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Research article

Fixed point theorem for orthogonal contraction of Hardy-Rogers-type mapping on O-complete metric spaces

  • In this paper, we introduce the notion of an orthogonal (F, ψ)-contraction of Hardy-Rogers-type mapping and prove some fixed point theorem for such contraction mappings in orthogonally metric spaces. Our result extend and improve the main result of the paper by Sawangsup et al. [Fixed point theorems for orthogonal F-contraction mappings on O-complete metric spaces, J. Fixed Point Theory Appl. (2020) 22:10].

    Citation: Qing Yang, Chuanzhi Bai. Fixed point theorem for orthogonal contraction of Hardy-Rogers-type mapping on O-complete metric spaces[J]. AIMS Mathematics, 2020, 5(6): 5734-5742. doi: 10.3934/math.2020368

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  • In this paper, we introduce the notion of an orthogonal (F, ψ)-contraction of Hardy-Rogers-type mapping and prove some fixed point theorem for such contraction mappings in orthogonally metric spaces. Our result extend and improve the main result of the paper by Sawangsup et al. [Fixed point theorems for orthogonal F-contraction mappings on O-complete metric spaces, J. Fixed Point Theory Appl. (2020) 22:10].


    Fixed point theory plays a fundamental role in mathematics and applied science, such as optimization, mathematical models and economic theories. Also, this theory has been applied to show the existence and uniqueness of the solutions of differential equations, integral equations and many other branches of mathematics, see [1,2]. A prominent result in fixed point theory is the Banach contraction principle [3]. Since the appearance of this principle, there has been a lot of activity in this area. Bakhtin [4] in 1989 introduced the notion of a b-metric space (Bms). Shoaib et. al [5] proved certain fixed point results in rectangular metric spaces. Multivalued mappings in various types of metric spaces have been extensively studied by many researchers to establish fixed point results and their applications, see for instance [6,7,8,9,10,11,12].

    In 1965, Zadeh [13] introduced the concept of a fuzzy set theory to deal with the unclear or inexplicit situations in daily life. Using this theory, Kramosil and Michálek [14] defined the concept of a fuzzy metric space (Fms). Grabiec [15] gave contractive mappings on a Fms and extended fixed point theorems of Banach and Edelstein in such a space. Successively, George and Veeramani [16] slightly modified the notion of a Fms introduced by Kramosil and Michálek [14] and then obtained a Hausdorff topology and a first countable topology on it. Many fixed point results have been established in a Fms. For instance, see [17,18,19,20,21,22,23,24,25] and the references therein. Recently, some coupled fuzzy fixed-point results on closed ball are established in fuzzy metric spaces [26]. The notion of generalized fuzzy metric spaces is studied in [27].

    The notion of a fuzzy b-metric space (Fbms) was defined in [28]. The notion of a Hausdorff Fms is introduced in [29]. Fixed point theory for multivalued mapping in fuzzy metric spaces has been extended in many directions. For a multivalued mapping (Mvp) in a complete Fms, some fixed point results are establish in [30]. Some fixed point results for a Mvp in a Hausdorff fuzzy b-metric space (Hfbms) are proved in [31]. In this article, we prove some fixed point results for a Mvp using Geraghty type contractions in a Hfbms. Results in [31,32] and [30] turn out to be special cases of our results.

    Throughout the article, refers to a non-empty set, N represents the set of natural numbers, R corresponds to the collection of real numbers, CB() and ˆC0() represent the collection of closed and bounded subsets and compact subsets of , respectively.

    Let us have a look at some core concepts that will be helpful for the proof of our main results.

    Definition 1.1. [33] For a real number b1, the triplet (,Θfb,) is called a Fbms on if for all ψ1,ψ2,ψ3 and γ>0, the following axioms hold, where is a continuous t-norm and Θfb is a fuzzy set on ××(0,):

    [Fb1:] Θfb(ψ1,ψ2,γ)>0;

    [Fb2:] Θfb(ψ1,ψ2,γ)=1 if and only if ψ1=ψ2;

    [Fb3:] Θfb(ψ1,ψ2,γ)=Θfb(ψ2,ψ1,γ);

    [Fb4:] Θfb(ψ1,ψ3,b(γ+β))Θfb(ψ1,ψ2,γ)Θfb(ψ2,ψ3,β) γ,β0;

    [Fb5:] Θfb(ψ1,ψ2,.):(0,)[0,1] is left continuous.

    The notion of a Fms in the sense of George and Veeramani [16] can be obtained by taking b=1 in the above definition.

    Example 1.1. For a Bms (,Θb,), define a mapping Θfb:××(0,)[0,1] by

    Θfb(ψ1,ψ2,γ)=γγ+db(ψ1,ψ1).

    Then (,Θfb,) is a Fbms.

