Research article

Numerical approximation of a variable-order time fractional advection-reaction-diffusion model via shifted Gegenbauer polynomials

  • Received: 11 April 2022 Revised: 11 June 2022 Accepted: 16 June 2022 Published: 23 June 2022
  • MSC : 65N35, 34A08

  • The fractional advection-reaction-diffusion equation plays a key role in describing the processes of multiple species transported by a fluid. Different numerical methods have been proposed for the case of fixed-order derivatives, while there are no such methods for the generalization of variable-order cases. In this paper, a numerical treatment is given to solve a variable-order model with time fractional derivative defined in the Atangana-Baleanu-Caputo sense. By using shifted Gegenbauer cardinal function, this approach is based on the application of spectral collocation method and operator matrices. Then the desired problem is transformed into solving a nonlinear system, which can greatly simplifies the solution process. Numerical experiments are presented to illustrate the effectiveness and accuracy of the proposed method.

    Citation: Yumei Chen, Jiajie Zhang, Chao Pan. Numerical approximation of a variable-order time fractional advection-reaction-diffusion model via shifted Gegenbauer polynomials[J]. AIMS Mathematics, 2022, 7(8): 15612-15632. doi: 10.3934/math.2022855

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  • The fractional advection-reaction-diffusion equation plays a key role in describing the processes of multiple species transported by a fluid. Different numerical methods have been proposed for the case of fixed-order derivatives, while there are no such methods for the generalization of variable-order cases. In this paper, a numerical treatment is given to solve a variable-order model with time fractional derivative defined in the Atangana-Baleanu-Caputo sense. By using shifted Gegenbauer cardinal function, this approach is based on the application of spectral collocation method and operator matrices. Then the desired problem is transformed into solving a nonlinear system, which can greatly simplifies the solution process. Numerical experiments are presented to illustrate the effectiveness and accuracy of the proposed method.



    Fixed point theory is derived from the investigation of the solution of some types of differential equations using the method of successive approximation. Indeed, Banach contraction principle [7] is a reformulation of the successive approximation techniques originally used by some earlier mathematicians, namely Cauchy, Liouville, Picard, Lipschitz and so on. In recent time, fixed point techniques have been used extensively to find solutions of both linear and nonlinear equations arising in many areas of sciences. The original idea of fixed point theorem due to Banach has been refined in different directions. See, for example, Rhoades [25] or Taskovic [27].

    In 2002, Branciari [11] analyzed the existence of fixed point of a single-valued mapping defined on a complete metric space satisfying general contractive condition of integral type. The condition is analogous to Banach-Caccioppoli's Principle. With different method of proof, however, the result in [11] reduces to the Banach contraction theorem if the integrand of the contractive condition in [11] is set identically to one. The first extension of Branciari's result was presented by Rhoades [25]. Thereafter, more than a handful of researchers extended the result of Branciari and obtained fixed point theorems for single-valued and multi-valued mappings involving several other contractive conditions of integral type in various spaces, such as metric spaces, modular spaces, fuzzy metric spaces, cone metric spaces, symmetric spaces, Hausdorff topological spaces and so on (see, for example, [1,2,9,18,23,25]). On this development, Jachymski [18] argued that most available contractive conditions of integral type are direct analogues of classical ones and hence provided a new contractive condition which does not coincide with the earlier results. By using the notion of modular spaces, Beygmohammadi and Razani [9] proved two fixed point theorems for mappings satisfying a general contractive condition of integral type in modular spaces, thereby generalizing fixed point theorem for a quasicontraction mapping given by Khamsi [20]. Murthy and Kumam [23] introduced the concept of fuzzy metric spaces and presented common fixed point of self maps satisfying an integral type contractive condition which improved the results of Branciari [11] and Rhoades [25]. Altun and Torkoglu [2] obtained fixed point theorem for weakly compatible mappings satisfying contractive condition of integral type in Hausdorff d-complete topological spaces by neglecting some possible assumptions in the result of Hicks and Rhoades [17]. On the other hand, Weiss [29] and Butnairu[12] originated the study of fixed point of fuzzy mappings. Thereafter, the idea of fuzzy set-valued mappings was introduced by Heilpern [16], which is a fuzzy generalization of Banach fixed point theorem and Nadler's [24] fixed point theorem for multivalued mappings. After wards, several authors [3,4,5,6,10] studied fixed points of fuzzy mappings satisfying various contractive conditions.

    Along the lane, the applications of mathematics witnessed tremendous developments as a result of the introduction of soft set theory by Molodstov [22]. The method of handling problems in classical mathematics is in the opposite of the technique of soft set theory. In conventional mathematics, to describe any system or object, we first construct its mathematical model and then attempt to obtain the exact solution. If the exact solution is too complicated, then we define the notion of approximate solution. On the other hand, in soft set theory, the initial description of an object takes an approximate nature with no restriction, and the notion of exact solution is not essential. In other words, to describe an object in soft set theory, any convenient parametrization tools which may be words/sentences, numbers, mappings/functions, to mention a few, may be used. Thereby, making the theory more easier and flexible in terms of applications in every day life. In [22], Moldstov highlighted several directions for the applications of soft set, such as smoothness of functions, game theory, Riemann-integration, operation research, probability and so on. At present, researchers have been extending the concepts of soft set into different fields of studies. For example, see [13,15,26] and references therein.

