Loading [MathJax]/jax/output/SVG/jax.js
Research article

Applying fixed point techniques to solve fractional differential inclusions under new boundary conditions

  • Received: 02 February 2024 Revised: 16 April 2024 Accepted: 19 April 2024 Published: 29 April 2024
  • MSC : 34A05, 34B15, 47H10

  • Many scholars have lately explored fractional-order boundary value issues with a variety of conditions, including classical, nonlocal, multipoint, periodic/anti-periodic, fractional-order, and integral boundary conditions. In this manuscript, the existence and uniqueness of solutions to sequential fractional differential inclusions via a novel set of nonlocal boundary conditions were investigated. The existence results were presented under a new class of nonlocal boundary conditions, Carathéodory functions, and Lipschitz mappings. Further, fixed-point techniques have been applied to study the existence of results under convex and non-convex multi-valued mappings. Ultimately, to support our findings, we analyzed an illustrative example.

    Citation: Murugesan Manigandan, Kannan Manikandan, Hasanen A. Hammad, Manuel De la Sen. Applying fixed point techniques to solve fractional differential inclusions under new boundary conditions[J]. AIMS Mathematics, 2024, 9(6): 15505-15542. doi: 10.3934/math.2024750

    Related Papers:

    [1] Fahad Alsharari, Ahmed O. M. Abubaker, Islam M. Taha . On r-fuzzy soft γ-open sets and fuzzy soft γ-continuous functions with some applications. AIMS Mathematics, 2025, 10(3): 5285-5306. doi: 10.3934/math.2025244
    [2] Dina Abuzaid, Samer Al-Ghour . Supra soft Omega-open sets and supra soft Omega-regularity. AIMS Mathematics, 2025, 10(3): 6636-6651. doi: 10.3934/math.2025303
    [3] Rui Gao, Jianrong Wu . Filter with its applications in fuzzy soft topological spaces. AIMS Mathematics, 2021, 6(3): 2359-2368. doi: 10.3934/math.2021143
    [4] Arife Atay . Disjoint union of fuzzy soft topological spaces. AIMS Mathematics, 2023, 8(5): 10547-10557. doi: 10.3934/math.2023535
    [5] Ibtesam Alshammari, Islam M. Taha . On fuzzy soft β-continuity and β-irresoluteness: some new results. AIMS Mathematics, 2024, 9(5): 11304-11319. doi: 10.3934/math.2024554
    [6] Orhan Göçür . Amply soft set and its topologies: AS and PAS topologies. AIMS Mathematics, 2021, 6(4): 3121-3141. doi: 10.3934/math.2021189
    [7] Rehab Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, Ismail Ibedou . Soft closure spaces via soft ideals. AIMS Mathematics, 2024, 9(3): 6379-6410. doi: 10.3934/math.2024311
    [8] Tareq M. Al-shami, Zanyar A. Ameen, A. A. Azzam, Mohammed E. El-Shafei . Soft separation axioms via soft topological operators. AIMS Mathematics, 2022, 7(8): 15107-15119. doi: 10.3934/math.2022828
    [9] G. Şenel, J. I. Baek, S. H. Han, M. Cheong, K. Hur . Compactness and axioms of countability in soft hyperspaces. AIMS Mathematics, 2025, 10(1): 72-96. doi: 10.3934/math.2025005
    [10] Dina Abuzaid, Samer Al Ghour . Three new soft separation axioms in soft topological spaces. AIMS Mathematics, 2024, 9(2): 4632-4648. doi: 10.3934/math.2024223
  • Many scholars have lately explored fractional-order boundary value issues with a variety of conditions, including classical, nonlocal, multipoint, periodic/anti-periodic, fractional-order, and integral boundary conditions. In this manuscript, the existence and uniqueness of solutions to sequential fractional differential inclusions via a novel set of nonlocal boundary conditions were investigated. The existence results were presented under a new class of nonlocal boundary conditions, Carathéodory functions, and Lipschitz mappings. Further, fixed-point techniques have been applied to study the existence of results under convex and non-convex multi-valued mappings. Ultimately, to support our findings, we analyzed an illustrative example.



    Zadeh [50] was the first to come up with the unprecedented theory of fuzzy set (F-set) for dealing with some types of uncertainties where conventional tools fail. This theory brought a grand paradigmatic change in mathematics and offered a convenient framework to model a huge number of empirical problems. On the other hand, this theory has its inherent difficulties, which are possibly attributed to the inadequacy of the parameterization tool and pre-requirement of membership function, as pointed out by Molodtsov in his pioneering work [41]. He introduced the concept of soft sets (S-sets) as a remarkable mathematical tool for coping with vagueness that is free from the aforementioned difficulties. Then, the S-set theory has been applied in many fields by many authors [16,35,36]. One of these fields that attracted a lot of attention is the abstract topological structures that were displayed by Shabir-Naz [48] and Çağman et al. [20]. Some divergences between classical and soft topologies were illuminated in [7].

    Over time, complicated issues have appeared that need combining parameterization of S-sets with the membership degree of F-sets. To tackle such dilemmas, Maji et al. [37] put forward a new paradigm known as a fuzzy soft set (FS-set) and demonstrated how this paradigm is applied [38]. Since then, the FS-set theory and its applications have been studied by several intellectuals [5,19,24,25]. To cover more situations and expand the range of applications, the concept of FS-set was generalized to (a,b)-Fuzzy soft sets by [11]. Kharal and Ahmad [34] defined the concept of mappings of FS-classes. Subsequently, the study of topological structure over the family of FS-sets was started by Tanay-Kandemir [49]. Mukherjee et al.[42] introduced the notions of FSδ-open and FSδ-closed sets, FSδ-closure and FSδ-interior operators, and FSδ-continuity. Kandil et al. provided the concepts of fuzzy soft connected and fuzzy soft hyperconnected spaces in [31,32], respectively. Various concepts in fuzzy soft settings have been considered, such as disjoint union of fuzzy soft topological structures [6] and filters [26].

    Since the importance of separation axioms in topological spaces, it was investigated topologies over the different types of uncertainty spaces. Kandil and El-Etriby [29] structured separation axioms in the spaces of fuzzy topologies, then Kandil and El-Shafei [30] familiarized the axioms of regularity in fuzzy topologies and FRi-proximities. Saleh et al. [45] displayed stronger types of separation and regularity axioms in the spaces of fuzzy topologies using fuzzy pre-open sets. In fuzzy soft topological spaces, separation axioms have been presented and discussed by many authors; see, for example, [1,2,39,40]. Kandil et al. [33] scrutinized the characterizations of separation axioms and regularity inspired by quasi-coincident and neighborhood systems. Recently, Saleh et al. [46,47] have described another sorts of FS-separation axioms and regularity axioms. In soft setting, a wide class of separation axioms have been offered by Al-shami and his coauthors [12,13,17,21,22,23]. They successfully exploited these axioms to address some real-life situations as given in [8,9]. Alcantud [3] conducted an interesting work to describe the relationships between topological structures in soft and fuzzy settings.

