Research article

Convergence results for cyclic-orbital contraction in a more generalized setting with application

  • Received: 02 March 2024 Revised: 22 April 2024 Accepted: 24 April 2024 Published: 29 April 2024
  • MSC : 34B15, 41A65, 46B20, 47H10

  • In the realm of double-controlled metric-type spaces, we investigated obtaining fixed points using the application of cyclic orbital contractive conditions. Diverging from conventional approaches utilized in standard metric spaces, our technique took a unique route due to the unique features of our structure. We demonstrated the significance of our outcomes through exemplary cases, clarifying the breadth of our results through comprehensive investigations. Significantly, our work not only improved and broadened earlier findings in the literature, but also offered unique notions that were discussed in our explanatory notes. Towards the end of our inquiry, we used insights obtained from previous discoveries to develop a second-order differential equation. This equation was an effective tool for dealing with the second class of Fredholm integral problems. In conclusion, this investigation extended our examination of double-controlled metric type spaces by providing new insights on fixed point theory, expanding on prior debates and building a substantial road towards solving a class of integral equations.

    Citation: Haroon Ahmad, Sana Shahab, Wael F. M. Mobarak, Ashit Kumar Dutta, Yasser M. Abolelmagd, Zaffar Ahmed Shaikh, Mohd Anjum. Convergence results for cyclic-orbital contraction in a more generalized setting with application[J]. AIMS Mathematics, 2024, 9(6): 15543-15558. doi: 10.3934/math.2024751

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  • In the realm of double-controlled metric-type spaces, we investigated obtaining fixed points using the application of cyclic orbital contractive conditions. Diverging from conventional approaches utilized in standard metric spaces, our technique took a unique route due to the unique features of our structure. We demonstrated the significance of our outcomes through exemplary cases, clarifying the breadth of our results through comprehensive investigations. Significantly, our work not only improved and broadened earlier findings in the literature, but also offered unique notions that were discussed in our explanatory notes. Towards the end of our inquiry, we used insights obtained from previous discoveries to develop a second-order differential equation. This equation was an effective tool for dealing with the second class of Fredholm integral problems. In conclusion, this investigation extended our examination of double-controlled metric type spaces by providing new insights on fixed point theory, expanding on prior debates and building a substantial road towards solving a class of integral equations.



    It is noted from the literature that many topologies with important applications in mathematics have been defined using new mathematical structures, for example, filter [20], ideal [19] and partially ordered [25]. The grill is one of the classic structures presented by Choquet [11] around 1947, which is a good tool for studying topological notions and has a wide range of important applications in topological theory (see [9,10,30]). Some structures based on both "filter" and "grill" are defined and investigated, as in [22,23], while others are based on "partially ordered set" as [7]. A grill's associated topology has been defined and investigated by Roy and Mukherjee [26]. They presented the basic topological concepts in terms of this new structure and studied the fundamental theorems.

    Moreover, some operators have been defined and their properties studied with clarity to determine whether they satisfy Kuratowskis closure axioms (see [27,28,29]). The notion of continuity occupies a great place of study in the wide range of literature [18,24,32].

    Császár [12] established the theory of generalizations in 1997 by presenting the notion of generalized open sets. Furthermore, he introduced the generalized topological spaces (breifly GTS) around 2002 [13], which differ from topological spaces in the condition of intersection. In more specific terms, they are closed under arbitrary unions.

    In [13,14,31,33], a lot of its characteristics are studied in detail, such as the generalized interior of a set A, the generalized closure of a set A, the generalized neighbourhood, separation axioms, and so on. The notion of continuity gets a lot of attention under the new appellations "generalized continuity", "θ-continuity", "γ-continuity", etc. (see [13,16]). Various kinds of generalized topology appeared and were studied, such as "extremely disconnected generalized topology", "δ- and θ-modifications generalized topology", etc. (see [15,16,33]).