    Not long ago, Shagari and Azam [28] studied the concept of soft set-valued maps and introduced the notions of e-soft fixed points and E-soft fixed point of maps whose range set is a family of soft sets. Consequently, they presented analogues of fixed point theorems of Nadler's and Edelstein's type. Motivated by the ideas of [28], the main aim of this paper is to establish e-soft fixed point theorem of soft set-valued map defined on a complete metric space under some generalized contractive conditions of integral type. In particular, our result extends the main result of Shagari and Azam [28] into integral contractive condition. Consequently, several other well-known results are deduced as corollaries. Moreover, as an application, our main result is used to investigate existence conditions for a solution of nonlinear neutral differential equation.

    In this section, we present preliminary concepts which are relevant to the aims of this paper. These basic concepts are recalled from [4,16,22,28].

    Throughout, N and R represent the set of natural and real numbers, respectively. Let (X,ϱ) be a metric space and X be the set of all nonempty closed and bounded subsets of X. For Θ,ΦX, the Hausdorff metric H on X induced by ϱ is defined as:

    H(Θ,Φ)=max{suplΘϱ(l,Φ),supkΦϱ(Θ,k)},

    where ϱ(l,Θ)=infkΘϱ(l,k).

    The theory of fuzzy set was launched by Zadeh [30]. The notion provides suitable framework for representing and analyzing vague ideas by allowing partial memberships. A fuzzy set Θ in a universal set X is a function with domain X and values in [0,1]=I. For lX, the function value Θ(l) is called the membership value of lX in the fuzzy set Θ. The α-level set of Θ, denoted by [Θ]α, is defined as follows:

    [Θ]α={l:Θ(l)α},ifα(0,1],

    and [Θ]0={l:Θ(l)>α}, is called the strong α-level set.

    Example 2.1. A dealer of certain domestic animals wants to know the different determinants of his customers' satisfaction. An indicated quality shown by a fraction of his customers is `maturity'. If X={1,2,3,4,5,6,7,8,9,10} is the set of all domestic animals, then the fuzzy set Θ={some animals are matured } can be seen graphically in Figure 1.

    Figure 1.  Graphical representation of the fuzzy set in Example 2.1.

    Notice that from Figure 1, the α-level set and strong α-level set for α=0.5, are respectively

    [Θ]α={4,5,7,8,10}and[Θ]α={4,5,8,10}.

    Let ϝX be the collection of all fuzzy sets in a reference set X.

    Definition 2.1. Let X be an arbitrary set and Y a metric space. A mapping Υ:XϝY is called a fuzzy mapping. A fuzzy mapping Υ is a fuzzy subset of X×Y with the grade of membership Υ(l)(k) of k in Υ(l). A point uX is a fuzzy fixed point of Υ if u[Υu]α, for some α[0,1].

    Soft set theory was introduced by Molodstov [22]. The concept provides a parametrization point of view for modeling uncertainty and soft computing. Let X be the universal set and E be the set of all parameters related to the elements of X. In this case, each parameter is a word or sentence. Denote by P(X), the power set of X.

    Definition 2.2.[22] A pair (F,Θ) is called a soft set over X, where ΘE and F is a set-valued mapping F:ΘP(X). In this way, a soft set over X is a parameterized family of subsets of X.

    Example 2.2. Suppose a soft (F,E) gives the section(s) of research papers that is/are more interested to reviewers. Let the universal set of reviewers under consideration be given by

    X={l1,l2,l3,l4,l5}

    and the set of all parameters be represented by

    E={e1,e2,e3,e4}={title,abstract,introduction,results}.

    Define F:EP(X) by F(e1)={l1,l2},F(e2)={l2,l4},F(e3)={l5},F(e4)={l1,l2,l3}. So, the soft set (F,E) is a parameterized family {F(ei):i=1,2,3,4} of P(X).

    Denote by [P(X)]E, the family of soft set over X under E. Consider two soft sets (F,Θ) and (G,Φ), (a,b)Θ×Φ. Assume that F(a),G(b)X. For ϵ>0, define Nϱ(ϵ,F(a)), S(a,b)EX(F,G) and Eϱ(Fa,Gb) as follows:

    Nϱ(ϵ,F(a))={lX:ϱ(l,k)<ϵ,forsomekF(a)}
    Eϱ(Fa,Gb)={ϵ>0:F(a)Nϱ(ϵ,G(b)),G(b)Nϱ(ϵ,F(a))},
    S(a,b)EX(F,G)=infEϱ(Fa,Gb).

    Define a distance function SEX:[P(X)]E×[P(X)]ER by

    SEX(F,G)=sup(a,b)¯ΘׯΦS(a,b)EX(F,G),where
    ¯ΘׯΦ={(a,b)Θ×Φ:F(a),G(b)X}.

    Definition 2.3.[28] A mapping Υ:X[P(X)]E is called a soft set-valued map. A point uX is called an e-soft fixed point of Υ if u(Υu)(a(u)), for some a(u)E. This is also written as uΥu, for short. If DomΥl=E, and u(Υu)(e) for all eE, then u is said to be an E-soft fixed point of Υ. We shall denote the set of all E-soft fixed point of Υ by EFix(Υ).

    Notice that if Υ:X[P(X)]E is a soft set-valued map, then Υl:EP(X) is a soft set, for all lX.

    Example 2.3. Let X=R and E=N. Define Υ:X[P(X)]E by

    (Υl)(e)={l1+l,l2+l},

    for all lX and eE. Then Υ is a soft set-valued map.

    Example 2.4. Let X=R and E={1,3,10}. Define Υ:X[P(X)]E by

    Υl:={(1,{l}),(3,{l2}),(10,{l2,6l})},

    for all lX. Then Υ is a soft set-valued map.

    Example 2.5. Let X={a,b,c,p,q,r} and E={1,2,3}. Define Υ:X[P(X)]E by

    (Υl)(e)={{a,b,r},if e=1{b,p},if e=2{a,b,c,q},if e=3.