    Following Grabiec [15], the notions of G-Cauchyness and completeness are defined as follows:

    Definition 1.2. [15]

    (i) If for a sequence {ψn} in a Fbms (,Θfb,), there is ψ such that

    limnΘfb(ψn,ψ,γ)=1,γ>0,

    then {ψn} is said to be convergent.

    (ii) If for a sequence {ψn} in a Fbms (,Θfb,), limnΘfb(ψn,ψn+q,γ)=1 then {ψn} is a G-Cauchy sequence for all γ>0 and positive integer q.

    (iii) A Fbms is G-complete if every G-Cauchy sequence is convergent.

    Definition 1.3. [30] Let B be any nonempty subset of a Fms (,Θfb,) and γ>0, then we define FΘfb(ϱ1,B,γ), the fuzzy distance between an element ϱ1 and the subset B, as follows:

    FΘfb(ϱ1,B,γ)=sup{Θf(ϱ1,ϱ2,γ):ϱ2B}.

    Note that FΘfb(ϱ1,B,γ)=FΘfb(B,ϱ1,α).

    Lemma 1.1. [31] Consider a Fbms (,Θfb,) and let CB() be the collection of closed bounded subsets of . If ACB() then ψA if and only if FΘfb(A,ψ,γ)=1γ>0.

    Definition 1.4. [31] Let (,Θfb,) be a Fbms. Define HFΘfb on ˆC0()׈C0()×(0,) by

    HFΘfb(A,B,γ)=min{ infψAFΘfb(ψ,B,γ),infϱBFΘfb(A,ϱ,γ)},

    for all A,B^C0() and γ>0.

    For Geraghty type contractions, follow [33] to define a class FΘb of all functions β:[0,)[0,1b) for b1, as

    FΘb={β:[0,)[0,1b)|limnβ(γn)=1blimnγn=0}. (1.1)

    Lemma 1.2. [31] Let (,Θfb,) be a G-complete Fbms. If ψ,ϱ and for a function βFΘfb

    Θfb(ψ,ϱ,β(Θfb(ψ,ϱ,γ))γ)Θfb(ψ,ϱ,γ),

    then ψ=ϱ.

    Lemma 1.3. [31] Let (^C0(),HFΘfb,) be a Hfbms where (Θfb,) is a Fbm on . If for all A,B^C0(), for each ψA and for γ>0 there exists ϱψB, satisfying FΘfb(ψ,B,γ)=Θfb(ψ,ϱψ,γ), then

    HFΘfb(A,B,γ)Θfb(ψ,ϱψ,γ).

    In this section, we develop some fixed point results by using the idea of a Hfbms. Furthermore, an example is also presented for a deeper understanding of our results.

    Recall that, given a multivalued mapping Ξ:ˆC0(), a point ψ is said to be a fixed point of Ξ if ψΞψ.

    Theorem 2.1. Let (,Θfb,) be a G-complete Fbms and (^C0(),HFΘfb,) be a Hfbms. Let Ξ:^C0() be a Mvp satisfying

    HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)Θfb(ψ,ϱ,γ), (2.1)

    for all ψ,ϱ, where βFΘfb as defined in (1.1). Then Ξ has a fixed point.

    Proof. Choose {ψn} for ψ0 as follows: Let ψ1 such that ψ1Ξψ0 by the application of Lemma 1.3, we can choose ψ2Ξψ1 such that for all γ>0,

    Θfb(ψ1,ψ2,γ)HFΘfb(Ξψ0,Ξψ1,γ).

    By induction, we have ψr+1Ξψr satisfying

    Θfb(ψr,ψr+1,γ)HFΘfb(Ξψr1,Ξψr,γ)rN.

    By the application of (2.1) and Lemma 1.3, we have

    Θfb(ψr,ψr+1,γ)HFΘfb(Ξψr1,Ξψr,γ)Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))HFΘfb(Ξψr2,Ξψr1,γβ(Θfb(ψr1,ψr,γ)))Θfb(ψr2,ψr1,γβ(Θfb(ψr1,ψr,γ))β(Θfb(ψr2,ψr1,γ)))HFΘfb(Ξψ0,Ξψ1,γβ(Θfb(ψr1,ψr,γ))β(Θfb(ψr2,ψr1,γ))β(Θfb(ψ1,ψ2,γ)))Θfb(ψ0,ψ1,γβ(Θfb(ψr1,ψr,γ))β(Θfb(ψr2,ψr1,γ))β(Θfb(ψ0,ψ1,γ))). (2.2)

    For any qN, writing q(γq)=γq+γq++γq and using [Fb4] repeatedly,

    Θfb(ψr,ψr+q,γ)Θfb(ψr,ψr+1,γqb)Θfb(ψr+1,ψr+2,γqb2)Θfb(ψr+2,ψr+3,γqb3)Θfb(ψr+q1,ψr+q,γqbq).