    The dual structure of a grill was presented around 2022. It was called "primal". This new structure has a lot of interesting properties, studied in detail by Acharjee et al. [1].

    Furthermore, a new topology has been defined by using this new structure, called "primal topology" and some of its topological properties have been investigated. Some articles have appeared that study primal topological spaces in different fields, such as vector spaces induced by metric or soft environments (see [8,21]).

    In 1990, [17] different kinds of topological operators were defined based on the structure "ideal". Then, other studies followed the same approach. Al-Omari and Noiri studied some types of operators based on the grill notion (see [4,5,6]). Moreover, [1,3] present types of operators have been defined by using the notion of primal with deep studies of their various properties.

    In this paper, our aim is to define a new structure based on both the notions of "primal" and "generalized". This new space will be called "generalized primal topological spaces".

    Also, the authors will introduce some operators based on the notion of generalized primal neighbourhood such as cl(A) and Φ(A), with a study of their fundamental characteristics and give an answer to the important question, "do they satisfy Kuratowskis closure axioms?" A lot of nice properties with examples will be present.

    Thereafter, we will define a new form of generalized primal topological spaces induced by the previous operators, with a comparison between the structures.

    We will go over the fundamental definitions and conclusions in this section. We represent the power set of Xϕ by 2X throughout our work.

    Definition 2.1. [11] Suppose that Xϕ. A collection GX of 2X is named a grill if the next hold:

    (i) ϕG.

    (ii) For two subsets A and B of X, with AB, we have BG, if AG.

    (iii) For two subsets A and B of X, AG or BG, if ABG.

    Definition 2.2. [12] Let Xϕ. A family g of 2X is named a generalized topology (GT) on X if the next hold:

    (i) ϕg.

    (ii) The countable union of Gig, for iIϕ is belongs to g.

    A generalized topological space is the pair (X,g).

    Remark 2.1. According to [12], we note that:

    (i) This space's elements are identified as g-open and their own complements are identified as g-closed.

    (ii) Cg(X) represents the set of all g-closed sets of X.

    (iii) cgA represents the closure of AX, which is described as the intersection of all g-closed sets that contain A.

    (iv) igA represents the interior of AX, which is described as the union of all g-open sets that contained in A.

    (v) cg(cgA)=cgA, ig(igA)=igA and igAAcgA.

    (vi) A is g-open if A=igA, A is g-closed if A=cgA and cgA=X(ig(XA)).

    Definition 2.3. [12] In a generalized topological space (X,g). Consider an operator ψ: X22X satisfies xU for Uψ(x). Then, Uψ(x) is known as a generalized neighbourhood of a point x in the space X, or briefly (GN).

    Furthermore, ψ is known as a generalized neighbourhood system on a space X.

    Definition 2.4. [12] The collection of all generalized neighbourhood systems on X is denoted by Ψ(X).

    Definition 2.5. [1] Suppose that Xϕ. A family P of 2X is named a primal on X, when the next hold:

    (i) XP.

    (ii) For A,BX with BA. If AP, then BP.

    (iii) For A,BX, then AP or BP, whenever ABP.

    Definition 2.6. A primal topological space is defined as the pair (X,τ) with a primal P on X. Moreover, it is represented by (X,τ,P).

    Corollary 2.1. [1] Suppose that Xϕ. A family P of 2X is named a primal on X, if and only if the next hold:

    (i) XP.

    (ii) For A,BX with BA. If BP, then AP.

    (iii) For A,BX, then ABP, whenever AP and BP.

    Theorem 2.1. [1] If the collection G forms a grill on X, then the set {A:AcG} is a primal on X.

    Theorem 2.2. [1] The union of two primals on X gives a primal on X.

    Remark 2.2. According to [1], the intersection of two primals on X need not be a primal on X.

    In this part, we establish a new category of generalized topology called "generalized primal topology" and define it as:

    Definition 3.1. A generalized primal topology is a generalized topology g with a primal P on X. The triples (X,g,P) denote the generalized primal topological space.