    Then Υ is a soft set-valued map. Notice that b(Υb)(e), for all eE and DomΥb=E. Hence, the set of all e-soft fixed point of Υ is given by EFix(Υ)={b}. The soft set-valued in Example 2 can also be seen graphically in Figure 2.

    Figure 2.  Graphical representation of the soft set-valued map in the Example 2.5.

    Definition 2.4. Denote by Ψ, the class of functions φ:[0,)[0,) which satisfy the following conditions:

    (ⅰ) φ is Lebesgue integrable, summable on each compact subset of [0,),

    (ⅱ) φ is nonnegative and such that for each η>0, η0φ(t)ϱt>0.

    In this section, we analyze the existence of e-soft fixed point of soft set-valued map defined on complete metric spaces satisfying some generalized contractive inequalities of integral type.

    Theorem 3.1. Let (X,ϱ) be a complete metric space, E be the parameter set and Υ:X[P(X)]E be a soft set-valued map. Let φΨ and τ:[0,)(0,1) be a decreasing function. Suppose that for lX, there exists a(l)DomΥl such that (Υl)(a(l))X. If for all l,kX,

    S(e(l),e(k))EX(Υl,Υk)0φ(t)ϱtτ(ϱ(l,k))(ϱ(l,k)0φ(t)ϱt) (3.1)

    then there exists uX such that {u}Υu.

    Proof. Let lX be arbitrary and rename it as l:=l0, then by hypothesis, (Υl0)(a(l0)) is a nonempty closed and bounded subset of X. Hence, there exists some l1X such that l1(Υl0)(e(l0)), l0l1. Setting l=l0 and k=l1 in (3.1), we have

    S(e(l0),e(l1))EX(Υl0,Υl1)0φ(t)ϱtτ(ϱ(l0,l1))(ϱ(l0,l1)0ψ(t)dt)<ϱ(l0,l1)0φ(t)ϱt. (3.2)

    For this l1X, by hypothesis, (Υl1)(a(l1))X. Therefore, we can find some l2X such that l2(Υl1)(a(l1)),l1l2. Notice that from (3.1) and (3.2), we have

    ϱ(l1,l2)0φ(t)ϱtS(e(l0),e(l1))EX(Υl0,Υl1)0φ(t)ϱt<ϱ(l0,l1)0φ(t)ϱt,

    which implies

    ϱ(l1,l2)<ϱ(l0,l1).

    Continuing in this fashion, we obtain a sequence {lj}jN such that lj+1(Υlj)(a(lj)) and

    0<ϱ(lj,lj+1)<ϱ(lj1,lj)<<ϱ(l0,l1).

    This shows that the sequence {ϱ(lj,lj+1)}jN is decreasing and bounded below, and thus converges to its infimum, say m. Hence, mϱ(lj,lj+1), for all jN. Suppose m>0, then from (3.1), and the decreasingness of τ, we have

    ϱ(lj,lj+1)0φ(t)ϱtS(e(lj1),e(lj))EX(Υlj1,Υlj)0φ(t)ϱtτ(ϱ(lj1,lj))(ϱ(lj1,lj)0φ(t)ϱt)τ(m)(ϱ(lj1,lj)0φ(t)ϱt). (3.3)

    Taking limit in (3.3) as j, yields

    m0φ(t)ϱtτ(m)(m0φ(t)ϱt)<m0φ(t)ϱt,

    which is a contradiction. So m=0. Next, we shall show that {lj}jN is a Cauchy sequence. Assume contrary, therefore, there exists η>0 and infinitely many pairs (lβ,lk) with ϱ(lβ,lk)η. The subsequence of pairs {(lβp,lkp)}pN, where lp<kp for all pN is chosen to satisfy the condition:

    ϱ(lβp,lkp)η,ϱ(lβp,lt)<η,forallt{lp+2,,kp1}.

    It follows that

    ηϱ(lβp,lkp)ϱ(lβp,lβp1)+ϱ(lβp1,lkp)<η+ϱ(lβp,lβp1). (3.4)

    As p in (3.4), we have

    ηlimpϱ(llp,lkp)η,

    which implies ϱ(lβp,lkp)η as p. Now, considering

    ϱ(lβp,lkp)ϱ(lβp,lβp1)+ϱ(lβp1,lkp1)+ϱ(lkp1,lkp),

    we conclude that

    limpϱ(lβp,lkp)=limpϱ(lβp1,lkp1)=η.

    Therefore, there exists p0N such that for all p>p0,

    ϱ(lβp1,lkp1)η3,whichimplies

    τ(ϱ(lβp1,lkp1))τ(η3) for p>p0. Now, from (3.1), we have

    ϱ(lβp,lkp)0φ(t)ϱtS(e(lβp1),e(lkp1))EX(Υllp1,Υlkp1)0φ(t)ϱtτ(ϱ(lβp1,lkp1))(ϱ(lβp1,lkp1)0φ(t)ϱt)τ(η3)(ϱ(lβp1,lkp1)0φ(t)ϱt). (3.5)

    Letting p in (3.5), yields

    m0φ(t)ϱtτ(η3)(m0φ(t)ϱt)<m0φ(t)ϱt,

    which is a contradiction. Hence, the sequence {lj}jN is Cauchy. The completeness of X implies that there exists uX such that lju as j. Let δj=ϱ(lj,{u}). Then by (3.1),

    S(e(lj),e(u))EX(Υlj,Υu)0φ(t)ϱtτ(ϱ(lj,u))(ϱ(lj,{u})0φ(t)ϱt)<ϱ(lj,{u})0φ(t)ϱt=δj0φ(t)dt,

    that is,

    S(e(lj+1),e(u))EX(lj+1,Υu)0φ(t)ϱt<δj0φ(t)ϱt. (3.6)

    Since δj0 as j, therefore, from (3.6) S(e(lj+1),e(u))EX(lj+1,Tu)=0 as j. Now, using

    S(e(u),e(u))EX({u},Υu)ϱ({u},lj+1)+S(e(lj+1),e(u))EX(lj+1,Υu)=δj+S(e(lj+1),e(u))EX(lj+1,Υu). (3.7)

    As j in (3.7), S(e(u),e(u))EX({u},Υu)=0, which implies {u}Υu.