    Using (2.2) and [Fb5], we get

    Θfb(ψr,ψr+q,γ)Θfb(ψ0,ψ1,γqbβ(Θfb(ψr1,ψr,γ))β(Θfb(ψr2,ψr1,γ))β(Θfb(ψ0,ψ1,γ)))Θfb(ψ0,ψ1,γqb2β(Θfb(ψr,ψr+1,γ))β(Θfb(ψr1,ψr,γ))β(Θfb(ψ0,ψ1,γ)))Θfb(ψ0,ψ1,γqb3β(Θfb(ψr+1,ψr+2,γ))β(Θfb(ψr,ψr+1,γ))β(Θfb(ψ0,ψ1,γ)))Θfb(ψ0,ψ1,γqbqβ(Θfb(ψr+q2,ψr+q1,γ))β(Θfb(ψr+q3,ψr+q2,γ))β(Θfb(ψ0,ψ1,γ))).

    That is,

    Θfb(ψr,ψr+q,γ)Θfb(ψ0,ψ1,br1γq)Θfb(ψ0,ψ1,br1γq)Θfb(ψ0,ψ1,br1γq)Θfb(ψ0,ψ1,br1γq).

    Taking limit as r, we get

    limnΘfb(ψr,ψr+q,γ)=111=1.

    Hence, {ψr} is G-Cauchy sequence. By the G-completeness of , there exists ϕ such that

    Θfb(ϕ,Ξϕ,γ)Θfb(ϕ,ψr+1,γ2b)Θfb(ψr+1,Ξϕ,γ2b)Θfb(ϕ,ψr+1,γ2b)HFΘfb(Ξψr,Ξϕ,γ2b)Θfb(ϕ,ψr+1,γ2b)Θfb(ψr,ϕ,γ2bβ(Θfb(ψr,ϕ,γ)))1asr.

    By Lemma 1.1, it follows that ϕΞϕ. That is, ϕ is a fixed point for Ξ.

    Remark 2.1.

    (1) If we take β(Θfb(ψ,ϱ,γ))=k with bk<1, we get Theorem 3.1 of [31].

    (2) By setting ^C0()= the mapping Ξ:^C0() becomes a single valued and we get Theorem 3.1 of [32]. Notice that when Ξ is a singlevalued map, Ξψ becomes a singleton set and the fact that HFΘfb(Ξψ,Ξϱ,γ)=Θfb(Ξψ,Ξϱ,γ) indicates that the fixed point will be unique as proved in [32].

    (3) Set b=1 and ^C0()= and let k(0,1) be such that β(Θfb(ψ,ϱ,γ))=k then we get the main result of [15].

    The next example illustrates Theorem 2.1.

    Example 2.1. Let =[0,1] and define a G-complete Fbms by

    Θfb(ψ,ϱ,γ)=γγ+(ψϱ)2,

    with b1. For βFb, define a mapping Ξ:^C0() by

    Ξψ={{0}if ψ=0,{0,β(Θfb(ψ,ϱ,γ))ψ2}otherwise.

    For ψ=ϱ,

    HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)=1=Θfb(ψ,ϱ,γ).

    If ψϱ, then following cases arise.

    For ψ=0 and ϱ(0,1], we have

    HFΘfb(Ξ0,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)=min{ infaΞ0FΘfb(a,Ξϱ,β(Θfb(ψ,ϱ,γ))γ),infbΞϱFΘfb(Ξ0,b,β(Θfb(ψ,ϱ,γ))γ))}=min{infaΞ0FΘfb(a,{0,β(Θfb(ψ,ϱ,γ))ϱ2},β(Θfb(ψ,ϱ,γ))γ),infbΞϱFΘfb({0},b,β(Θfb(ψ,ϱ,γ))γ)}=min{inf{FΘfb(0,{0,β(Θfb(ψ,ϱ,γ))ϱ2},β(Θfb(ψ,ϱ,γ))γ)},inf{FΘfb({0},0,β(Θfb(ψ,ϱ,γ))γ),FΘfb({0},β(Θfb(ψ,ϱ,γ))ϱ2,β(Θfb(ψ,ϱ,γ))γ)}}=min{inf{sup{FΘfb(0,0,β(Θfb(ψ,ϱ,γ))γ),FΘfb(0,β(Θfb(ψ,ϱ,γ))ϱ2,β(Θfb(ψ,ϱ,γ))γ)}},inf{FΘfb(0,0,β(Θfb(ψ,ϱ,γ))γ),FΘfb(0,β(Θfb(ψ,ϱ,γ))y2,β(Θfb(ψ,ϱ,γ))γ)}}=min{inf{sup{1,γγ+ϱ24}},inf{1,γγ+ϱ24}}=min{inf{1},γγ+ϱ24}=min{1,γγ+ϱ24}=γγ+ϱ24.