    This space's elements are identified as (g,P)-open sets, and their own complement known as (g,P)-closed sets.

    C(g,P)(X) denoted the set of all (g,P)-closed sets on X, and cl(g,P)(A) represents the closure of AX, which is described as the intersection of all (g,P)-closed sets that contain A.

    Example 3.1. Consider X={a,b,c}, g={ϕ,{a,b},{a,c},X} and the primal set P={ϕ,{a}}. Hence, (X,g,P) is a generalized primal topological space X.

    Definition 3.2. In a generalized primal topological space (X,g,P). Consider an operator ψ: X22X satisfies xU for Uψ(x). Then, Uψ(x) is known as a generalized primal neighbourhood for a point x in the space X.

    Remark 3.1. ψ is known as a generalized primal neighbourhood system on a space X. The collection of all generalized primal neighbourhood systems on X is denoted by Ψ(X).

    Definition 3.3 In a generalized primal topological space (X,g,P). Consider an operator (.): 2X2X defined by

    A(X,g,P)={xX:AcUcP, Uψ(x)},

    where U is a generalized primal neighbourhood of xX.

    Remark 3.2. In a generalized primal topological space (X,g,P):

    (i) We cannot say that AA, or AA the following example shows that.

    (ii) With an additional condition, the relationship AA is always true, which is proved in the following theorems.

    Example 3.2. Consider

    X={a,b,c},   g={ϕ,{a},{b},{a,b},X}

    and

    P={ϕ,{a},{b},{c},{a,b}}.

    Let A={a,b}. Then, A={c}. Therefore, AA and AA.

    Theorem 3.1. Consider (X,g,P) as a generalized primal topological space. Hence, AA, whenever Ac is (g,P)-open.

    Proof. Let Ac be (g,P)-open. Let xA with xA. Then, Ac is an open generalized primal neighbourhood of x briefly (Acψ(x)) and AcUcP, for all Uψ(x). Hence,

    Ac(Ac)c=XP,

    which is a contradicts with the fact that XP. Therefore, AA.

    Theorem 3.2. Consider (X,g,P) as a generalized primal topological space. Hence, we have:

    (i) ϕ=ϕ.

    (ii) A is (g,P)-closed, for AX, i.e., cl(g,P)(A)=A.

    (iii) For AX, (A)A.

    (iv) AB, whenever, AB for A,BX.

    (v) For A,BX, AB=(AB).

    Proof. (i) It is clear that ϕc=X, but XP. Thus, we are done.

    (ii) Since Acl(g,P)(A) always true. We need to show that cl(g,P)(A)A. Consider xcl(g,P)(A) and Uψ(x). Thus,

    UAϕyX

    satisfies yUyA. Then,  Vψ(y) we have

    VcAcPUcAcP.

    Therefore, xA. Hence, cl(g,P)(A)A.

    (iii) From Theorem 3.1 and (ii) we can conclusion that (A)A, that is A is (g,P)-closed in X implies (A)c is (g,P)-open.

    (iv) Suppose that AB and xA. Then, for all Uψ(x) we have AcUcP. Hence, BcUcP. Therefore, xB.

    (v) From (iv) we have A(AB) and B(AB). Then,

    AB(AB).

    To prove the opposite direction, consider xAB implies xA and xB. Then, there exists U;Vψ(x) satisfies AcUcP and BcVcP. Consider a set W=UV, then Wψ(x) such that AcWcP and BcWcP. Then,

    (AB)cWc=(AcWc)(BcWc)P.

    Hence, x(AB). Therefore,

    (AB)AB.

    (vi) The proof follows in the same way.

    Remark 3.3. In a generalized primal topological space (X,g,P), the opposite inclusion of the Theorem 3.2 (vi) is not always true; the following example shows that.

    Example 3.3. Consider

    X={a,b,c},  g={ϕ,{a},{b},{a,b},X}

    and

    P={ϕ,{a},{b},{c},{a,b}}.