    To go along this line of research, we are writing this paper, which contributes to the understanding of fuzzy soft separability properties and produces some categories of fuzzy soft topological spaces. It is well known that the environment of the current work widens other known generalizations such as fuzzy topology and soft topology; this means the results and relationships obtained in these frames are special cases of their counterparts investigated herein. This is attributed to that the frameworks of soft and fuzzy topologies are produced by "fuzzy soft topology" by replacing the membership function with the characteristic function in the case of fuzzy topology and restricting the set of parameters by a singleton set in the case of soft topology. Hence, the paper enhances the body of knowledge and provides a comprehensive insight to study the properties and characteristics of topological structures.

    After this introduction, the reader may pursue the content of this research as follows. In Section 2, we requisition the definitions and findings that are needful to go along with the results obtained herein. In the next sections, Sections 3 and 4, we delve into the topic of separation properties in fuzzy soft topological spaces and propose a new set of axioms called FSδ-separation (FS-δTi, where i=0,1,2,3,4) and FSδ-regularity (FS-δRi, where i=0,1.2,3). These separations are structured by utilizing the ideas of fuzzy soft δ-open sets and the quasi-coincident relation. We provide various characterizations of these properties and present a range of results, theorems, and relationships related to these notions. In Section 5, we look at the interplay between FSδ-compactness and FSδ-separation axioms and analyze the relationships between them. In the end, we outline the master contributions of this manuscript and suggest a road map for future direction in Section 6.

    Here, we recall the basic definitions that will be needed in this sequel. In the present work, U refers to the universe set,  E is the set of all parameters for U, I=[0,1], and FS- refers to fuzzy soft.

    Definition 2.1. [50] An F-set A of U is a mapping A:UI. IU refers to the set of all F-sets on U. An F-point xλ, λ(0,1] is an F-set in U given by xλ(y)=λ at x=y and xλ(y)=0 for all yU. For αI, α_IU refers to the F-constant function where α_(x)=α   xU.

    Definition 2.2. [37] An FS-set hE=(f,E) on U is the set of ordered pairs hE={(e,h(e):eE,h(e)IU}.

    In this content, FSS(UE) refers to the set of all FS-sets on U. ˜αEFSS(UE) defined by ˜αE={(e,α_):eE α_IU} is called an FS-constant set.

    Definition 2.3. [18,43] An FS-point xeα on UE is an FS-set on U defined by xeα(e)=xα if e=e and xeα(e)=0_  if  eE{e}, where xα is the F-point in U. FSP(UE) refers to the set of all FS-points in U. An FS-point xeα˜fE if αf(e)(x).

    Definition 2.4. [43,49] The triplet (U,τ,E) is called a fuzzy soft topological space (briefly, an FSTS), where U is an initial universal set, E is a fixed set of parameters, and τ is a family of FS-sets on U such that τ is closed under arbitrary union and finite intersection and 0E,1E belong to τ. The elements in τ are called fuzzy soft open sets (briefly, FSO-sets) and the complements of them are called fuzzy soft closed sets (briefly, FSC-sets).

    Definition 2.5. [18] The FS-sets hE and gE are called quasi-coincident, denoted by fEqgE if there are eE, uU such that h(e)(u)+g(e)(u)>1. If hE is not quasi-coincident with gE, then we write hE˜qgE.

    Proposition 2.1. [18,46] Let xer,yetFSP(UE), fE, gE, hEFSS(UE), and {fiE:iJ}FSS(UE), then

    (ⅰ) fE˜qgE fEgcE and fE˜qfcE.

    (ⅱ) fEgE=0EfE˜qgE.

    (ⅲ) fE˜qgE,hEgEfE˜qhE.

    (ⅳ) fEqgExerqgE, for some xer ˜fE.

    (ⅴ) fEgE(xerqfExerqgE),xer.

    (ⅵ) If xerq(iJ fiE), then xer{q}fiE,  iJ.

    (ⅶ) xyxer˜qyet,  r, tI.

    (ⅷ) xer˜qyetxy  or (x=y and r+t1).

    Definition 2.6. [42] An FS-set hE in (U,τ,E) is called q-neighborhood (briefly, q-nbd) of xeα if there is an FSO-set gE such that xeαqgEhE.

    Definition 2.7. [34] Let FSS(UE) and FSS(VK) be two classes of all FS-sets over U and V, respectively. Let u:UV and p:EK be two maps, then fup:FSS(UE)FSS(VK) is called a fuzzy soft map (or an FS-map) for which:

    (ⅰ) If fEFSS(UE), then the image of fE denoted by fup(fE) is the FS-set on V given by fup(fE)(k)=sup{u(f(e)):ep1(k)}  if p1(k), and fup(fE)(k)=˜0V otherwise, for all kK.

    (ⅱ) If gKFSS(VK), then the preimage of gK denoted by f1up(gK) is the FS-set on U defined by, f1up(gK)(e)=u1(g(p(e))) for all eE.

    The FS-map fup is called one-to-one (onto), if u and p are one-to-one (onto). For more details about the properties of FS-maps; see, [34].

    Definition 2.8. [46] Let (U,τ,E) be an FSTS and YU. Let hYE be an FS-set on YE such that hYE:EIY, hYE(e)IY and hYE(e)(x)=1 if xY, hE(e)(x)=0 if xY. Let τY={hYEgE:gEτ}, then τY is a fuzzy soft topology (in short, FST) on Y and (Y,τY,E) is called an FS-subspace of (U,τ,E). If hYEτ (resp., hYEτc), then (Y,τY,E) is called an FS-open (resp., closed) subspace of (U,τ,E).

    Definition 2.9. [18,42] For an FS-set hE in (U,τ,E), we have:

    (ⅰ) The FS-closure cl(hE) of hE is the intersection of all FSC-sets containing hE, and the FS-interior int(hE) of hE is the union of all FSO-sets contained in hE.

    (ⅱ) hE is said to be a fuzzy soft regular open set (FSRO-set) if hE=int(cl(hE)). The complement of an FSRO-set is called a fuzzy closed regular set (FSRC-set). FSRO(UE) refers to the set of all FSRO-sets and FSRC(UE) refers to the set of all FSRC-sets.

    (ⅲ) hE is said to be a fuzzy soft δ-neighborhood (briefly, FSδ-nbd) of xeα if and only if there is FSRO q-nbd gE of xeα such that gEfE.