    It follows that

    HFΘfb(Ξ0,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)>Θfb(0,ϱ,γ)=γγ+ϱ2.

    For ψ and ϱ(0,1], after simplification we have

    HFΘfb(S(ψ),Ξϱ,β(Θfb(ψ,ϱ,γ))γ)=min{sup{γγ+ψ24,γγ+(ψϱ)24},sup{γγ+ϱ24,γγ+(ψϱ)24}}γγ+(ψϱ)24>γγ+(ψϱ)2=Θfb(ψ,ϱ,γ).

    Thus, for all cases, we have

    HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)Θfb(ψ,ϱ,γ).

    Since all conditions of Theorem 2.1 are satisfied and 0 is a fixed point of Ξ.

    Theorem 2.2. Let (,Θfb,) be a G-complete Fbms with b1 and (^C0(),HFΘfb,) be a Hfbms. Let Ξ:^C0() be a Mvp satisfying

    HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)min{FΘfb(ϱ,Ξϱ,γ)[1+FΘfb(ψ,Ξψ,γ)]1+Θfb(ψ,ϱ,γ),Θfb(ψ,ϱ,γ)}, (2.3)

    for all ψ,ϱ, where βFΘfb as given in (1.1). Then Ξ has a fixed point.

    Proof. Choose {ψn} for ψ0 as follows: Let ψ1 such that ψ1Ξψ0. By the application of Lemma 1.3 we can choose ψ2Ξψ1 such that

    Θfb(ψ1,ψ2,γ)HFΘfb(Ξψ0,Ξψ1,γ),γ>0.

    By induction, we have ψr+1Ξψr satisfying

    Θfb(ψr,ψr+1,γ)HFΘfb(Ξψr1,Ξψr,γ),rN.

    By the application of (2.3) and Lemma 1.3 we have

    Θfb(ψr,ψr+1,γ)HFΘfb(Ξψr1,Ξψr,γ)min{FΘfb(ψr,Ξψr,γβ(Θfb(ψr1,ψr,γ)))[1+FΘfb(ψr1,Ξψr1,γβ(Θfb(ψr1,ψr,γ)))]1+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))[1+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))]1+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))},Θfb(ψr,ψr+1,γ)min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}. (2.4)

    If

    min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),

    then (2.4) implies

    Θfb(ψr,ψr+1,γ)Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))).

    The result is obvious by Lemma 1.2.

    If

    min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),

    then from (2.4) we have

    Θfb(ψr,ψr+1,γ)Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))Θfb(ψr2,ψr1,γβ(Θfb(ψr1,ψr,γ))β(Θfb(ψr2,ψr1,γ)))Θfb(ψ0,ψ1,γβ(Θfb(ψr1,ψr,γ))β(Θfb(ψr2,ψr1,γ))β(Θfb(ψ0,ψ1,γ)).

    The rest of the proof can be done by proceeding same as in Theorem 2.1.

    Remark 2.2.

    (1) If we take β(Θfb(ψ,ϱ,γ))=k with bk<1, we get Theorem 3.2 of [31].

    (2) By taking b=1 and for some 0<k<1 setting β(Θfb(ψ,ϱ,γ))=k in Theorem 2.2, we get the main result of [30].

    Theorem 2.3. Let (,Θfb,) be a G-complete Fbms with b1 and (^C0(),HFΘfb,) be a Hfbms. Let Ξ:ˆC0() be a Mvp satisfying

    HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)min{FΘfb(ϱ,Ξϱ,γ)[1+FΘfb(ψ,Ξψ,γ)+FΘfb(ϱ,Ξψ,γ)]2+Θfb(ψ,ϱ,γ),Θfb(ψ,ϱ,γ)} (2.5)

    for all ψ,ϱ, where βFΘfb, the class of functions defined in (1.1). Then Ξ has a fixed point.

    Proof. Choose {ψn} for ψ0 as follows: Let ψ1 such that ψ1Ξψ0. by the application of Lemma 1.3 we can choose ψ2Ξψ1 such that

    Θfb(ψ1,ψ2,γ)HFΘfb(Ξψ0,Ξψ1,γ),γ>0.

    By induction, we have ψr+1Ξψr satisfying

    Θfb(ψr,ψr+1,γ)HFΘfb(Ξψr1,Ξψr,γ),rN.