    Remark 3.1. Theorem 3.1. will fail if we admit zero value almost everywhere near zero for the mapping φ. We illustrate this observation with an example as follows.

    Example 3.1. Let X=E={1,2,3} and

    φ(t)={1+ln(t)4,if t>00,ift[0,2].

    Define a soft set-valued map Υ:X[P(X)]E by

    (Υl)(e)={{1},if l1,eE{2},if l{1,2},eE,ifl,eX.

    Define τ:[0,)(0,1) by τ(t)=λ. Let ϱ:E2[0,) be the Euclidean metric restricted to E so that (E,ϱ) is a complete metric space. We see that for each l,kX and for all eE, S(e(l),e(k))XX(Υl,Υk)2. Consequently,

    S(e(l),e(k))(Υl,Υk)EX0φ(t)ϱt20φ(t)ϱt=0λϱ(l,k)0φ(t)ϱt.

    Thus, Inequality (3.1) is satisfied for all λ(0,1), but Υ has no e-soft fixed point in X. Similarly, one can construct an example to show that the mapping φΨ in Theorem 3.1 cannot admit negative value.

    Next, we provide an example to support the hypotheses of Theorem 3.1.

    Example 3.2. Let X=E={1,2,3} and ϱ:X×XR be defined by

    ϱ(l,k)={0,if l=k1,if lkandl,k{1,3}2,if lkandl,k{1,2}3,if lkandl,k{2,3}.

    Define a soft set-valued map Υ:X[P(X)]E by

    (Υl)(e)={{1},if e=3andl=1{3},if e=2andl=3,if e=1foralllX.

    Take φ(t)=ln(1+t)1+t,t[0,). Clearly, φΨ. It can be seen that for all l,kX, eE, and τ(t)(0,1),t0,

    S(e(l),e(k))EX(Υl,Υk)0φ(t)ϱtτ(t)ϱ(l,k)0ln(1+t)1+tϱtτ(t)2[ln2(1+t)]ϱ(l,k)0.

    Hence, all the conditions of Theorem 3.1. are satisfied. In this case, {1,3} is the set of all e-soft fixed points of Υ in X for e=2,3.

    The following Corollary is the main result of [28,Theorem 3.1] with f=IX, the identity mapping on X.

    Corollary 3.1.[28] Let (X,ϱ) be a complete metric space, E be the parameter set and Υ:X[P(X)]E be a soft set-valued map. Suppose that for lX, there exists a(l)DomΥl such that (Υl)(a(l))X. If there exists λ(0,1) such that for all l,kX,

    S(e(l),e(k))EX(Υl,Υk)λϱ(l,k), (3.8)

    then there exists uX such that uΥu.

    Proof. Define τ:[0,)(0,1) by τ(t)=λ and take φ:=1 in Theorem 3.1.

    Corollary 3.2. Let (X,ϱ) be a complete metric space, E be the parameter set and Υ:X[P(X)]E be a soft set-valued map; let φΨ and τ:[0,)(0,1) be a decreasing function. Suppose that for lX, there exists a(l)DomΥl such that (Υl)(a(l))X. If for all l,kX,

    SEX(Υl,Υk)0φ(t)ϱtτ(ϱ(l,k))(ϱ(l,k)0φ(t)ϱt) (3.9)

    then there exists uX such that {u}Υu.

    Proof. Since

    S(e(l),e(k))EX(Υl,Υk)0φ(t)ϱtSEX(Υl,Υk)0φ(t)ϱt,

    therefore, using (3.10), we get

    S(e(l),e(k))EX(Υl,Υk)0φ(t)ϱtSEX(Υl,Υk)0φ(t)ϱtτ(ϱ(l,k))(ϱ(l,k)0φ(t)ϱt).

    Hence, Theorem 3.1. can be applied to find uX such that {u}Υu.

    Corollary 3.3. Let (X,ϱ) be a complete metric space, E be the parameter set and Υ:X[P(X)]E be a soft set-valued map; let τ:[0,)(0,1) be a decreasing function. Suppose that for lX, there exists a(l)DomΥl such that (Υl)(a(l))X. If for all l,kX,

    S(e(l),e(k))EX(Υl,Υk)τ(ϱ(l,k))(ϱ(l,k)) (3.10)

    then, there exists uX such that {u}Υu.

    Proof. Putφ(t):=1 in Theorem 3.

    Corollary 3.4. Let (X,ϱ) be a complete metric space, E be the parameter set and Υ:X[P(X)]E be a soft set-valued mapping; let τ:[0,)(0,1) be a decreasing function. Suppose that for lX, there exists a(l)DomΥl such that (Υl)(a(l))X. If for all l,kX,

    SEX(Υl,Υk)τ(ϱ(l,k))(ϱ(l,k)) (3.11)

    then there exists uX such that {u}Υu.

    Proof. Set φ(t):=1 in Corollary 3.

    In connection with Theorem 3.1., we have the next result.