    Definition 2.10. [42] Let hE be an FS-set in (U,τ,E), then:

    (ⅰ) An FS-point xeα is called an FSδ-cluster point of hE if and only if every FSRO q-nbd fE of xeα, fEqhE. The set of all FSδ-cluster points of hE is called the FSδ-closure of hE, denoted by clδ(hE); that is, clδ(hE)={gEFSRC(UE):hEgE}.

    (ⅱ) An FS-set hE is called a fuzzy soft δ-closed set (FSδC-set) if and only if hE=clδ(hE). The complement of an FSδC-set is called a fuzzy soft δ-open set (FSδO-set). FSδC(UE) refers to the set of all FSδC-sets and FSδO(UE) refers to the set of all FSδO-sets.

    (ⅲ) The FSδ-interior intδ(hE) of hE is defined by intδ(hE)=˜1Eclδ(hcE); that is, intδ(hE)={gEFSRO(UE):gEhE}. Consequently, hE is FSδ-open if and only ifhE=intδ(hE).

    Notation. For xer in FSP(UE), Oxer refers to an FSδO-set containing xer. In general, OhE refers to an FSδO-set containing an FS-set hE.

    Result 1. [42] Every FSRO-set is an FSδO-set and every FSδO-set is an FSO-set. Moreover, if hE is anFS-semi open set in (U,τ,E), then cl(hE)=clδ(hE).

    Result 2. [42] If hE is an FSO-set in (U,τ,E), thencl(hE) is an FSRC-set; that is, {cl(gE):gEτ}={hE:hEFSRC(UE)}, and for any FS-set hE in (U,τ,E), clδ(hE)={cl(gE):hEgE gEτ}.

    Theorem 2.1. [42] For any FS-sets fE and gE in (U,τ,E), we have:

    (ⅰ) clδ(0E)=0E and clδ(1E)=1E.

    (ⅱ) clδ(fE) is an FSδC-set, that is, clδ(clδ(fE))=clδ(fE).

    (ⅲ) cl(fE)clδ(fE) and if fEτ, then cl(fE)=clδ(fE).

    Result 3. [42] The FSδ-closure operator on (U,τ,E) satisfies the Kuratowski closure axioms so that there is one topology on U. This topology is defined as follows:

    The set of all FSδO-sets of (U,τ,E) forms an FS-topology, denoted by τδ. It is called an FSδ-topology on U, and the triplet (U,τδ,E) is called an FSδ-topological space. Moreover, τδτ.

    Definition 2.11. [42] An FS-map fup: (U,τ,E)(V,δ,K) is called:

    (ⅰ) FSδ-open if fup(hE) is an FSδO-set in V for all FSδO-sets hE in U.

    (ⅱ) FSδ-closed if fup(gE) is an FSδC-set in V for all FSδC-sets gE in U.

    Theorem 2.2. [42] Let fup: (U,τ,E)(V,δ,K) be an FS-map, then the next items are equivalent:

    (ⅰ) fup is FSδ-continuous.

    (ⅱ) f1up(gK) is an FSδO-set in (U,τ,E) for all FSδO-sets gK in (V,δ,K).

    (ⅲ) f1up(gK) is an FSδC-set in (U,τ,E) for all FSδC-sets gK in (V,δ,K).

    Definition 2.12. [46] An FSTS (U,τ,E) is said to be:

    (ⅰ) FST0 if for any xer,yetFSP(UE) with xer˜qyet, then xer˜qcl(yet) or cl(xer)˜qyet.

    (ⅱ) FST1 if for any xer,yetFSP(UE) with xer˜qyet, then xer˜qcl(yet) and cl(xer)˜qyet.

    (ⅲ) FST2 if for any xer,yetFSP(UE) with xer˜qyet, there are FSO-sets Oxer,Oyetτ such that Oxer˜qOyet.

    Definition 2.13. [44] For FSTS(U,τ,E) and hEFSS(UE), then:

    (ⅰ) A family A={liE:iJ} of FSδO-sets is called an FSδ- open cover of hE if for all xer ˜hE there is i0J such that xer ˜fi0E.

    (ⅱ) hE is called an FSδ-compact set if every FSδ-open cover of hE has a finite FSδ-open subcover. In general, (U,τ,E) is FSδ-compact if 1E itself is FSδ-compact.

    Here, we are going to give the definitions of a new class of separation axioms called FSδ-separation axioms (or FS-δTi i=0,1,2) and study some their properties.

    Definition 3.1. An FSTS (U,τ,E) is said to be:

    (ⅰ) FS -δT0 if for any xer,yetFSP(UE) with xer˜qyet, there is an FSδO-set Oxer such that Oxer˜qyet, or there is an FSδO-set Oyet such that Oyet˜qxer.

    (ⅱ) FS -δT1 if for any xer,yetFSP(UE) with xer˜qyet, there are FSδO-sets Oxer,Oyet such that yet˜qOxer andxer˜qOyet.

    (ⅲ) FS -δT2 if for any xer,yetFSP(UE) with xer˜qyet, there are FSδO-sets Oxer,Oyet such that Oxer˜qOyet.

    Lemma 3.1. For FSTS (U,τ,E), xerFSP(UE), and hEFSS(UE), then:

    (ⅰ) xer˜intδ(hE) there is an FSδO-set Oxer such that OxerhE.

    (ⅱ) xerqclδ(hE)OxerqhE for any FSδO-set Oxer in (U,τ,E).

    (ⅲ) gEqhEgEqclδ(hE) for any FSδO-set gE in (U,τ,E).

    In the next results, we give some characterizations of FS-δTi space, i=0,1,2.

    Theorem 3.1. An FSTS (U,τ,E) is FS-δT0 if and only if for any xer, yetFSP(UE) with xer˜qyet implies xer˜qclδ(yet) or clδ(xer)˜qyet.

    Proof. Let (U,τ,E) be FS-δT0 and xer˜qyet for any xer, yetFSP(UE). Then there is an FSδO-set Oxer such that yetqOxer or there is an FSδO-set Oyet such that xer˜qOyet. From (ⅱ) of the above lemma, we get xer˜qclδ(yet) or clδ(xer)˜qyet.

    Conversely, let xer˜qyet. By given xeα˜qclδ(yet) or clδ(xer)˜qyeβ, and again from (ⅱ) of the above lemma, there is an FSδO-set Oxer such that Oxer˜qyet or there is Oyet with Oyet˜qxer. Hence, (U,τ,E) is FS-δT0.

    Remark 3.1. Clearly, every FS-δT0 is FST0. The converse is not necessarily true.

    Example 3.1. Let U=[0,1] and E={e}, then the class τ={0E,1E}{fiE:iN} is an FST on U, where

    fiE(e)(u)={1_,       u=0,   11i, 0<u1i,1_,      1i<u1.