    By the application of (2.5) and Lemma 1.3, we have

    Θfb(ψr,ψr+1,γ)HFΘfb(Ξψr1,Ξψr,γ)min{FΘfb(ψr,Ξψr,γβ(Θfb(ψr1,ψr,γ)))[1+FΘfb(ψr1,Ξψr1,γβ(Θfb(ψr1,ψr,γ)))+FΘfb(ψr,Ξψr1,γβ(Θfb(ψr1,ψr,γ)))]2+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))[1+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))+Θfb(ψr,ψr,γβ(Θfb(ψr1,ψr,γ)))]2+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))[1+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))+1]2+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))[2+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))]2+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}. (2.6)

    If

    min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),

    then (2.6) implies

    Θfb(ψr,ψr+1,γ)Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),

    and the proof follows by Lemma 1.2.

    If

    min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))).

    Then from (2.6) we have

    Θfb(ψr,ψr+1,γ)Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))β(Θfb(ψr2,ψr1,γ))β(Θfb(ψ0,ψ1,γ))).

    The rest of the proof is same as in Theorem 2.1.

    Remark 2.3. Theorem 3.3 of [31] becomes a special csae of the above theorem by setting β(Θfb(ψ,ϱ,γ))=k where k is chosen such that bk<1.

    Theorem 2.4. Let (,Θfb,) be a G-complete Fbms with b1 and (^C0(),HFΘfb,) be a Hfbms. Let Ξ:ˆC0() be a multivalued mapping satisfying

    HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)min{FΘfb(ψ,Ξψ,γ)[1+FΘfb(ϱ,Ξϱ,γ)]1+FΘfb(Ξψ,Ξϱ,γ),FΘfb(ψ,Ξϱ,γ)[1+FΘfb(ψ,Ξψ,γ)]1+Θfb(ψ,ϱ,γ),FΘfb(ψ,Ξψ,γ)[2+FΘfb(ψ,Ξϱ,γ)]1+Θfb(ψ,Ξϱ,γ)+FΘfb(ϱ,Ξψ,γ),Θfb(ψ,ϱ,γ)}, (2.7)

    for all ψ,ϱ, where βFfb, the class of functions defined in (1.1). Then Ξ has a fixed point.

    Proof. In the same way as Theorem 2.1, we have

    Θfb(ψ1,ψ2,γ)HFΘfb(Ξψ0,Ξψ1,γ),γ>0.

    By induction, we obtain ψr+1Ξψr satisfying

    Θfb(ψr,ψr+1,γ)HFΘfb(Ξψr1,Ξψr,γ),nN.

    Now, by (2.7) together with Lemma 1.3, we have

    Θfb(ψr,ψr+1,γ)HFΘfb(Ξψr1,Ξψr,γ)min{FΘfb(ψr1,Ξψr1,γβ(Θfb(ψr1,ψr,γ)))[1+FΘfb(ψr,Sψr,γβ(Θfb(ψr1,ψr,γ)))]1+FΘfb(Ξψr1,Ξψr,γβ(Θfb(ψr1,ψr,γ))),FΘfb(ψr,Ξψr,γβ(Θfb(ψr1,ψr,γ)))[1+FΘfb(ψr1,Ξψr1,γβ(Θfb(ψr1,ψr,γ)))]1+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),FΘfb(ψr1,Ξψr1,γβ(Θfb(ψr1,ψr,γ)))[2+FΘfb(ψr1,Ξψr,γβ(Θfb(ψr1,ψr,γ)))]1+FΘfb(ψr1,Ξψr,γβ(Θfb(ψr1,ψr,γ)))+FΘfb(ψr,Ξψr1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}min{Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))[1+Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))]1+Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))[1+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))]1+Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))[2+Θfb(ψr1,ψr+1,γβ(Θfb(ψr1,ψr,γ)))]1+Θfb(ψr1,ψr+1,γβ(Θfb(ψr1,ψr,γ)))+Θfb(ψr,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))},Θfb(ψr,ψr+1,γ)min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}. (2.8)

    If

    min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),

    then (2.8) implies

    Θfb(ψr,ψr+1,γ)Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))).

    Then the proof follows by Lemma 1.2.

    If

    min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),

    then from (2.6) we have

    Θfb(ψr,ψr+1,γ)Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))Θfb(ψ0,ψ1,γβ(Θfb(ψr1,ψr,γ))β(Θfb(ψr2,ψr1,γ))β(Θfb(ψ0,ψ1,γ))).

    The rest of the proof is similar as in Theorem 2.1.

    Remark 2.4. Again by taking β(Θfb(ψ,ϱ,γ))=k with kb<1, we get Theorem 3.4 of [31].

    Theorem 2.5. Let (,Θfb,) be a G-complete Fbms with b1 and (^C0(),HFΘfb,) be a Hfbms. Let Ξ:ˆC0() be a Mvp satisfying

    HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)min{HFΘfb(Ξψ,Ξϱ,γ).Θfb(ψ,ϱ,γ),HFΘfb(ψ,Ξψ,γ).HFΘfb(ϱ,Ξϱ,γ)})max{HFΘfb(ψ,Ξψ,γ),HFΘfb(ϱ,Ξϱ,γ)}, (2.9)

    for all ψ,ϱ, where βFfb. Then Ξ has a fixed point.