    Theorem 3.2. Let (X,ϱ) be a complete metric space, E be the parameter set and Υ:X[P(X)]E be a soft set-valued map; let φΨ and τ:[0,)(0,1) be a decreasing function. Suppose that for lX, there exists a(l)DomΥl such that (Υl)(a(l))X. If for all l,kX,

    S(e(l),e(k))EX(Υl,Υk)0φ(t)ϱtτ(ϱ(l,k))max{ϱ(l,k)0φ(t)ϱt,ϱ(l,Υl)0φ(t)ϱt,ϱ(k,Υk)0φ(t)ϱt}, (3.12)

    then, there exists uX such that {u}Υu.

    Proof. Let l0X and ln+1(Υln)(a(ln)), for some a(ln)E and for all nN. We shall prove that if ln=ln+1 for nN, then there exists uX such that ln=ln+1=u(Υu)(a(u)). Assume that lnln+1, for some nN so that u(Υu)(a(u)), for all a(u)E. Applying Inequality (3.12) with l=ln1 and k=kn, we have

    S(e(ln1),e(ln))(Υln1,Υln)EX0φ(t)dtτ(ϱ(ln1,ln))×max{ϱ(ln1,ln)0φ(t)ϱt,ϱ(ln1,Υln1)0φ(t)ϱt,ϱ(ln,Υln)0φ(t)ϱt}τ(ϱ(ln1,ln))×max{ϱ(ln1,ln)0φ(t)ϱt,ϱ(ln1,ln)0φ(t)ϱt,ϱ(ln,ln+1)0φ(t)ϱt}τ(ϱ(ln1,ln))×max{ϱ(ln1,ln)0φ(t)ϱt,ϱ(ln,ln+1)0φ(t)ϱt}.

    If

    max{ϱ(ln1,ln)0φ(t)ϱt,ϱ(ln,ln+1)0φ(t)ϱt}=ϱ(ln1,ln)0φ(t)ϱt,

    then

    S(e(ln1),e(ln))(Υln1,Υln)EX0φ(t)ϱtτ(ϱ(ln1,ln))ϱ(ln1,ln)0φ(t)ϱt. (3.13)

    From (3.13), Theorem 3.1. can be applied to obtain the conclusion of Theorem 3.2. But suppose

    max{ϱ(ln1,ln)0φ(t)ϱt,ϱ(ln,ln+1)0φ(t)ϱt}=ϱ(ln,ln+1)0φ(t)ϱt.

    Then,

    ϱ(ln,ln+1)0φ(t)ϱtS(e(ln1),e(ln))(Υln1,Υln)EX0φ(t)ϱtτ(ϱ(ln1,ln))ϱ(ln,ln+1)0φ(t)ϱt.<ϱ(ln,ln+1)0φ(t)ϱt,

    gives a contradiction. Consequently, in either case, one obtains u(Υu)(a(u)), for some a(u)E.

    As an application of e-soft fixed point theorem (Theorem 3.1) of the previous section, in this section, we obtain fixed points of fuzzy and multivalued mappings satisfying some integral type conditions of Heilpern and Nadler's type (see, [16,24]).

    Definition 4.1.[16] A fuzzy set Θ in X is called an approximate quantity if and only if its α-level set is a compact convex subset of X for each α[0,1] and suplXΘ(l)=1. The set of all approximate quantities in X is denoted by W(X).

    Theorem 4.1. Let (X,ϱ) be a complete metric space and F:XW(X) be a fuzzy mapping. If there exist λ(0,1) and φΨ such that

    ϱ(Fl,Fk)0φ(t)ϱtλϱ(l,k)0φ(t)ϱt, (4.1)

    for each l,kX, then there exists uX such that {u}F(u).

    Proof. Let τ:[0,)(0,1) be a mapping defined by τ(t)=λ,t0, and E=[0,1] be a parameter set. Consider a soft set-valued map

    ΥF:X[P(X)][0,1],defined by
    ΥFl(1)={tX:(Fl)(t)1}=[Fl]1.

    Then

    S(1,1)EX(ΥFl,ΥFk)0φ(t)ϱt=infEϱ((Υl)(1),(Υk)(1))0φ(t)ϱt=H([Fl]1,[Fk]1)0φ(t)ϱtϱ(Fl,Fk)0φ(t)ϱt.

    By (4.1), we obtain

    S(1,1)EX(ΥFl,ΥFk)0φ(t)ϱtϱ(Fl,Fk)0φ(t)ϱtλϱ(l,k)0φ(t)ϱtτ(t)ϱ(l,k)0φ(t)ϱt.

    Consequently, by Theorem 3.1, the conclusion holds.

    Corollary 4.1. (see [16]) Let (X,ϱ) be a complete metric space and F:XW(X) be a fuzzy mapping. If there exists λ(0,1) such that

    ϱ(Fl,Fk)λϱ(l,k), (4.2)

    for each l,kX, then there exists uX such that {u}F(u).

    Proof. Put φ(t):=1 in Theorem 4.

    Corollary 4.2. Let (X,ϱ) be a complete metric space and Υ:XX be a multivalued mapping satisfying the following conditions: There exist λ(0,1) and φΨ such that

    H(Υl,Υk)0φ(t)ϱtλϱ(l,k)0. (4.3)

    Then there exists uX such that uΥu.

    Proof. Let τ:[0,)(0,1) be a mapping defined by τ(t)=λ,t0. Let E={e1,e2} and consider a soft set-valued map Υ:X[P(X)]E, defined by

    Υl(e)={X,if e=e1Υl,if e=e2.