    One can check that (U,τ,E) is FST0. On other hand, clearly {cl(hE):hEτ}={gE:gEFSRC(UE)}, and for any lE in (U,τ,E), we have clδ(lE)={gEFSRC(UE):lEgE}. Since cl(fiE)=1E for all iN, cl(1E)=1E, and cl(0E)=0E, then FSRC(UE)={0E,1E}. Therefore, FSδC(UE)={0E,1E}=FSδO(UE). Thus, (U,τ,E) is not FS-δT0.

    Theorem 3.2. Let fup:(U,τ,E)(V,δ,K) be one-to-one and FSδ-continuous. If (V,δ,K) is FS-δT0, then (U,τ,E) also is FS-δT0.

    Proof. Let xer˜qyet for any xer, yetFSP(UE). Since fup is one-to-one, then fup(xer)˜qfup(yet). Since (V,δ,K) is FS-δT0, there is an FSδO-set Ofup(xer) such that fup(yet)˜qOfup(xer), or there is an FSδO-set Ofup(yet) such that fup(xer)˜qOfup(yet). Since fup is FSδ-continuous, we have f1up(Ofup(xer)) as an FSδO-set in (U,τ,E) with yet˜qf1up(Ofup(xer)), or there is an FSδO-set f1up(Ofup(yet)) in (U,τ,E) with xer˜qf1up(Ofup(yet)). Hence, (U,τ,E) is FS-δT0.

    Theorem 3.3. An FSTS (U,τ,E) is FS-δT1 if and only if for any xer,yetFSP(UE) with xer˜qyet implies xer˜qclδ(yet) and clδ(xer)˜qyet.

    Proof. By a similar way to that in Theorem 3.3.

    Theorem 3.4. For an FSTS (U,τ,E), the next items are equivalent:

    (ⅰ) (U,τ,E) is FS-δT1.

    (ⅱ) clδ(xer)=xer for all xerFSP(UE).

    Proof. (i)(ii). Let (U,τ,E) be FS-δT1 and xer,yetFSP(UE) with xer˜qyet, then there is an FSδO-set Oyet such that xer˜qOyet. This impliesOyet(xer)c; thus, (xer)c is an FSδO-set and is, xer is an FSδC-set for all xeαFSP(UE). Hence, clδ(xer)=xer.

    (ii)(i). Let xer,yetFSP(UE) with xer˜qyet, then xeα(yeβ)c and yeβ(xeα)c(since FS-points xer,yet are FSδC-sets). Now, take Oxer=(yeβ)c and Oyet=(xeα)c. Thus, there are FSδO-sets Oxeα and Oyet such that xer˜q(xer)c=Oyet and yet˜q(yet)c=Oxer. The result holds.

    Theorem 3.5. If (V,δ,K) is an FS-δT1 and fup:(U,τ,E)(V,δ,K) are one-to-one and FSδ-continuous, then so is (U,τ,E).

    Proof. Let (V,δ,K) be FS-δT1 and xer˜qyet for any xer, yetFSS(UE). Since fup is one-to-one, we have fup(xer)˜qfup(yet). Since (V,δ,K) is FS-δT1, there are FSδO-sets Ofup(xer), Ofup(yet)δ such that fup(yet)˜qOfup(xer) and fup(xer)˜qOfup(yet). Since fup is FSδ-continuous, we have f1up(Ofup(xer)) and f1up(Ofup(yet)) as FSδO-sets in (U,τ,E) with yet˜qf1up(Ofup(xer)) and xer˜qf1up(Ofup(yet)). Hence, (U,τ,E) is FS-δT1.

    Theorem 3.6. If FSTS(U,τ,E) is FS-δT2, then xer={clδ(hE):xer˜hE}.

    Proof. Let (U,τ,E) be FS-δT2 and xerFSP(UE), then for any xer˜qyet, there are FSδO-sets hE=Oxer and Oyet such that hE˜qOyet. From (ⅱ) of Lemma 3.2, we have yet˜qclδ(hE) and yet˜q{clδ(hE):xer˜hE}. From (ⅴ) of Proposition 2.2, we have {clδ(hE):xer˜hE}xer, but xer˜{clδ(hE):xer˜hE}. The result holds.

    Proposition 3.1. Every FS-δTi is FS-δTi1, i=1,2.

    Proof. It is obvious.

    The next example shows that the converse of the above proposition is not necessarily true.

    Example 3.2. Let U={x,y}, E={e}, and τ={0E ,1E}{xer:r(0,1]}, then τ is an FST on U. It is easy to check that (U,τ,E) is FS-δT0. Indeed, all members in τ are FSRO-sets, so they are FSδO-sets. For any xer,yet with xer˜qyet, there is an FSδO-set Oxer=xer such that Oxer=xer˜qyet. On the other hand, the unique FSδO-set containing yet is 1E, but 1Eqxer. Therefore, (U,τ,E) is not FS-δT1.

    Theorem 3.7. Let (U,τ,E) be FS-δT1 and hE be any FSδO-set. If hcE is also FSδO-set in (U,τ,E), then (U,τ,E) is FS-δT2.

    Proof. Let xer˜qyet for any xer, yetFSP(UE). Since (U,τ,E) is FS-δT1, there is an FSδO-set Oxer such that Oxer˜qyet, or there is anFSδO-set Oyet such that Oyet˜qxer. Let us assume that Oxer˜qyet, then yet(Oxer)c, which is an FSδO-set by assumption, and Oxer˜q(Oxer)c. This completes the proof.

    Proposition 3.2. If every crisp FS-point xe1 is FSδO-set in (U,τ,E), then (U,τ,E) is FS-δT2.

    Proof. It is obvious.

    Theorem 3.8. If (V,δ,K) is FS-δT2 and fup:(U,τ,E)(V,δ,K) is one-to-one and FSδ-continuous, then (U,τ,E) is FS-δT2.

    Proof. It follows by using a similar way to that in Theorem 3.9.

    Theorem 3.9. Every FS-subspace (Y,τY,E) of FS-δTi(U,τ,E) is FS-δTi, i=0,1,2.

    Proof. As a sample, we prove the case i=1. The proof of the rest of the cases is similar. Let xer, yetFSP(YE) with xer˜qyet, then also xer , yet FSP(UE) with xeα˜qyeβ. Since (U,τ,E) is FS-δT1, there is FSδO-sets Oxer, Oyet such that yet˜qOxer and xer˜qOyet. Thus, OxerhYE and OyeβhYE are FSδO-sets in (Y,τY,E). Take Oxer=OxerhYE and Oyet=OyethYE, then yet˜qOxer and xer˜qOyet. Hence, the result holds.