    Proof. In the same way as Theorem 2.1, we have

    Θfb(ψ1,ψ2,γ)HFΘfb(Ξψ0,Ξψ1,γ),γ>0.

    By induction we have ψr+1Ξψr satisfying

    Θfb(ψr,ψr+1,γ)HFΘfb(Ξψr1,Ξψr,γ),nN.

    Now by (2.7) together with Lemma 1.3 and some obvious simplification step, we have

    Θfb(ψr,ψr+1,γ)HFΘfb(Ξψr1,Ξψr,γ)min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))).Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))).Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))}max{Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))}Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))).Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))max{Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))} (2.10)

    If

    max{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),

    then (2.10) implies

    Θfb(ψr,ψr+1,t)Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))

    Then the proof follows by Lemma 1.2.

    If

    max{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),

    then from (2.10) we have

    Θfb(ψr,ψr+1,γ)Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))Θfb(ψ0,ψ1,γβ(Θfb(ψr1,ψr,γ))β(Θfb(ψr2,ψr1,γ))β(Θfb(ψ0,ψ1,γ))).

    The remaining proof follows in the same way as in Theorem 2.1.

    Theorem 2.6. Let (,Θfb,) be a G-complete Fbms with b1 and (^C0(),HFΘfb,) be a Hfbms. Let Ξ:ˆC0() be a Mvp satisfying

    HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)Γ1(ψ,ϱ,γ)Γ2(ψ,ϱ,γ), (2.11)

    where,

    {Γ1(ψ,ϱ,γ)=min{HFΘfb(Ξψ,Ξϱ,γ),HFΘfb(ψ,Ξψ,γ),HFΘfb(ϱ,Ξϱ,γ),Θfb(ψ,ϱ,γ)}Γ2(ψ,ϱ,γ)=max{HFΘfb(ψ,Ξϱ,γ),HFΘfb(Ξψ,ϱ,γ)}}, (2.12)

    for all ψ,ϱ, and βFfb. Then Ξ has a fixed point.

    Proof. In the same way as Theorem 2.1, we have

    Θfb(ψ1,ψ2,γ)HFΘfb(Ξψ0,Ξψ1,γ),γ>0.

    By induction we have ψr+1Ξψr satisfying

    HFΘfb(ψr,ψr+1,γ)=Fθ(Ξψr1,Ξψr,γ)Γ1(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))Γ2(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))) (2.13)

    Now,

    Γ1(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))=min{HFΘfb(Ξψr1,Ξψr,γβ(Θfb(ψr1,ψr,γ))),HFΘfb(ψr1,Ξψr1,γβ(Θfb(ψr1,ψr,γ))),HFΘfb(ψr,Ξψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}.Γ1(ψr1,ψr1,γβ(Θfb(ψr1,ψr,γ)))=min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}. (2.14)
    Γ2(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))=max{HFΘfb(ψr1,Ξψr,γβ(Θfb(ψr1,ψr,γ))),HFΘfb(Ξψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=max{Θfb(ψr1,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr,ψr,γβ(Θfb(ψr1,ψr,γ)))}=max{Θfb(ψr1,ψr+1,γβ(Θfb(ψr1,ψr,γ))),1}.         Γ2(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))=1. (2.15)

    Using (2.14) and (2.15) in (2.13) we have

    Θfb(ψr,ψr+1,γ)min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}1,Θfb(ψr,ψr+1,γ)min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}. (2.16)

    If

    min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),

    then (2.16) implies

    Θfb(ψr,ψr+1,γ)Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))

    Then the proof follows by Lemma 1.2

    If

    min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),

    then from (2.16), we have

    Θfb(ψr,ψr+1,γ)Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))Θfb(ψ0,ψ1,γβ(Θfb(ψr1,ψr,γ))β(Θfb(ψr2,ψr1,γ))β(Θfb(ψ0,ψ1,γ))).

    The remaining proof is similar as in Theorem 2.1.

    Remark 2.5. If we set ^C0()= the map Ξ becomes a singlevalued and we get Theorem 3.11 of [32]. Again as stated in Remark 2.1, the corresponding fixed point will be unique.