    Then,

    S(e2(l),e2(k))(Υl,Υk)EX0φ(t)ϱt=infEϱ((Υl)(e2(l)),(Υk)(e2(k)))0φ(t)ϱt=infEϱ(Υl,Υk)0φ(t)ϱtH(Υl,Υk)0φ(t)ϱt.

    Using (4.3), the last inequality can be written as

    S(e2(l),e2(k))EX0φ(t)ϱtH(Υl,Υk)0φ(t)ϱtλd(l,k)0φ(t)ϱtτ(t)ϱ(l,k)0φ(t)ϱt.

    Thus, the conclusion follows by Theorem 3.

    Corollary 4.3. (see [24]) Let (X,ϱ) be a complete metric space and Υ:XX be a multivalued mapping satisfying the following conditions: there exist λ(0,1) such that

    H(Υl,Υk)λϱ(l,k). (4.4)

    Then, there exists uX such that uΥu.

    Proof. Set φ(t):=1 in Theorem 4.

    Consider the nonlinear neutral delay differential equation with an unbounded delay:

    l(t)=a(t)l(t)+b(t)l2(tr(t))+c(t)l(t)(tr(t))l(tr(t)), (5.1)

    where a(t),b(t) are continuous, c(t) is continuously differentiable and r(t)>0 for all tR and is twice continuously differentiable. For details in this direction, the interested reader may consult [8,14].

    Lemma 5.1.[14] Suppose that r(t)1 for all tR. Then l(t) is a solution of (5.1) if and only if

    l(t)=(l(0)12c(0)1r(0)l2(r(0)))et0a(s)ϱs+12l2(tr(t))c(t)1r(t)12t0(h(u)2b(u))l2(ur(u))etua(s)ϱsϱu, (5.2)

    where

    h(u)=r(u)c(u)+(c(u)+c(u)a(u)(1r(u)))(1r(u))2.

    Let ρ:(,0]R be a given continuous function. We say that l(t) is a solution of (5.1) if it satisfies (5.1) for t0 and l(t)=ρ(t) for t0. Denote by C(R,R), the space of all continuous functions from R to R and define the set Mρ by

    Mρ={l:RR,l(t)=ρ(t),t0,lC(R,R)}.

    Then, equipped with the uniform norm ., Mρ is a Banach space.

    Theorem 5.1. Consider Eqs. (5.1) and (5.2). Suppose the following conditions hold:

    (i) there exists L>0 such that lL, for all lMρ;

    (ii) there exists λ(0,1) such that

    (|c(t)1r(t)|+t0|h(u)2b(u)|etua(s)ϱu)(Llk)0φ(t)ϱtλlk0φ(t)ϱt;

    then problem (5.1) has a solution in Mρ.

    Proof. Let E=(0,) and X=Mρ. Then under the Chebyshev metric ϱ induced by the uniform norm, (X,ϱ) is a complete metric space. Consider an arbitrary function fl:XX defined by

    fl(t)=(l(0)12c(0)1r(0)l2(r(0)))et0a(s)ϱs+12l2(tr(t))c(t)1r(t)12t0(h(u)2b(u))l2(ur(u))etua(s)ϱsdu. (5.3)

    Let τ:[0,)(0,1) be given by τ(t)=λ,t0. Define a soft set-valued map Υ:X[Υ(X)]E by

    (Υl)(e)={{lX:lfl(t)},if 0<e<1100{lX:l(t)=fl(t)},if e=1100if e>1100.

    Then for lX, there exists eE such that

    (Υl)(e)={fl(t)}.

    Now, for l,kX with lk, we have

    S(e(l),e(k))EX(Υl,Υk)=infEϱ((Υl)(e),(Υk)(e))=|flfk|.

    Therefore,

    S(e(l),e(k))EX(Υl,Υk)0φ(t)ϱt=|flfk|0φ(t)ϱt12{|c(t)1r(t)|+|h(u)2b(u)|etuϱsϱu0}l2k20φ(t)ϱt12(2L){|c(t)1r(t)|+|h(u)2b(u)|etuϱsϱu0}lk0φ(t)ϱt{|c(t)1r(t)|+|h(u)2b(u)|etuϱsϱu0}(Llk)0φ(t)ϱtλlk0φ(t)ϱt.

    Consequently, Theorem 3.1. implies that there exists uX such that {u}Υu, which solves problem (5.1).

    Two classical theorems involving fixed points are Banach and Brouwer's Theorems. Banach fixed point theorem states that if X is a complete metric space and Υ is a contraction on X, then Υ has a unique fixed point in X. In Brouwer's fixed point theorem, X is required to be a closed unit ball in a Euclidean space. Then, any contraction Υ on X has a fixed point. But in this case, uniqueness of fixed point is not guaranteed. In Banach theorem, a metric on X is used with the assumption that Υ is a contraction. The unit ball in a Euclidean space is also a metric space and the metric topology determines the continuity of continuous functions, however, the main idea of Brouwer's theorem is a topological property of the unit ball, namely, the unit ball is compact and contractible. Banach theorem and Brouwer theorem tell us a difference between two main branches of fixed point theory, metric fixed point theory and topological fixed point theory. It is not easy to differentiate two fixed point theories in an exact way, or to determine a certain topics belonging to which branch. Generally, fixed point theory is regarded as a branch of topology. But due to deep influence on topics related to nonlinear analysis or dynamic systems, many areas of fixed point theory can be thought of as a branch of analysis.

