 and =Γ, then we get Definition 2.2 in [18].

    ● Definition 3.2 reduces to Definition 3.1, if we put =Γ.

    Example 3.3. Let Ξ=[0,). Define the mappings ,Γ:ΞCBmb(Ξ) and α,β,η:Ξ[0,) by λ={3λ}, Γλ={λ2}, η(λ)=λ, λΞ,

    α(λ)={e3λ2,ifλ>0,1,otherwise,andβ(λ)={52λ,ifλ>0,1,otherwise.

    For all λ>0, we get

    α(λ)=e3λ2λ=η(λ)β(λ)=β(3λ)=56λ3λ=η(λ).

    Similarly,

    β(λ)=52λλ=η(λ).

    Otherwise, for λ=0 the conditions of definition are satisfied. Then the pair (,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 ,Γ:ΞCBmb(Ξ) is called a pair of generalized (α,β)-Geraghty ϝ -contraction mappings if there exist φξ and ϝϝs so that for all λ,γΞ, s1 and τR+ with Hmb(λ,Γγ)>0,

    β(λ)α(Γγ)η(λ)η(Γγ)τ+ϝ(sHmb(λ,Γγ))ϝ(φ(Mmb(λ,γ))Mmb(λ,γ)), (3.1)

    where

    Mmb(λ,γ)=max{mb(λ,γ),mb(λ,γ),mb(γ,Γγ),mb(λ,λ)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 ,Γ:ΞCBmb(Ξ) be two multivalued mappings satisfy the postulates below:

    (1) the pair (,Γ) is generalized (α,β)-Geraghty ϝ-contraction;

    (2) the pair (,Γ) is η-cyclic (α,β)-admissible;

    (3) either there is λ0Ξ so that α(Γλ0)η(Γλ0) or γ0Ξ so that β(γ0)η(γ0).

    Then and Γ have a common FP λΞ.

    Proof. Let λ0Ξ so that α(λ0)η(λ0), by axiom (2) λ1λ0 and λ2Γλ1 so that

    α(λ0)η(λ0)β(λ1)β(λ0)η(λ0)

    and

    α(λ2)α(Γλ1)η(Γλ1).

    Therefore

    α(Γλ1)β(λ0)η(Γλ1)η(λ0),

    Since F is right continuous, then from Remark 2.11, we have

    ϝ(smb(λ1,Γλ1))=infγΓλ1ϝ(smb(λ1,γ)).

    Thus, there is γ=λ2Γλ1, so that

    ϝ(smb(λ1,λ2))ϝ(sHmb(λ0,Γλ1))ϝ(φ(Mmb(λ0,λ1))Mmb(λ0,λ1))τ, (3.3)

    where

    Mmb(λ0,λ1)=max{mb(λ0,λ1),mb(λ0,λ0),mb(λ1,Γλ1),mb(λ0,λ0)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 λ3λ2 so that

    ϝ(smb(λ2,λ3))ϝ(φ(Mmb(λ1,λ2)))Mmb(λ1,λ2))τϝ(φ(mb(λ1,λ2)))mb(λ1,λ2))τ.

    Continuing with the same scenario, we construct a sequence {λn} in Ξ so that λ2n+1λ2n and λ2n+2Γλ2n+1. Since the pair (,Γ) is η-cyclic (α,β) -admissible, we have

    β(λ2n)α(Γλ2n+1)η(λ2n)η(Γλ2n+1),n0.

    Subsequently, by (3.1), we get

    τ+ϝ(smb(λ2n+1,λ2n+2))τ+ϝ(sHmb(λ2n,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, n0, then, we can write

    τ+ϝ(sρ2n+1)ϝ(ρ2n),n0.

    By (ϝ4), one can obtain

    τ+ϝ(μ2n+1)ϝ(μ2n),n0. (3.5)

    Repeating the inequality (3.5), we obtain

    ϝ(μ2n+1)ϝ(μ2n)τ...ϝ(μ0)(2n+1)τ,n0. (3.6)

    Letting n in (3.6), we have

    limnϝ(μ2n+1)=.

    It follows from (ϝ2) that

    limnμ2n+1=0.