    Theorem 2.7. Let (,Θfb,) be a G-complete Fbms with b1 and (^C0(),HFΘfb,) be a Hfbms. Let Ξ:ˆC0() be a Mvp satisfying

    HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)Γ1(ψ,ϱ,γ)Γ2(ψ,ϱ,γ)Γ3(ψ,ϱ,γ), (2.17)

    where

    {Γ1(ψ,ϱ,γ)=min{HFΘfb(Ξψ,Ξϱ,γ).Θfb(ψ,ϱ,γ),HFΘfb(ψ,Ξψ,γ).HFΘfb(ϱ,Ξϱ,γ)}Γ2(ψ,ϱ,γ)=max{HFΘfb(ψ,Ξψ,γ).HFΘfb(ψ,Ξϱ,γ),HFΘfb(ϱ,Ξψ,γ))2}Γ3(ψ,ϱ,γ)=max{HFΘfb(ψ,Ξψ,γ),HFΘfb(ϱ,Ξϱ,γ)}}, (2.18)

    for all ψ,ϱ, and βFfb. Then Ξ has a fixed point.

    Proof. In the same way as Theorem 2.1, we have

    Θfb(ψ1,ψ2,γ)HFΘfb(Ξψ0,Ξψ1,γ),γ>0.

    By induction we have ψr+1Ξψr satisfying

    Θfb(ψr,ψr+1,γ)=HFΘfb(Ξψr1,Ξψr,γ)Γ1(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))Γ2(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))Γ3(ψ,ϱ,γβ(Θfb(ψr1,ψr,γ))). (2.19)
    Γ1(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))=min{HFΘfb(Ξψr1,Ξψr,γβ(Θfb(ψr1,ψr,γ))).Fθ(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),HFΘfb(ψr1,Ξψr1,γβ(Θfb(ψr1,ψr,γ))).HFΘfb(ψr,Ξψr,γβ(Θfb(ψr1,ψr,γ)))}=min{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))).Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))).Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))).Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))). (2.20)

    Similarly,

    Γ2(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))=max{HFΘfb(ψr1,Ξψr1,γβ(Θfb(ψr1,ψr,γ))),HFΘfb(ψr1,Ξψr,γβ(Θfb(ψr1,ψr,γ))),(HFΘfb(ψr,Ξψr1,γβ(Θfb(ψr1,ψr,γ))))2}=max{Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr+1,γβ(Θfb(ψr1,ψr,γ))),(Θfb(ψr,ψr,γβ(Θfb(ψr1,ψr,γ))))2}=max{Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr+1,γβ(Θfb(ψr1,ψr,γ))),1}.

    It follows that

    Γ2(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))=1. (2.21)
    Γ3(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))=max{HFΘfb(ψr1,Ξψr1,γβ(Θfb(ψr1,ψr,γ))),HFΘfb(ψr,Ξψr,γβ(Θfb(ψr1,ψr,γ)))}=max{Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))}. (2.22)

    Using (2.20), (2.21) and (2.22) in (2.19), we have

    Θfb(ψr,ψr+1,t)Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))).Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))max{Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ)))}. (2.23)

    If

    max{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))),

    then (2.23) implies

    Θfb(ψr,ψr+1,γ)Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))).

    It is obvious by Lemma 1.2.

    If

    max{Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr1,ψr,γ))),

    then from (2.23), we have

    Θfb(ψr,ψr+1,γ)Θfb(ψr1,ψr,γβ(Θfb(ψr1,ψr,γ))).

    Continuing in this way, we will get

    Θfb(ψr,ψr+1,t)Θfb(ψr1,ψr,γβ(Fθ(ψr1,ψr,γ)))Θfb(ψ0,ψ1,γβ(Θfb(ψr1,ψr,γ)).β(Θfb(ψn2,ψr1,γ))β(Θfb(ψ0,ψ1,γ))).

    The rest of the proof follows in the same way as in Theorem 2.1.

    Remark 2.6. By setting ^C0()=, the mapping Ξ:^C0() becomes a self (singlevalued) mapping and we get Theorem 3.13 of [32].

    An application of Theorem 2.1 is presented here. Recall that the space of all continuous realvalued functions on [0,1] is denoted by C([0,1],R). Now set =C([0,1],R) and define the G-complete Fbm on by

    Θfb(ψ,ϱ,γ)=esupu[0,1]|ψ(u)ϱ(u)|2γ,γ>0andψ,ϱ.

    Consider

    ψ(u)u0G(u,v,ψ(v))dv+h(u)for allu,v[0,1],whereash,ψC([0,1],R). (3.1)

    Here G:[0,1]×[0,1]×RPcv(R) is multivalued function and Pcv(R) represents the collections of convex and compact subsets of R. Moreover, for each ψ in C([0,1],R) the operator G(,,ψ) is lower semi-continuous.

    For the integral inclusion given in (3.1), define a multivalued operator S:^C0() by

    Sψ(u)={w:wu0G(u,v,ψ(v))dv+h(u),u[0,1]}.