    Here, we introduce the definitions of a new class of regularity axioms, namely, FSδ-regularity axioms (or FS-δRi, i=0,1,2,3), and investigate some its properties.

    Definition 4.1. An FSTS (U,τ,E) is said to be:

    (ⅰ) FS -δR0 if for any xer, yetFSP(UE) with xer˜qclδ(yet) implies clδ(xer)˜qyet.

    (ⅱ) FS -δR1 if for any xer, yetFSP(UE) with xer˜qclδ(yet), there are FSδO-sets Oxer and Oyet such that Oxer˜qOyet.

    In the next results, some descriptions of FS-δRi spaces for i=0,1 are investigated.

    Theorem 4.1. In an FSTS (U,τ,E), the next items are equivalent:

    (ⅰ) (U,τ,E) is FS -δR0 .

    (ⅱ) clδ(xer)Oxer for any FSδO-set Oxer.

    (ⅲ) clδ(xer) {Oxer:OxerFSδOS(UE)} for all  xerFSP(UE).

    Proof. (ⅰ)(ⅱ) Let (U,τ,E) be FS-δR0 and yetqclδ(xer), then xerqclδ(yet). From (ⅱ) of Lemma 3.2, we have yetqOxer, and by (ⅴ) of Proposition 2.2, we get clδ(xer)Oxer for any FSδO-set Oxer. The result holds.

    (ⅱ) (ⅲ) It is clear.

    (ⅲ)(ⅰ) Let clδ(xer){Oxer:OxerFSδO(UE)}Oxer for any Oxer and let xer,yetFSP(UE) with xer˜qclδ(yet), then xeα[clδ(yet)]c=Oxer, which is an FSδO-set containing xer. So by hypothesis, clδ(xer)Oxer=[clδ(yet)]c=intδ[(yet)c](yet)c. Thus, clδ(xer)˜qyet. Hence, (U,τ,E) is FS-δR0.

    Theorem 4.2. An FSTS (U,τ,E) is FS-δR0 if and only if hE is an FSδC-set with xeα˜qhE, and there is an FSδO-set OhE containing hE such that xeα˜qOhE.

    Proof. Let (U,τ,E) be FS-δR0 and hEFSδC(UE) with xer˜qhE, then xerhcE=Oxer. From (ⅱ) of Theorem 4.2, we have clδ(xer)hcE=Oxeα and hE[clδ(xer)]c=OhE. Since xerclδ(xer), then [clδ(xer)]c(xer)c. Therefore, xer˜q[clδ(xer)]c=OhE. The result holds.

    The converse part is obvious.

    Theorem 4.3. In an FSTS (U,τ,E), the next properties are equivalent:

    (ⅰ) (U,τ,E) is FS-δR0 .

    (ⅱ) If gE is FSδC-set with xer˜qgE, then clδ(xer)˜qgE.

    (ⅲ) If xer˜qclδ(yet), then clδ(xer)˜qclδ(yet).

    Proof. (ⅰ)(ⅱ) Let gE be an FSδC-set with xer˜qgE. Since (U,τ,E) is FS-δR0, then by the above theorem there is an FSδO-set OgE such that xer˜qOgE. From (ⅱ) of Lemma 3.2, we have clδ(xer)˜qgE.

    (ⅱ) (ⅲ) It is obvious.

    (ⅲ) (ⅰ) Let xer,yetFSP(UE) with xer˜qclδ(yet). By given clδ(xer)˜qclδ(yet), since yetclδ(yet), we have clδ(xer)˜qyet. Thus (U,τ,E) is FS-δR0.

    Proposition 4.1. An FSTS (U,τ,E) is FS-δR1 if and only if for any xeryetFSP(UE) with xer˜qclδ(yet), there are FSδO-sets Oclδ(xer) and Oclδ(yet) such that Oclδ(xer)˜qOclδ(yet).

    Proof. It follows from that of the above theorem and from (ⅱ) of Theorem 4.2.

    Theorem 4.4. Every FS-subspace (Y,τY,E) of FS-δRi is FS-δRi, i=0,1.

    Proof. As a sample, we prove the case i=1. The proof of the rest case is similar.

    Let xer , yet be FS-points in (YE) with xer˜qclδ(yet), then also xer , yetFSP(UE) and xer˜qclδ(yet). Since (U,τ,E) is FS-δR1, there are FSδO-sets Oxer,Oyet such that Oxer˜qOyet. Take Oxer=OxerhYE and Oyet=OyethYE, then Oxer,Oyet are FSδO-sets in (Y,τY,E) and Oxer˜qOyet. Hence, (Y,δYE) is FS-δR1.

    Proposition 4.2. For FSTS (U,τ,E), every FS-δTi is FS-δRi1, i=1,2.

    Proof. It is obvious.

    The next example shows that the converse of the above proposition is not necessarily true.

    Example 4.1 Let U{u} and E={e1,e2}. The family τ={0E,1EhE={(e1,u0.5),(e2,u0.5)} is an FST on U. One can check that (U,τ,E) is FS-δR0, but is not FS-δT0. Indeed, for xe10.7˜qxe10.2, the unique FSδO-set containing ue10.7 is 1E, but 1Eque10.2.

    Theorem 4.5. An FSTS (U,τ,E) is FS-δTi if and only if it is both FS-δTi1 and FS-δRi1,i=1,2.

    Proof. As a sample, we prove the case i=2. The proof of the rest case is similar. Necessity follows from the Proposition 3.11 and 4.7.

    Conversely, let (U,τ,E) be FS-δT1 and FS-δR1, and let xer,yetFSP(UE) withxer˜qyet. By Theorem 3.7, we have xer˜qcl(yet). Since (U,τ,E) is FS-δR1, there are FSδO-sets Oxer,Oyet such that Oxer˜qOyet. Therefore, (U,τ,E) is FS-δT2.

    Definition 4.2. An FSTS (U,τ,E) is said to be:

    (ⅰ) FS δ-regular(or FS-δR2) if for any FSδC-set hE and any FS-point xer with xer˜qhE, there are FSδO-sets Oxer and OhE such that Oxer˜qOhE.

    (ⅱ) FS δ-normal(or FS-δR3) if for any FSδC-sets hE and gE with hE˜qgE, there are FSδO-sets OhE and OgE such that OhE˜qOgE.

    (ⅲ) FS -δT3(resp., FS-δT4) if it is FS-δR2(resp., FS- δR3) and FS-δT1.

    Theorem 4.6. For an FSTS (U,τ,E), the next items are equivalent:

    (ⅰ) (U,τ,E) is FS-δR2.

    (ⅱ) For any xerFSP(UE) and any FSδO-set Oxer, there is an FSδO-set Oxer containing xer such that clδ(Oxer)Oxer.