    In the setting of metric fixed point theory, here in this paper, fixed point theorems of soft-valued maps via some integral type contractive conditions are proved. As a consequence of our main result, some fixed point theorems in the framework of fuzzy set-valued and multivalued mappings are derived. Following the fundamental role of Banach fixed point theorem in the study of existence and uniqueness of solutions of equations, we investigated some conditions for the existence of solution of nonlinear neutral differential equations. We hope that the presented ideas herein will motivate the interested researcher(s) and hence, extend the concepts to other related areas such as topological fixed point theory. Moreover, the soft set component of this work can also be extended to other models such as N-soft set, fuzzy soft set, intuitionistic fuzzy soft set, intuitionistic neutrosophic soft set, rough sets, and so on.

    The authors are thankful to the editors and the anonymous reviewers for their valuable suggestions and comments on the manuscript.The first author gratefully acknowledges with thanks the World Academy of Science (TWAS), Italy, and COMSATS University, Islamabad, Pakistan, for providing him with full-time postgraduate fellowship award (FR: 3240293231).

    The authors declare that they have no competing interests.



    [1] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. http://dx.doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
    [2] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
    [3] X. J. Yang, H. M. Srivastava, J. A. Tenreiro Machado, A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Therm. Sci., 20 (2016), 753–756. https://doi.org/10.2298/TSCI151224222Y doi: 10.2298/TSCI151224222Y
    [4] S. T. Sutar, K. D. Kucche, On nonlinear hybrid fractional diferential equations with Atangana-Baleanu-Caputo derivative, Chaos Soliton. Fract., 143 (2021), 110557. https://doi.org/10.1016/j.chaos.2020.110557 doi: 10.1016/j.chaos.2020.110557
    [5] X. M. Gu, H. W. Sun, Y. L. Zhao, X. Zheng, An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order, Appl. Math. Lett., 120 (2021), 107270. https://doi.org/10.1016/j.aml.2021.107270 doi: 10.1016/j.aml.2021.107270
    [6] M. Hassouna, E. H. El Kinani, A. Ouhadan, Global existence and uniqueness of solution of Atangana-Baleanu-Caputo fractional differential equation with nonlinear term and approximate solutions, Int. J. Differ. Equations, 2021 (2021), 5675789. https://doi.org/10.1155/2021/5675789 doi: 10.1155/2021/5675789
    [7] J. Gómez-Aguilar, R. Escobar-Jiménez, M. López-López, V. Alvarado-Martínez, Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media, J. Electromagn. Waves Appl., 30 (2016), 1937–1952. https://doi.org/10.1080/09205071.2016.1225521 doi: 10.1080/09205071.2016.1225521
    [8] S. Ullah, M. A. Khan, M. Farooq, Modeling and analysis of the fractional HBV model with Atangana-Baleanu derivative, Eur. Phys. J. Plus, 133 (2018), 313. https://doi.org/10.1140/epjp/i2018-12120-1 doi: 10.1140/epjp/i2018-12120-1
    [9] O. J. Peter, A. S. Shaikh, M. O. Ibrahim, K. S. Nisar, D. Baleanu, I. Khan, et al., Analysis and dynamics of fractional order mathematical model of covid-19 in Nigeria using Atangana-Baleanu operator, Comput. Mater. Con., 66 (2020), 1823–1848. http://dx.doi.org/10.32604/cmc.2020.012314 doi: 10.32604/cmc.2020.012314
    [10] C. F. Lorenzo, T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn., 29 (2002), 57–98. https://doi.org/10.1023/A:1016586905654 doi: 10.1023/A:1016586905654
    [11] H. Sun, W. Chen, H. Wei, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 193 (2011), 185. https://doi.org/10.1140/epjst/e2011-01390-6 doi: 10.1140/epjst/e2011-01390-6
    [12] M. A. Abdelkawy, M. A. Zaky, A. H. Bhrawy, D. Baleanu, Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom. Rep. Phys., 67 (2015), 773–791.
    [13] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods in fluid dynamics, Springer, Berlin, 1987.
    [14] J. Solís-Pérez, J. Gómez-Aguilar, A. Atangana, Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws, Chaos Soliton. Fract., 114 (2018), 175–185. https://doi.org/10.1016/j.chaos.2018.06.032 doi: 10.1016/j.chaos.2018.06.032
    [15] X. Li, Y. Gao, B. Wu, Approximate solutions of Atangana-Baleanu variable order fractional problems, AIMS Math., 5 (2020), 2285–2294. https://doi.org/10.3934/math.2020151 doi: 10.3934/math.2020151
    [16] M. H. Heydari, Z. Avazzadeh, A. Atangana, Shifted Jacobi polynomials for nonlinear singular variable-order time fractional Emden-Fowler equation generated by derivative with non-singular kernel, Adv. Differ. Equations, 2021 (2021), 188. https://doi.org/10.1186/s13662-021-03349-1 doi: 10.1186/s13662-021-03349-1
    [17] A. H. Bhrawy, M. A. Zaky, Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dyn., 80 (2015), 101–116. https://doi.org/10.1007/s11071-014-1854-7 doi: 10.1007/s11071-014-1854-7
    [18] T. El-Gindy, H. Ahmed, M. Melad, Shifted Gegenbauer operational matrix and its applications for solving fractional differential equations, J. Egypt. Math. Soc., 26 (2018), 72–90. https://doi.org/10.21608/JOMES.2018.9463 doi: 10.21608/JOMES.2018.9463
    [19] M. Usman, M. Hamid, T. Zubair, R. U. Haq, W. Wang, M. Liu, Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials, Appl. Math. Comput., 372 (2020), 124985. https://doi.org/10.1016/j.amc.2019.124985 doi: 10.1016/j.amc.2019.124985
    [20] F. Soufivand, F. Soltanian, K. Mamehrashi, An operational matrix method based on the Gegenbauer polynomials for solving a class of fractional optimal control problems, Int. J. Industrial Electron. Control Optim., 4 (2021), 475–484. https://doi.org/10.22111/IECO.2021.39546.1371 doi: 10.22111/IECO.2021.39546.1371
    [21] M. Heydari, A. Atangana, A cardinal approach for nonlinear variable-order time fractional schr¨odinger equation defined by Atangana-Baleanu-Caputo derivative, Chaos Soliton. Fract., 128 (2019), 339–348. https://doi.org/10.1016/j.chaos.2019.08.009 doi: 10.1016/j.chaos.2019.08.009
    [22] Y. V. Mukhartova, M. A. Davydova, N. F. Elansky, O. V. Postylyakov, S. A. Zakharova, A. N. Borovski, On application of nonlinear reaction-diffusion-advection models to simulation of transport of chemically-active impurities, Remote Sensing Technologies and Applications in Urban Environments IV, 11157 (2019), 180–187. https://doi.org/10.1117/12.2535489 doi: 10.1117/12.2535489
    [23] F. Heße, F. A. Radu, M. Thullner, S. Attinger, Upscaling of the advection-diffusion-reaction equation with Monod reaction, Adv. Water Resour., 32 (2009), 1336–1351. https://doi.org/10.1016/j.advwatres.2009.05.009 doi: 10.1016/j.advwatres.2009.05.009
    [24] A. Hamdi, Identification of point sources in two-dimensional advection-diffusion-reaction equation: Application to pollution sources in a river. Stationary case, Inverse Probl. Sci. Eng., 15 (2007), 855–870. https://doi.org/10.1080/17415970601162198 doi: 10.1080/17415970601162198
    [25] A. Rubio, A. Zalts, C. El Hasi, Numerical solution of the advection-reaction-diffusion equation at different scales, Environ. Modell. Softw., 23 (2008), 90–95. https://doi.org/10.1016/j.envsoft.2007.05.009 doi: 10.1016/j.envsoft.2007.05.009
    [26] K. Issa, B. M. Yisa, J. Biazar, Numerical solution of space fractional diffusion equation using shifted Gegenbauer polynomials, Comput. Methods Differ. Equations, 10 (2022), 431–444. https://dx.doi.org/10.22034/cmde.2020.42106.1818 doi: 10.22034/cmde.2020.42106.1818
    [27] U. Ali, A. Iqbal, M. Sohail, F. A. Abdullah, Z. Khan, Compact implicit difference approximation for time-fractional diffusion-wave equation, Alex. Eng. J., 61 (2022), 4119–4126. https://doi.org/10.1016/j.aej.2021.09.005 doi: 10.1016/j.aej.2021.09.005
    [28] M. A. Zaky, S. S. Ezz-Eldien, E. H. Doha, J. A. Tenreiro Machado, A. H. Bhrawy, An efficient operational matrix technique for multidimensional variable-Order time fractional diffusion equations, ASME J. Comput. Nonlinear Dyn., 11 (2016), 061002. https://doi.org/10.1115/1.4033723 doi: 10.1115/1.4033723
    [29] M. M. Izadkhah, J. Saberi-Nadjafi, Gegenbauer spectral method for time-fractional convection-difffusion equations with variable coefficients, Math. Methods Appl. Sci., 38 (2015), 3183–3194. https://doi.org/10.1002/mma.3289 doi: 10.1002/mma.3289
    [30] M. H. Heydari, A. Atangana, Z. Avazzadeh, M. R. Mahmoudi, An operational matrix method for nonlinear variable-order time fractional reaction-diffusion equation involving Mittag-Leffler kernel, Eur. Phys. J. Plus, 135 (2020), 237. https://doi.org/10.1140/epjp/s13360-020-00158-5 doi: 10.1140/epjp/s13360-020-00158-5
    [31] P. Pandey, S. Kumar, J. Gˊomez-Aguilar, Numerical solution of the time fractional reaction-advection-diffusion equation in porous media, J. Appl. Comput. Mech., 8 (2022), 84–96. https://doi.org/10.22055/JACM.2019.30946.1796 doi: 10.22055/JACM.2019.30946.1796
    [32] S. Kumar, D. Zeidan, An efficient Mittag-Leffler kernel approach for time-fractional advection-reaction-diffusion equation, Appl. Numer. Math., 170 (2021), 190–207. https://doi.org/10.1016/j.apnum.2021.07.025 doi: 10.1016/j.apnum.2021.07.025
    [33] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
    [34] M. Hosseininia, M. H. Heydari, Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2d reaction-diffusion equation involving Mittag-Leffler non-singular kernel, Chaos Soliton. Fract., 127 (2019), 400–407. https://doi.org/10.1016/j.chaos.2019.07.017 doi: 10.1016/j.chaos.2019.07.017
    [35] F. R. Lin, H. Qu, A Runge-Kutta Gegenbauer spectral method for nonlinear fractional differential equations with Riesz fractional derivatives, Int. J. Comput. Math., 96 (2018), 417–435. https://doi.org/10.1080/00207160.2018.1487059 doi: 10.1080/00207160.2018.1487059
    [36] H. Tajadodi, A numerical approach of fractional advection-diffusion equation with Atangana-Baleanu derivative, Chaos Soliton. Fract., 130 (2020), 109527. https://doi.org/10.1016/j.chaos.2019.109527 doi: 10.1016/j.chaos.2019.109527
    [37] S. Yadav, R. K. Pandey, Numerical approximation of fractional Burgers equation with Atangana-Baleanu derivative in Caputo sense, Chaos Soliton. Fract., 133 (2020), 109630. https://doi.org/10.1016/j.chaos.2020.109630 doi: 10.1016/j.chaos.2020.109630
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