    By (ϝ3), there is k(0,1) so that

    limnμk2n+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)τ0,n0.

    Taking the limit as n and since τ>0, we obtain

    limnμk2n+1(2n+1)=0.

    Thus, there is n1N so that

    μk2n+1(2n+1)1μ2n+11(2n+1)1knn1.

    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+3s[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+4s[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,λqq1i=psip+1mb(λi,λi+1)q1i=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+4mbλ2n+1,λ2n+4s(Mbλ2n+1,λ2n+2mbλ2n+1,λ2n+2)+s2(Mbλ2n+2,λ2n+3mbλ2n+2,λ2n+3)+s3(Mbλ2n+3,λ2n+4mbλ2n+3,λ2n+4).

    In general, for all q>p>n1 with p=2n+1, we obtain

    Mbλp,λqmbλp,λqq1i=psip+1(Mbλi,λi+1mbλi,λi+1)q1i=psip+1Mbλi,λi+1q1i=psip+1mb(λi,λi)q1i=psip+1mb(λi,λi+1)q1i=psimb(λi,λi+1)i=pμi.

    The convergence of the series i=pμi leads to

    limp,q(Mbλp,λqmbλp,λq)=0.

    Therefore, {λn} is an mb-Cauchy sequence in Ξ. Since Ξ is mb-complete, there exists λΞ so λnλ as n, implies λ2n+1λ and λ2n+2λ as n. Thus, we have

    limn(mb(λ2n+1,λ)mbλ2n+1,λ)=0. (3.7)

    Since limnmb(λ2n+1,λ2n+1)=0, then by (3.7), we get

    limnmb(λ2n+1,λ)=0. (3.8)

    It follows from (3.1), (3.4) and (3.8) that

    limnHmb(λ2n,Γλ)=0. (3.9)

    Since λ2n+1λ2n and

    mb(λ2n+1,Γλ)Hmb(λ2n,Γλ).

    Then after taking the limit as n, we obtain that

    limnmb(λ2n+1,Γλ)=0. (3.10)

    By (Mb2), one can write

    mbλ2n+1,Γλmb(λ2n+1,Γλ),

    that is

    limnmbλ2n+1,Γλ=0. (3.11)

    Now, utilizing (Mb4), we have

    mb(λ,Γλ)supγΓλmbλ,γmb(λ,Γλ)mbλ,Γλs[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)supγΓλmbλ,γ. (3.13)

    Using (Mb2), for all γΓλ, we get

    mbλ,γmb(λ,γ),

    yields

    mbλ,γmb(λ,γ)0.

    Thus

    sup{mbλ,γmb(λ,γ):γΓλ}0,

    this implies that

    supγΓλmbλ,γsupγΓλmb(λ,γ)0.

    Therefore

    supγΓλmbλ,γmb(λ,Γλ). (3.14)

    From (3.13) and (3.14), we obtain

    mb(λ,Γλ)=supγΓλmbλ,γ.

    Hence by Lemma 2.12, we get λ¯Γλ=Γλ. Similarly, we can easily conclude that λλ. Therefore λ is a common FP of 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(λ,Γγ)>0,

    α(λ)β(Γγ)η(λ)η(Γγ)τ+ϝ(Hm(λ,Γγ))ϝ(φ(M(λ,γ))M(λ,γ))

    where φξ, ϝϝs, τR+ and

    M(λ,γ)=max{m(λ,γ),m(λ,λ),m(γ,Γγ),m(λ,λ)m(γ,Γγ)1+m(λ,γ)}.

    Moreover, under the same conditions (1)–(3) of Theorem 3.5, 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 and Γ in complete MbMS, provided that the stipulations (1)–(3) of Theorem 3.5 hold.

    ● If we consider =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+|λγ|p,λ,γΞ.

    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,κmb(κ,γ)mbκ,γ.

    This means (Ξ,mb) is not MMS. Define ,Γ:ΞCBmb(Ξ) by

    λ={{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 λΞ. Now, for λ(0,1], we have α(λ)η(λ) implies

    β(λ)=β(λ64)=5λ64+1λ64+1=η({λ64})=η(λ).

    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)2λ48+1=η(λ48)=η(Γλ).

    Hence the pair (,Γ) is η-cyclic (α,β)-admissible mappings. Consider φ(t)=16p, so for λ,γ(0,1], one can write

    ϝ(sHmb(λ,Γγ))=ϝ(2p(max{supaλmb(a,Γγ),supbΓγmb(λ,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(λ,Γγ))ln(16pmb(λ,γ))=pln(6)+ln(mb(λ,γ)),

    which implies that

    ln(sHmb(λ,Γγ))ln(mb(λ,γ))τ.

    Since mb(λ,γ)Mb(λ,γ), and using the definition of φ, then we obtain

    ϝ(sHmb(λ,Γγ))ϝ(φ(Mb(λ,γ)))τϝ(φ(Mmb(λ,γ))Mmb(λ,γ))τ.

    Otherwise, the inequality below holds

    ϝ(sHmb(λ,Γγ)ϝ(φ(Mmb(λ,γ))Mmb(λ,γ))τ.

    Analogously, for each λ,γΞ, 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 and Γ.

    This part is a reduction of the previous part by taking and Γ are single-valued mappings.

    Definition 4.1. Let Ξ and α,β,η:Ξ[0,) be given functions. The mapping Γ:ΞΞ is called η-cyclic (α,β)-admissible if for some λΞ,

    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 ,Γ:ΞΞ are called η-cyclic (α,β)-admissible if for some λΞ,

    α(λ)η(λ)β(λ)η(λ),

    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:ΞΞ satisfies the following condition: There are φξ and ϝϝs so that for all λ,γΞ, s1 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 λ0Ξ 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 ,Γ:ΞΞ satisfy the assumption below: There are φξ and ϝϝs so that for all λ,γΞ, s1 and τ>1,

    α(λ)β(γ)η(λ)η(γ)τ+ϝ(smb(λ,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) (,Γ) is a pair of η-cyclic (α,β)-admissible;

    (ii) there is λ0Ξ so that α(λ0)η(λ0) or γ0Ξ so that β(γ0)η(γ0).

    Then and Γ have a common FP λΞ.

    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, and Γ be self-mappings defined on Ξ. If there are φξ and ϝϝs so that for all λ,γΞ, s1 and τ>1,

    τ+ϝ(smb(λ,Γγ))ϝ(φ(Mmb(λ,γ))Mmb(λ,γ)), (4.1)

    where Mmb(λ,γ) is described as (3.2). Then and Γ have a common FP λΞ.

    Note. The pair (,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]×RR 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(1s),0ts1,s(1t),0st1.

    Let Ξ=C([0,1],R) be the set of all continuous functions defined on [0,1]. Define a function m:Ξ×ΞR+ by

    mb(λ,γ)=maxtI(|λ(t)+γ(t)2|)2,λ,γΞ.

    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,z2R, 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+z12|2,|z2+Γz22|2,|z1+z12|2|z2+Γz22|2s+|z1+z22|2};

    (2) there is s1 so that 10G(t,r)dr12eτ7s, 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 λΞ.

    Proof. Describe the operators ,Γ:ΞΞ as

    λ(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 and Γ. Let λ,γΞ, by our assumption, for all t[0,1], we get

    [|λ(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[12eτ7s]2=ω(t)12eτ7sMmb(λ,γ).

    Consequently, we get

    smb(λ,Γγ)3ω(t)7eτMmb(λ,γ)eτφ(Mmb(λ,γ))Mmb(λ,γ),

    which implies that

    τ+ln(smb(λ,Γγ))ln[φ(Mmb(λ,γ))Mmb(λ,γ)],

    where ϝ(λ)=lnλϝs and φ(t)=3ω(t)7, for all t[0,1]. Thus, all stipulations of Corollary 4.5 are fulfilled. Therefore, the operators 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×ΞR+. It is defined as

    mb(λ,γ)=maxtI(|λ(t)+γ(t)2|)2,λ,γΞ.

    On a complete metric space, the classical theorem holds true, but the first metric space requirement is not met as follows:

    forλ,γΞ,ifλ=γ,thenmb(λ,λ)=maxtI(|λ(t)+λ(t)2|)2=maxtI(|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.



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