    Proof. (i)(ii) Let xerFSP(UE) and Oxer be any FSδO-set containing xer, then Ocxer=hE is an FSδC-set. Clearly, Oxer˜qOcxer andxer˜qOcxer. Since (U,τ,E) is FS-δR2, there are FSδO-sets Oxer,OOcxer such that Oxer ˜qOOcxer=OhE, then OxerOchE and clδ(Oxer)OchE. Clearly, OcxerOOcxer=OhE, so OchEOxer. Therefore, clδ(Oxer)Oxer.

    (ii)(i) Let xerFSP(UE) and gE be any FSδC-set with xer˜qgE, then xer˜gcE=Oxer which is an FSδO-set containing xer. So there is an FSδO-set Oxer such that clδ(Oxer)Oxer=gcE, which implies gE[clδ(Oxer)]c=OgE. Clearly, clδ(Oxer)˜q[clδ(Oxer)]c=OgE and Oxer˜qOgE. Thus, the result holds.

    Theorem 4.7. An FSTS (U,τ,E) is FS-δR2 if and only if for any FSδC-set hE with xer˜qhE, there are FSδO-sets Oxer,OhE such that cl(Oxer)˜qcl(OhE).

    Proof. Let xerFSP(UE) and hE be an FSδC-set with xer˜qhE. Since (U,τ,E) is FS-δR2, there are FSδO-sets Oxer,OhE such that OhE˜qOxer. From (ⅲ) of Lemma 3.2, we obtain cl(OhE)˜qOxer, that is, cl(OhE)˜qxer. Again (U,τ,E) is FS-δR2, and there are FSδO-sets Oxer,Ocl(OhE) such that Oxer˜qOcl(OhE). By (ⅲ) of Lemma 3.2, we have cl(Oxer)˜qOcl(OhE). Now, put Oxer=OxerOxer. Since (U,τ,E) is FS-δR2 and Oxer is an FSδO-set, then by the above theorem, there is an FSδO-set Oxer such that clδ(Oxer)Oxer that is, cl(Oxer)Oxer. Since cl(OhE)˜qOxer, then cl(OhE)˜qcl(Oxer).

    Conversely, it follows from hypothesis.

    Theorem 4.8. For an FSTS (U,τ,E), the next items are equivalent:

    (ⅰ) (U,τ,E) is FS-δR3.

    (ⅱ) For any FSδC-set hE and any FSδO-set OhE, there is an FSδO-set OhE containing hE such that clδ(OhE)OhE.

    Proof. Let (U,τ,E) be FS-δR3, hE be an FSδC-set, and OhE be any FSδO-set containing hE, then OchE is an FSδC-set. Since OhE˜qOchE, that is, hE˜qOchE, (U,τ,E) is FS-δR3, there are FSδO-sets OhE,OOchE such that OhE˜qOOchE, then OhE(OOchE)c and clδ(OhE)(OOchE)c. Since OchEOOchE, then (OOchE)cOhE and clδ(OhE)(OOchE)cOhE. The result holds.

    Conversely, let fE,gE be two FSδC-sets with fE˜qgE, then fEgcE=OfE which is an FSδO-set containing fE. By hypothesis, there is an FSδO-set OfE such that clδ(OfE)gcE=OfE, then gE[clδ(OfE)]c=OgE. Since clδ(OfE)˜q[clδ(OfE)]c=OgE, then OgE˜qOfE. The result holds.

    Theorem 4.9. An FSTS (U,τ,E) is FS-δR3 if and only if for any two FSδC-sets hE,gE with hE˜qgE, there are FSδO-sets OhE,OgE such that cl(OhE)˜qcl(OgE).

    Proof. It is analogous to that in Theorem 4.12.

    Saleh et. al [44] introduced and studied a new type of FS-compactness, namely, FSδ-compactness. In this section, we study more properties and investigate the relations between FSδ-compact and FSδ-separation axioms, which are introduced in this work.

    To begin we show that the axioms FS-δRi, i=1,2,3 and FS-δTi, i=1,2,3,4 are harmonic.

    Theorem 5.1. For FSTS (U,τ,E), we have:

    FSδR3FSδR0FSδR2FSδR1FSδR0.

    Proof. Let (U,τ,E) be FS-δR3, FS-δR0,xerFSP(UE) for any FSδC-set fE with xer˜qhE. Since (U,τ,E) is FS-δR0, then clδ(xer)˜qfE where clδ(xer),hE are FSδC-sets. Again, (U,τ,E) is FS-δR3, so there are FSδO-sets Oclδ(xer), OhE such that Oclδ(xer)˜qOhE. Put Oclδ(xer)=Oxer, and we have Oxer˜qOhE. Thus, (U,τ,E) is FS-δR2. The proof for the rest of the cases is obvious.

    Theorem 5.2. For an FSTS (U,τ,E), we have:

    FSδT4FSδT3FSδT2FSδT1FSδT0

    Proof. Let (U,τ,E) be FS-δT4, then it is both FS-δR3 and FS-δT1. From Proposition 4.7, we have (U,τ,E) is FSR0. Let us assume that xerFSP(UE),hE is an FSδC-set with xer˜qhE, then by Theorem 4.4, clδ(xer)˜qhE, where clδ(xer),hE are FSδC-sets. Since (U,τ,E) is FS-δR3, there are FSδO-sets Oclδ(xer),OhE such that Oclδ(xer)˜qOhE. Take Oclδ(xer)=Oxer, and we have Oxer˜qOhE. Thus, (U,τ,E) is FS-δR2. Hence, we obtain the result.

    The proof of the rest of the cases follows from the above theorem and Proposition 3.11.

    From the above theorems, Definition 4.10, and Proposition 4.7, we obtain the next result.

    Corollary 5.1. For an FSTS (U,τ,E), the next implications hold.

    FSδT4FSδT3FSδT2FSδT1FSδT0FSδR3FSδR0FSδR2FSδR1FSδR0.

    Theorem 5.3. Let (U,τ,E) be FS-δT3 and gE be an FSδ-compact set, then for any FSδC-set hE with hE˜qgE, there are FSδO-sets OhE,OgE such that OhE˜qOgE.

    Proof. Let (U,τ,E) be FS-δT3 and gE be an FSδ-compact set, then for any FSδC-set hE with hE˜qgE, we have for any yet˜gE, there are FSδO-sets Oyet,OhE such that Oyet˜qOhE. Clearly, {Oyet:yet˜gE} is FSδ-open cover of gE. Since gE is FSδ-compact, there is a finite FSδ-open subcover of gE, say, {Oiyet:i=1,2,, n}. One can verify that OgE=ni=1Oiyet and OhE=ni=1OihE have the required property.