    Let A={a,b} and B={c}. Then,

    A={c}=B.

    Thus, AB={c} and (AB)=ϕ. Therefore,

    AB(AB).

    Theorem 3.3. Consider (X,g,P) as a generalized primal topological space. Hence, for two subsets A and B of X, AB(AB), whenever A is (g,P)-open.

    Proof. If A is (g,P)-open and xAB, then xA and xB implies for all

    Uψ(x):BcUcP.

    Thus,

    (AB)cUc=Bc(AU)cP.

    Hence, x belongs to (AB). Therefore, we are done.

    Theorem 3.4. Consider (X,g,P) as a generalized primal topological space. Suppose C(g,P)(X){X} is a primal on X. Then, for all (g,P)-open sets U, we have UU.

    Proof. If U=ϕ, then by Theorem 3.2 (i) we have U=ϕ, hence UU. By hypotheses, if C(g,P)(X){X} is primal, then X=X since Xc=ϕ. From Theorem 3.3, we get

    U=UX(UX)=U,

    (g,P)-open set U. Therefore, we are done.

    Lemma 3.1. Consider (X,g,P) as a generalized primal topological space. Hence, A=ϕ, whenever AcX is not a primal.

    Proof. Let xA. Then, AcUcP for all Uψ(x). Since Ac is not primal, we get a contradiction. Therefore, A=ϕ.

    Theorem 3.5. Consider (X,g,P) as a generalized primal topological space. Then,

    AB=(AB)B,

    for two subsets A and B of X.

    Proof. We can represent A as

    A=[(AB)(AB)].

    From Theorem 3.2 (v), we get

    A=(AB)(AB)(AB)(AB).

    This implies

    A(AB)B.

    Thus,

    AB(AB)B.

    On the other hand, since (AB)A, (AB)A from Theorem 3.2 (iv). Hence,

    (AB)BAB.

    Therefore, we are done.

    Corollary 3.1. Consider (X,g,P) as a generalized primal topological space. If A,BX and Bc is not a primal, then

    (AB)=A=(AB).

    Proof. From Lemma 3.1 we get B=ϕ. Thus,

    (AB)=AB=Aϕ=A.

    Also, by Theorem 3.5, (AB)=A.

    Definition 3.4. In a generalized primal topological space (X,g,P). Consider an operator cl: 2X2X defined by cl(A)=AA, for a subset A of X.

    Remark 3.4. It is shown by the following theorem that the map cl is a Kuratowskis closure operator.

    Theorem 3.6. Consider (X,g,P) as a generalized primal topological space. Hence, we have:

    (i) cl(ϕ)=ϕ.

    (ii) For a subset A of X, Acl(A).

    (iii) cl(cl(A))=cl(A).

    (iv) cl(A)cl(B), whenever AB for A,BX.

    (v) For A,BX, cl(A)cl(B)=cl(AB).

    Proof. (i) Since ϕ=ϕ,

    cl(ϕ)=ϕϕ=ϕ.

    (ii) From the definition cl(A)=AA, we have Acl(A).

    (iii) From (ii) Acl(A) implies

    cl(A)cl(cl(A)).

    Conversely, since (A)A by Theorem 3.2 (iii). Then,

    cl(cl(A))=cl(A)(cl(A)).

    Thus,

    cl(cl(A))=cl(A)(AA).

    Hence,

    cl(cl(A))=cl(A)A(A)cl(A)AA=cl(A).

    Therefore, we are done.

    (iv) If AB, then from Theorem 3.2 (iv) we get AB. Hence,

    AABB

    implies cl(A)cl(B).

    (v) Since

    (AB)=AB

    is holded and cl(A)=AA. Then,

    cl(A)cl(B)=(AA)(BB).

    Thus,

    cl(A)cl(B)=(AB)(AB)=cl(AB).

    Throughout this section, we will define a new kind of generalized primal topology on X that is stronger than the previous structure that is described in Definition 3.1.

    Definition 4.1. In a generalized primal topological space (X,g,P) the collection

    g={AX:cl(Ac)=Ac}

    is a generalized primal topology on X induced by an operator cl.

    Proposition 4.1. Consider (X,g,P) as a generalized primal topological space. Hence, g is weaker than g.

    Proof. Suppose that Ag. Then, Ac is (g,P)-closed in X implies (Ac)Ac. Hence

    cl(Ac)=Ac(Ac)Ac.

    But Accl(Ac). Then, cl(Ac)=Ac. Therefore, Ag.

    Theorem 4.1. Consider (X,g,P) and (X,g,Q) as two generalized primal topological spaces. If PQ, then g induced by primal P is finer than g induced by primal Q.

    Proof. Let Ag induced by primal Q. Then, Ac(Ac) with respect to Q equal to Ac implies (Ac) with respect to QAc. Let xAc. Then, x(Ac) with respect to Q implies Uψ(x) such that

    Uc(Ac)c=UcAQ,

    but PQ. Then, UcAP. Hence, x(Ac) with respect to P. Thus, (Ac) with respect to P Ac implies

    cl(Ac)=Ac(Ac)

    with respect to P equal to Ac. Therefore, Ag induced by P.

    Theorem 4.2. Consider (X,g,P) as a generalized primal topological space. Hence, Ag for all xA, Uψ(x) satisfies (UcA)P.

    Proof. Let Ag. Then,

    Agcl(Ac)=Ac.

    Thus,

    AgAc(Ac)=Ac.

    However, (Ac)Ac. Then,

    AgA((Ac))c.

    Thus,

    Agx(Ac), xA.

    Therefore,

    AgUc(Ac)c=UcAP,

    for some Uψ(x).

    Theorem 4.3. For a generalized primal topological space (X,g,P) the collection

    BP={GP:Gg and PP}

    is a base for g on X.

    Proof. Consider BBP. Then, there exists Gg and PP such that B=GP. Since gg, then Gg. Since cl(Pc)=Pc, then Pg. Hence, Bg. Consequently, BPg. Let Ag and xA. Hence, by Theorem 4.2, Uψ(x) satisfies UcAP. Consider

    B=U(UcA).

    Then, BBP such that xBA.

    Definition 4.2. In a generalized primal topological space (X,g,P). Consider an operator Φ: 2X2X identify as

    Φ(A)={xX:  Uψ(x) satisfay (UA)cP},

    for AX.

    The following theorem defines the relationship between the previously investigated maps (.) and (Φ).

    Theorem 4.4. Consider (X,g,P) as a generalized primal topological space. Hence,

    Φ(A)=X(XA),

    for a subset A of X.

    Proof. Suppose that xΦ(A), then Uψ(x) such that (UA)cP. Hence,

    (U(XA))c=(UcA)P.

    Thus, x(XA) implies

    xX(XA).

    Hence,

    Φ(A)X(XA).

    Conversely, suppose that

    xX(XA)

    implies x(XA). Then, Uψ(x) we have

    Uc(XA)c=(UA)cP.

    Then, xΦ(A). Hence,

    X(XA)Φ(A).

    Therefore, we are done.

    Corollary 4.1. Consider (X,g,P) as a generalized primal topological space. Hence, Φ(A) is (g,P)-open.

    Proof. The proof follows from Theorem 3.2 (ii) and Theorem 4.4.

    Remark 4.1. Unlike the map cl, the map Φ does not satisfy the four Kuratowskis closure operator conditions, and this is proven in the following theory.

    Theorem 4.5. Consider (X,g,P) as a generalized primal topological space. Hence, the next hold:

    (i) Φ(A)Φ(B), whenever AB.

    (ii) If Ug, then UΦ(U).

    (iii) Φ(A)Φ(Φ(A)).

    (iv) Φ(A)=Φ(Φ(A))(XA)=((XA)).

    Proof. (i) Let xΦ(A). Then, (UA)cP, but AB. Then, (UB)cP. Hence, xΦ(B). Therefore, Φ(A)Φ(B).

    (ii) If Ug, then

    cl(XU)=XU

    implies

    (XU)(XU)=XU.

    Then, (XU) subset of XU. Hence,

    UX(XU)=Φ(U).

    (iii) Since Φ(A) is (g,P)-open and gg,

    Φ(A)gΦ(A)Φ(Φ(A))

    from (ii).

    (iv) Let Φ(A)=Φ(Φ(A)). Then

    Φ(Φ(A))=Φ(X(XA))=X(X(X(XA))).

    Hence, we have

    Φ(Φ(A))=X((XA)).

    Since Φ(A)=Φ(Φ(A)),

    (XA)=((XA)).

    Conversely, let

    (XA)=((XA)).

    From (iii) we get Φ(A)Φ(Φ(A)). Let xΦ(Φ(A)). Then, x belongs to X((XA)) implies x belongs to X(XA). Then, xΦ(A). Hence, Φ(Φ(A))Φ(A). Therefore, Φ(A)=Φ(Φ(A)).

    Extra properties of the function Φ will be studied in the next theorem.

    Theorem 4.6. Consider (X,g,P) as a generalized primal topological space. Hence, the next hold:

    (i) Φ(AB)=Φ(A)Φ(B), for A,BX.

    (ii) For AX, Φ(A)=XX, whenever AcP.

    Proof. (i) It is clear that Φ(AB)Φ(A) and Φ(AB)Φ(B). Then,

    Φ(AB)Φ(A)Φ(B).

    Conversely, suppose that xΦ(A)Φ(B). Implies there exists U;Vψ(x) satisfies (UA)cP and (VB)cP. Consider

    W=UVψ(x)

    implies (WA)cP and (WB)cP. Hence,

    (W(AB))c=(WA)c(WB)cP.

    Then, x belongs to Φ(AB). Hence,

    Φ(A)Φ(B)Φ(AB).

    (ii) By using Corollary 3.1, we have X=(XA). Hence,

    Φ(A)=X(XA)=XX.

    Theorem 4.7. For a generalized primal topological space (X,g,P). Consider the collection

    σ={AX:AΦ(A)},

    thus σ is a generalized primal topology induced by an operator Φ. Moreover, σ=g.

    Proof. It is clear that ϕΦ(ϕ). Then, ϕσ. If

    {Aα:αΛ}σ,

    then for each αΛ we have

    AαΦ(Aα)Φ(αΛAα).

    Hence,

    αΛAαΦ(αΛAα).

    Thus, σ is a generalized primal topology on X induced by Φ.

    Let Ug and xU. Then, from Theorem 4.3, there exists Vψ(x) and PP satisfy

    xVPU.

    Since P(VU)c, then (VU)cP. Hence, xΦ(U) implies UΦ(U). Therefore, gσ. On the other hand let Aσ, then AΦ(A) implies

    AX(XA)

    and

    (XA)XA.

    Thus, XA is (g,P)-closed. Then, Ag. Therefore, σg.

    Coinciding with the great spread of many literatures in various fields important mathematical structures appeared in the theory of topology coinciding with this scientific revolution. For example, the concept of generalized topology appeared, which is based on the concept of "generalized open set" and this space was more generalized than the topological space as the intersection condition was neglected. This space has been extensively studied; much literature has been written about it, and many properties and theories have been studied about it.

    In this paper, we have made a new contribution to the field of generalized topology by studying the concept of primal generalized topology, which has a lot of interesting properties. The results obtained in this paper are preliminary. Future research could give more insights by exploring further properties of generalized primal topology. This work opens up the door for possible contributions to this trend by combining primal structures with generalized structures in the theory of generalized topology. If possible, we are looking forward to connecting this notion with some ideas like supra-topology and infra-topology.

    The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

    The authors declare no conflicts of interest.



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