    Now for arbitrary ψ(C([0,1],R), denote Gψ(u,v)=G(u,v,ψ(v)) where u,v[0,1]. For the multivalued map Gψ:[0,1]×[0,1]Pcv(R), by Michael selection theorem [34], there exists a continuous selection gψ:[0,1]×[0,1]R such that gψ(u,v)Gψ(u,v) for each u,v[0,1]. It follows that

    u0gψ(u,v)dv+h(u)Sψ(u).

    Since gψ is continuous on [0,1]×[0,1] and h is continuous on [0,1], therefore both gψ and h are bounded realvalued functions. It follows that, the operator Sψ is nonempty and Sψ^C0().

    With the above setting, the upcoming outcome shows the existence of a solution of the integral inclusion (3.1).

    Theorem 3.1. Let =C([0,1],R) and define the multivalued operator S:^C0() by

    Sψ(u)={w:wu0G(u,v,ψ(v))dv+h(u),u[0,1]},

    where h:[0,1]R is continuous and the map G:[0,1]×[0,1]×RPcv(R) is defined in such a way that for every ψC([0,1],R), the operator G(,,ψ) is lower semi-continuous. Assume further that the given terms are satisfied:

    (i) There exists a continuous mapping f:[0,1]×[0,1][0,) such that

    HFΘfb(G(u,v,ψ(v))G(u,v,ϱ(v))f2(u,v)|ψ(v)ϱ(v)|2,

    for each ψ,ϱ and u,v[0,1].

    (ii) There exists βFΘ2, such that

    supu[0,1]u0f2(u,v)dvβ(Θfb(ψ,ϱ,γ)).

    Then (3.1) has a solution in .

    Proof. We will show that the operator S satisfies the conditions of Theorem 2.1. In particular we prove (2.1) as follows:

    Let ψ,ϱ be such that qSψ. As stated earlier, by selection theorem there is gψ(u,v)Gψ(u,v)=G(u,v,ψ(v)) for u,v[0,1] such that

    q(u)=u0gψ(u,v)dv+h(u),u[0,1].

    Further, the condition (ⅰ) ensures that there is some g(u,v)Gϱ(u,v) such that

    |gψ(u,v)g(u,v)f2(u,v)|ψ(v)ϱ(v)|2,u,v[0,1].

    Now consider the multivalued operator T defined as follows:

    T(u,v)=Gϱ(u,v){wR:|gψ(u,v)w|f2(u,v)|ψ(v)ϱ(v)|2}.

    Since, by construction, T is lower semi-continuous, it follows again by the selection theorem that there is continuous function gϱ(u,v):[0,1]×[0,1]R such that for each u,v[0,1], gϱ(u,v)T(u,v).

    Thus, we have

    r(u)=u0gϱ(u,v)dv+h(u)u0G(u,v,ϱ(v))dv+h(u),u[0,1].

    Therefore, for each u[0,1] we get

    esupt[0,1]|q(u)r(u))|2β(Θfb(ψ,ϱ,γ))γesupu[0,1]u0|gψ(u,v)gϱ(u,v)|2dvβ(Θfb(ψ,ϱ,γ))γesupu[0,1]u0f2(u,v)|ψ(v)ϱ(v)|2dvβ(Θfb(ψ,ϱ,γ))γe|ψ(v)ϱ(v)|2supu[0,1]u0f2(u,v)dvβ(Θfb(ψ,ϱ,γ))γeβ(Θfb(ψ,ϱ,γ))|ψ(v)ϱ(v)|2β(Θfb(ψ,ϱ,γ))γ=e|ψ(v)ϱ(v)|2γesupv[0,1]|ψ(v)ϱ(v)|2γ=Θfb(ψ,ϱ,γ).

    This implies that,

    Θfb(q,r,β(Θfb(ψ,ϱ,γ))γ)Θfb(ψ,ϱ,γ).

    Interchanging the roles of ψ and ϱ, we get

    HFΘfb(Sψ,Sϱ,β(Θfb(ψ,ϱ,γ))γ)Θfb(ψ,ϱ,γ).

    Hence, by Theorem 2.1, the operator S has a fixed point which in turn proves the existence of a solution of integral inclusion (3.1).

    In the present work, in the setting of a Hausdorff Fbms, some fixed fixed point results for multivalued mappings are established. The main result, that is, Theorem 2.1 shows that a multivalued mapping satisfying Geraghty type contractions on G -complete Hfbms has a fixed point. Example 2.1 illustrates the main result. Some other interesting fixed point theorems are also proved for the multivalued mappings satisfying certain contraction condition on G -complete Hfbms. The results proved in [30,31,32] turn out to be special cases of the results established in this work. For the significance of our results, an application is presented to prove the existence of solution of an integral inclusion.

    The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work grant code: 22UQU4331214DSR02

    The authors declare that they have no conflict of interest.



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