    Theorem 5.4. Let (U,τ,E) be FS-δT2,xerFSP(UE) and gE be an FSδ-compact set with xer˜qgE, then there are FSδO-sets Oxer and OgE such that Oxer˜qOgE.

    Moreover, if hE, gE are FSδ-compact sets with hE˜qgE, then there are FSδO-sets OhE,OgE such that OhE˜qOgE.

    Proof. It follows by a similar way to that in the above theorem.

    Theorem 5.5. Every FSδ-compact set in an FS-δT2 space is an FSδC-set.

    Proof. Let (U,τ,E) be FS-δT2 and gE be an FSδ-compact set. From the above theorem for any xerFSP(UE) with xer˜qgE, there is FSδO-set Oxer such that Oxer˜qgE; that is, for any xer˜gcE, there is FSδO-set Oxer such that Oxer˜gcE. Therefore, gcE is an FSδO-set in (U,τ,E). Thus, gE is an FSδC-set.

    Theorem 5.6. Let (U,τ,E) be FS-δR1, then (U,τ,E) is FS-δT2 if and only if every FSδ-compact set is an FSδC-set.

    Proof. The necessary parts follows directly from the above theorem. Conversely, if any FSδ-compact set is an FSδC-set, then (U,τ,E) is an FS-δT1 space. Since (U,τ,E) is FS-δR1 and FS-δT1, then by Theorem 4.9, we obtain that (U,τ,E) is FS-δT2.

    Theorem 5.7. For FSTS(U,τ,E), every FSδ-compact FS-δR1 space is FS-δR2 (FS-δR3).

    Proof. Let (U,τ,E) be an FSδ-compact, FS-δR1 space and let hE be an FSδC-set with xer˜qhE, then for any FS-point yet˜hE, we have xer˜qclδ(yet). Since (U,τ,E) is FS-δR1, there are FSδO-sets Oxer Oyet such that Oxer˜qOyet so that the family {Oyet:yet˜hE} is an FSδ-open cover of hE. Since (U,τ,E) is FSδ-compact, hE is FSδ-compact and there is a finite FSδ-open subcover of hE, say, {Oiyet:yet˜hE i=1,2,,n}. Take Oxer=ni=1Oixer and OhE=ni=1Oiyet, then Oxer, OhE are FSδO-sets with Oxer˜qOhE. The result holds.

    The proof of the rest case is analogous.

    Corollary 5.2. For FSδ-compact space (U,τ,E), the next items are equivalent:

    (ⅰ)  (U,τ,E) is FS-δR1.

    (ⅱ)  (U,τ,E) is FS-δR2.

    (ⅲ) (U,τ,E) is FS-δR0 and FS-δR3.

    Proof. It is obvious.

    Theorem 5.8. (U,τ,E) is FS-δTi (U,τδ,E) is FSTi, i=0,1,2.

    Proof. It follows directly from Result 3 and Definition 3.1.

    It is well known that separation axioms provide some categories for topological spaces and help to prove some interesting properties of compactness and connectedness. Therefore, we have written this article to shed light on the properties of separability in the framework of fuzzy soft topologies.

    We have defined and studied a new set of separation properties in fuzzy soft topological spaces, namely, FSδ-separation and regularity properties via FSδO-sets by using quasi-coincident relation for FS-points. Several basic desirable properties, relations, and results have been obtained with some necessary examples. The relationships between FSδ-compact spaces and FSδ-separation have been investigated as well. We have shown that the implications FS-δT4FS-δT3FS-δT2FS-δT1FS-δT0 hold true, but we cannot get examples to show that the converse in these implications may not be true in general, except the case FS-δT0FS-δT1.

    By and large, the results obtained in the manuscript frame "fuzzy soft topology" represent a wider view than that inspired by the frameworks of fuzzy and soft topologies, since these frames are created by replacing the membership function with the characteristic function in the case of fuzzy topology and restricting the set of parameters by a singleton set in the case of soft topology. The present results elucidate that the perspective on the theory of separation axioms adopted in this paper is very useful and will open up the door for further contributions. We plan in upcoming studies to generate fuzzy soft topologies by hybridizing F-set with the recent types of F-set like complemental fuzzy sets [4], (2,1)-fuzzy sets [10], (m,n)-fuzzy sets [15], nth power root fuzzy sets [14,27], and knm-Rung picture fuzzy sets [28]. One may examine the current concepts and the previous ones in these hybridizations.

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

    The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number "NBU-FPEJ-2024-2727-02".

    The authors also extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2023/R/1444).

    The authors declare that they have no competing interests.



    [1] B. Ahmad, S. K. Ntouyas, Existence results for Caputo-type sequential fractional differential inclusions with nonlocal integral boundary conditions, J. Appl. Math. Comput., 50 (2016), 157–174. https://doi.org/10.1007/s12190-014-0864-4 doi: 10.1007/s12190-014-0864-4
    [2] M. Kisielewicz, Stochastic differential inclusions and applications, In: Springer optimization and its applications, New York: Springer, 2013. https://doi.org/10.1007/978-1-4614-6756-4
    [3] H. A. Hammad, R. A. Rashwan, A. Nafea, M. E. Samei, M. De la Sen, Stability and existence of solutions for a tripled problem of fractional hybrid delay differential euations, Symmetry, 14 (2022), 2579. https://doi.org/10.3390/sym14122579 doi: 10.3390/sym14122579
    [4] H. A. Hammad, R. A. Rashwan, A. Nafea, M. E. Samei, S. Noeiaghdam, Stability analysis for a tripled system of fractional pantograph differential equations with nonlocal conditions, J. Vib. Control, 30 (2024), 632–647. https://doi.org/10.1177/10775463221149232 doi: 10.1177/10775463221149232
    [5] M. F. Danca, Synchronization of piecewise continuous systems of fractional order, Nonlinear Dyn., 78 (2014), 2065–2084. https://doi.org/10.1007/s11071-014-1577-9 doi: 10.1007/s11071-014-1577-9
    [6] Y. Cheng, R. P. Agarwal, D. O. Regan, Existence and controllability for nonlinear fractional differential inclusions with nonlocal boundary conditions and time-varying delay, FCAA, 21 (2018), 960–980. https://doi.org/10.1515/fca-2018-0053 doi: 10.1515/fca-2018-0053
    [7] S. K. Ntouyas, S. Etemad, J. Tariboon, Existence results for multi-term fractional differential inclusions, Adv. Diff. Equ., 2015 (2015), 140. https://doi.org/10.1186/s13662-015-0481-z doi: 10.1186/s13662-015-0481-z
    [8] A. Losta, Z. Opial, Application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations or noncompact acyclic-valued map, Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys., 13 (1965), 781–786.
    [9] S. Chakraverty, R. M. Jena, S. K. Jena, Computational fractional dynamical systems: Fractional differential equations and applications, John Wiley & Sons, 2022. https://doi.org/10.1002/9781119697060
    [10] B. Ahmad, V. Otero-Espinar, Existence of solutions for fractional differential inclusions with antiperiodic boundary conditions, Bound. Value Probl., 2009 (2009), 625347. https://doi.org/10.1155/2009/625347 doi: 10.1155/2009/625347
    [11] H. A. Hammad, M. Zayed, Solving systems of coupled nonlinear Atangana-Baleanu-type fractional differential equations, Bound. Value Probl., 2022 (2022), 101. https://doi.org/10.1186/s13661-022-01684-0 doi: 10.1186/s13661-022-01684-0
    [12] H. A. Hammad, H. Aydi, M. Zayed, On the qualitative evaluation of the variable-order coupled boundary value problems with a fractional delay, J. Inequal. Appl., 2023 (2023), 105. https://doi.org/10.1186/s13660-023-03018-9 doi: 10.1186/s13660-023-03018-9
    [13] Humaira, H. A. Hammad, M. Sarwar, M. De la Sen, Existence theorem for a unique solution to a coupled system of impulsive fractional differential equations in complex-valued fuzzy metric spaces, Adv. Differ. Equ., 2021 (2021), 242. https://doi.org/10.1186/s13662-021-03401-0 doi: 10.1186/s13662-021-03401-0
    [14] M. Subramanian, M. Manigandan, C. Tunc, T. N. Gopal, J. Alzabut, On the system of nonlinear coupled differential equations and inclusions involving Caputo-type sequential derivatives of fractional order, J. Taibah Uni. Sci., 16 (2022), 1–23. https://doi.org/10.1080/16583655.2021.2010984 doi: 10.1080/16583655.2021.2010984
    [15] M. Manigandan, S. Muthaiah, T. Nandhagopal, R. Vadivel, B. Unyong, N. Gunasekaran, Existence results for a coupled system of nonlinear differential equations and inclusions involving sequential derivatives of fractional order, AIMS Math., 7 (2022), 723–755. https://doi.org/10.3934/math.2022045 doi: 10.3934/math.2022045
    [16] A. Perelson, Modeling the interaction of the immune system with HIV, In: Mathematical and statistical approaches to AIDS epidemiology, Berlin: Springer, 1989,350–370. https://doi.org/10.1007/978-3-642-93454-4_17
    [17] A. S. Perelson, D. E. Kirschner, R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81–125. https://doi.org/10.1016/0025-5564(93)90043-a doi: 10.1016/0025-5564(93)90043-a
    [18] D. Baleanu, O. G. Mustafa, R. P. Agarwal, On Lp-solutions for a class of sequential fractional differential equations, Appl. Math. Comput., 218 (2011), 2074–2081. https://doi.org/10.1016/j.amc.2011.07.024 doi: 10.1016/j.amc.2011.07.024
    [19] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
    [20] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
    [21] Z. Wei, W. Dong, Periodic boundary value problems for Riemann-Liouville sequential fractional differential equations, Electron. J. Qual. Theory Differ. Equ., 87 (2011), 1–13. https://doi.org/10.14232/ejqtde.2011.1.87 doi: 10.14232/ejqtde.2011.1.87
    [22] H. A. Hammad, H. Aydi, H. Isik, M. De la Sen, Existence and stability results for a coupled system of impulsive fractional differential equations with Hadamard fractional derivatives, AIMS Math., 8 (2023), 6913–6941. https://doi.org/10.3934/math.2023350 doi: 10.3934/math.2023350
    [23] H. A. Hammad, M. De la Sen, Stability and controllability study for mixed integral fractional delay dynamic systems endowed with impulsive effects on time scales, Fractal Fract., 7 (2023), 92. https://doi.org/10.3390/fractalfract7010092 doi: 10.3390/fractalfract7010092
    [24] X. Li, D. Chen, On solvability of some p-Laplacian boundary value problems with Caputo fractional derivative, AIMS Math., 6 (2021), 13622–13633. https://doi.org/10.3934/math.2021792 doi: 10.3934/math.2021792
    [25] H. A. Hammad, P. Agarwal, S. Momani, F. Alsharari, Solving a fractional-order differential equation using rational symmetric contraction mappings, Fractal Fract., 5 (2021), 159. https://doi.org/10.3390/fractalfract5040159 doi: 10.3390/fractalfract5040159
    [26] Z. Wei, Q. Li, J. Che, Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative, J. Math. Anal. Appl., 367 (2010), 260–272. https://doi.org/10.1016/j.jmaa.2010.01.023 doi: 10.1016/j.jmaa.2010.01.023
    [27] M. Klimek, Sequential fractional differential equations with Hadamard derivative, Commun. Nonlinear Sci., 16 (2011), 4689–4697. https://doi.org/10.1016/j.cnsns.2011.01.018 doi: 10.1016/j.cnsns.2011.01.018
    [28] C. Bai, Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative, J. Math. Anal. Appl., 384 (2011), 211–231. https://doi.org/10.1016/j.jmaa.2011.05.082 doi: 10.1016/j.jmaa.2011.05.082
    [29] Y. K. Chang, J. J. Nieto, Some new existence results for fractional differential inclusions with boundary conditions, Math. Comput. Model., 49 (2009), 605–609. https://doi.org/10.1016/j.mcm.2008.03.014 doi: 10.1016/j.mcm.2008.03.014
    [30] S. Hu, N. S. Papageorgiou, Handbook of multivalued analysis (theory), Dordrecht: Kluwer Academic Publishers, 1997.
    [31] K. Deimling, Multivalued differential equations, New York: De Gruyter, 1992. https://doi.org/10.1515/9783110874228
    [32] I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, methods of their solution, and some of their applications, Elsevier, 1998.
    [33] A. Granas, J. Dugundji, Fixed point theory, New York: Springer, 2003.
    [34] H. Covitz, S. B. Nadler, Multi-valued contraction mappings in generalized metric spaces, Israel J. Math., 8 (1970), 5–11. https://doi.org/10.1007/BF02771543 doi: 10.1007/BF02771543
    [35] C. Castaing, M. Valadier, Convex analysis and measurable, multifunctions, In: Lecture Notes in Mathematics, Berlin: Springer, 1977. https://doi.org/10.1007/BFb0087685
  • This article has been cited by:

    1. Mohammed Abu Saleem, On soft covering spaces in soft topological spaces, 2024, 9, 2473-6988, 18134, 10.3934/math.2024885
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(942) PDF downloads(63) Cited by(4)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog