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New extension to fuzzy dynamic system and fuzzy fixed point results with an application

  • Received: 19 August 2022 Revised: 24 September 2022 Accepted: 03 October 2022 Published: 18 October 2022
  • MSC : 46T99, 47H10, 54H25

  • In this paper we introduce the notion of fuzzy dynamic system in b -metric-like space. By applying this, discuss some new refinements of the F-fuzzy Suzuki-type fixed point results for the fuzzy operators are presented. Also, establish the concept fuzzy dynamic system instead of the Piscard iterative sequence, which improves the existing results for such analysis as those presented here. Includes some tangible instances and an application are given to highlight the usability and validity of the theoretical results.

    Citation: Amjad Ali, Eskandar Ameer, Suhad Subhi Aiadi, Muhammad Tariq, Muhammad Arshad, Nabil Mlaiki, Wasfi Shatanawi. New extension to fuzzy dynamic system and fuzzy fixed point results with an application[J]. AIMS Mathematics, 2023, 8(1): 1208-1229. doi: 10.3934/math.2023061

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  • In this paper we introduce the notion of fuzzy dynamic system in b -metric-like space. By applying this, discuss some new refinements of the F-fuzzy Suzuki-type fixed point results for the fuzzy operators are presented. Also, establish the concept fuzzy dynamic system instead of the Piscard iterative sequence, which improves the existing results for such analysis as those presented here. Includes some tangible instances and an application are given to highlight the usability and validity of the theoretical results.



    The mathematical modelling of the collective motion through aggregation/diffusion phenomena has raised a lot of interest in the recent years and it has been deeply studied for its application in several areas, such as biology [9, 39, 47, 48], ecology [35, 42, 43], animal swarming [3, 4, 40, 46] sociology and economics, [49, 50, 51]. One of the common idea in this modelling approach is that a certain population composed by agents evolves according to long-range attraction and short-range repulsion forces between agents. We are interested in modelling the problem of predator-prey interactions, namely we consider two populations that attract (prey) and repel (predators) each others. The pioneering works for this problem are the ones by Lotka, [36] and Volterra[54], which describe the predator-prey interaction via reaction terms in a set of differential equations, possibly combined with diffusion terms, see [41] and the references therein.

    As in [21], in this paper we model predator-prey interactions via a transport terms rather than a reaction ones as follows: consider N predators located at X1,,XNRn, and M prey at Y1,,YMRn with masses miX>0 and miY>0 respectively. We assume that each agent of the same species interacts under the effect of a radial non-local force that is attractive in the long range and repulsive in the short range. Moreover, predators are attracted by the prey, while the latter are subject to a repulsive force from the predators, that is proportional to the previous one. This set of modelling assumptions leads to the following system of ODEs:

    {˙Xi(t)=Nk=1mkX(Sr1(Xi(t)Xk(t))+Sa1(Xi(t)Xk(t)))Mh=1mhYK(Xi(t)Yh(t)),˙Yj(t)=Mh=1mhY(Sr2(Xi(t)Xk(t))+Sa2(Yj(t)Yh(t)))+αMk=1mkXK(Yj(t)Xk(t)), (1)

    with i=1,,N and j=1,,M. The potentials Sa1 and Sa2 are called self-interaction and model the long-range attraction among agents of the same species. The potential K is responsible for the predator-prey interaction, and it is called cross-interaction potential. The coefficient α>0 models the escape propensity of prey from the predators. The short-range repulsion among particles of the same species is modelled by the non-local forces Sr1 and Sr2, and we can assume that it scales with the number of particles, Sri(z)=NβS(Nβ/nz) for a smooth functional S, see [40].

    The formal limit when the number of particles tends to infinity leads to the following system of partial differential equations

    {tρ=div(ρ(d1ρSa1ρKη)),tη=div(η(d2ηSa2η+αKρ)), (2)

    where ρ and η are the densities of predators and prey respectively. Through this limit the (non-local) short-range repulsion formally turns to a (local) nonlinear diffusion terms, being d1 and d2 positive constants modelling the spreading velocity, while the long-range attraction takes into account the non-local self-interactions. We can therefore lighten the notation by calling Sa1=Sρ and Sa2=Sη.

    The goal of this paper is to show that the model above catches one of the main features that occur in nature, namely the formation of spatial patterns where the predators are surrounded of empty zones and the prey aggregates around, that is usually observed in fish schools or in flock of sheeps, see [32, 38]. In the fully aggregative case, namely system (2) with d1=d2=0, the formation of these types of patterns has been studied in several papers, see [15, 27, 21, 46] and references therein.

    Existence theory for solutions to system of the form (2) can be performed in the spirit of [10, 20]. In particular, system (2) should be framed in the context of non symmetrizable systems, for which the Wasserstein gradient flow theory developed in [1] and adapted to systems in [22] does not work. In [10, 20, 22], the authors consider different choices for the diffusion part (no diffusion in [22], diagonal nonlinear diffusion in [10] and cross-diffusion with dominant diagonal in [20]), and the existence of solutions is proved via an implicit-explicit version of the JKO scheme [33].

    In the following, we reduce our analysis to the one-dimensional setting

    {tρ=x(ρx(d1ρSρρKη))tη=x(ηx(d2ηSηη+αKρ)). (3)

    We are interested in the existence of stationary solutions to (3), which are solutions to the following system

    {0=(ρ(d1ρSρρKη)x)x,0=(η(d2ηSηη+αKρ)x)x, (4)

    as well as their properties, e.g., symmetry, compact support, etc. We shall assume throughout the paper that d1=d2=θ for simplicity. Note that we can always assume this by a simple scaling argument. Indeed, it is enough to multiply the first equation in (4) by d2/d1, and setting d2=θ, ˜Sρ=d2d1Sρ, ˜K=d2d1K and ˜α=d1d2α to get

    {0=(ρ(θρ˜Sρρ˜Kη)x)x,0=(η(θηSηη+˜α˜Kρ)x)x.

    The stationary equation for the one species case, i.e.,

    tρ=x(ρx(θρSρ))

    is studied several papers, see [2, 7, 11, 16] and therein references. In [7] the Krein-Rutman theorem is used in order to characterise the steady states as eigenvectors of a certain non-local operator. The authors prove that a unique steady state with given mass and centre of mass exists provided that θ<KL1, and it exhibits a shape similar to a Barenblatt profile for the porous medium equation; see [52] and [24] for the local stability analysis. Similar techniques are used in [8] in order to partly extend the result to more general nonlinear diffusion, see also [34]. This approach is used in [6] in order to explore the formation of segregated stationary states for a system similar to (3) but in presence of cross-diffusion. Unfortunately, when dealing with systems, it is not possible to reproduce one of the major issues solved in [7], namely the one-to-one correspondence between the diffusion constant (eigenvalue) and the support of the steady state. A support-independent existence result for small diffusion coefficient θ is obtained in [6] by using the generalised version of the implicit function theorem, see also [5] where this approach is used in the one species case.

    In this paper we apply the aforementioned approach in order to show that stationary solutions to (3) are composed of multiple Barenblatt profiles. Let us introduce, for fixed zρ,zη>0, the following space

    M={(ρ,η)(LL1(R))2:ρ,η0,ρL1=zρ,ηL1=zη}.

    Definition 1.1. We say that a pair (ρ,η)M is a multiple bumps steady state to (3) if the pair (ρ,η) solves (4) weakly and there exist two numbers Nρ,NηN, and two families of intervals Iiρ=[liρ,riρ], for i=1,...,Nρ, and Ihη=[lhη,rhη], for h=1,...,Nη such that

    IiρIjρ=, for i,j=1,...,Nρ, ij and IhηIkη=, for h,k=1,...,Nη, hk,

    ρ and η are supported on

    supp(ρ)=Nρi=1Iiρandsupp(η)=Nηi=1Iiη,

    respectively and

    where, for i=1,...,Nρ and h=1,...,Nη, ρi and ηh are even w.r.t the centres of Iiρ and Ihη respectively, non-negative and C1 functions supported on that intervals.

    Example 1.1. A possible example of steady states as defined above is a steady state (ρ,η) consisting of three bumps for each one of the pair (ρ,η) (Nρ=Nη=3), namely, ρ1,ρ2,ρ3 and η1,η2,η3 respectively, with centers of masses {cmiρ:=riρliρ}Nρi=1 and {cmiη:=riηliη}Nηi=1 as solutions to system (17). A plot is carried out in Figure 1 which shows all the properties stated in Definition 1.1.

    Figure 1. 

    A possible example of a stationary solution to (3) with Nρ=Nη=3 is plotted as described in Definition 1.1

    .

    Remark 1.1. We remark that one should be careful with finding a shape of steady state. More precisely, one should choose the numbers Nρ,Nη and the centers of masses {cmiρ}Nρi=1 and {cmiη}Nηi=1 such that all the conditions required for the existence of steady states are satisfied, see Theorem 1.1.

    In order to simplify the notations, in the following we will denote with l{ρ,η} a generic index that recognise one of the two families. Throughout the paper we shall assume that the kernels satisfy the following:

    (A1) Sρ, Sη and K are C2(R) functions.

    (A2) Sρ, Sη and K are radially symmetric and decreasing w.r.t. the radial variable.

    (A3) Sρ, Sη and K are non-negative, with finite L1-norm supported on the whole real line R.

    Note that assumption (A2) together with the sign in front of the nonlocal terms Sρ and K (in the first equation) and Sη in system (3), give the effect of an attractive potential, i.e. a radial interaction potential G(x)=g(|x|) for some g:[0,+)R, such that g(r)>0 for r>0. For K in the second equation we obtain the effect of a repulsive potential, i.e. g(r)<0 for r>0.

    The main result of the paper is the following

    Theorem 1.1. Assume that Sρ, Sη and K are interaction kernels are under the assumptions (A1), (A2) and (A3). Consider Nρ,NηN and let zil be fixed positive numbers for i=1,2,,Nl, and l{ρ,η}. Consider two families of real numbers {cmiρ}Nρi=1 and {cmiη}Nηi=1 such that

    (i) {cmiρ}Nρi=1 and {cmiη}Nηi=1 are stationary solutions of the purely non-local particle system, that is, for i=1,2,,Nl, for l,h{ρ,η} and lh,

    Bil=Nlj=1Sl(cmilcmjl)zjl+αlNhj=1K(cmilcmjh)zjh=0, (5)

    (ii) the following quantities

    Dil=Nlj=1Sl(cmilcmjl)zjlαlNhj=1K(cmilcmjh)zjh, (6)

    are strictly positive, for all i=1,2,,Nl, l,h{ρ,η} and lh.

    where αρ=1 and αη=α. Then, there exists a constant θ0 such that for all θ(0,θ0) the stationary equation (4) admits a unique solution in the sense of Definition 1.1 of the form

    where

    each interval Iil is symmetric around cmil for all i=1,2,,Nl, l{ρ,η},

    ρi and ηj are C1, non-negative and even w.r.t the centres of Iiρ and Ijη respectively, with masses ziρ and zjη, for i=1,...,Nρ and j=1,...,Nη,

    the solutions ρ and η have fixed masses

    zρ=Nρi=1ziρandzη=Nηi=1ziη,

    respectively.

    The paper is structured as follows. In Section 2 we recall the basics notions on optimal transport and we introduce the p-Wasserstein distances in spaces of probability measures. Then, we recall the strategy for proving existence to systems of the form (2). The remaining part of the Section is devoted to a preliminary and partial existence analysis of steady states via the Krein-Rutman theorem of a particular type of stationary solutions that we call mixed steady state. Section 3 is devoted to the proof of Theorem 1.1 in which existence and uniqueness results for multiple bumps stationary solutions are proved in case of small diffusion coefficient using the Implicit Function Theorem. We conclude the paper with Section 4, complementing our results with numerical simulations that also show interesting stability issues of the stationary states, namely transitions between states and others effects such as traveling waves profiles.

    We start collecting preliminary concepts on the Wasserstein distance. Let P(Rn) be the space of probability measures on Rn and fix p[1,+). The space of probability measures with finite p-moment is defined by

    Pp(Rn)={μP(Rn):mp(μ)=Rn|x|pdμ(x)<}.

    For a measure μP(Rn) and a Borel map T:RnRk, denote with T#μP(Rn) the push-forward of μ through T, defined by

    Rkf(y)dT#μ(y)=Rnf(T(x))dμ(x)forallfBorelfunctionsonRk.

    We endow the space Pp(Rn) with the Wasserstein distance, see for instance [1, 45, 53]

    Wpp(μ,ν)=infγΓ(μ,ν){Rn×Rn|xy|pdγ(x,y)}, (7)

    where Γ(μ1,μ2) is the class of transport plans between μ and ν, that is the class of measures γP(Rn×Rn) such that, denoting by πi the projection operator on the i-th component of the product space, the marginality condition πi#γ=μi i=1,2 is satisfied.

    Since we are working in a 'multi-species' structure, we consider the product space Pp(Rn)×Pp(Rn) endowed with a product structure. In the following we shall use bold symbols to denote elements in a product space. For a p[1,+], we use the notation

    Wpp(μ,ν)=Wpp(μ1,ν1)+Wpp(μ2,ν2),

    with μ=(μ1,μ2),ν=(ν1,ν2)Pp(Rn)×Pp(Rn). In the one-dimensional case, given μP(R), we introduce the pseudo-inverse variable uμL1([0,1];R) as

    uμ(z)inf{xR:μ((,x])>z},z[0,1], (8)

    see [14].

    In the Section 1 we mentioned that the well-posedness of (3) can be stated according to the results in [10, 20] in an arbitrary space dimension n. In these papers, the existence of weak solutions is provided using an implicit-explicit version of the Jordan-Kinderlehrer-Otto (JKO) scheme [33, 22], that we will sketch in the following. A key point in this approach is to associate to (3) a relative energy functional

    F[μ,ν](ρ,η)=θ2Rnρ2+η2dx12RnρSρρdx12RnηSηηdxRnρKμdx+αRnηKνdx,

    for a fixed reference couple of measures (μ,ν). We state our definition of weak measure solution for (3), in the space P2(Rn)2:=Pp(Rn)×Pp(Rn).

    Definition 2.1. A curve μ=(ρ(),η()):[0,+)P2(Rn)2 is a weak solution to (3) if

    (i) ρ,ηL2([0,T]×Rn) for all T>0, and ρ,ηL2([0,+)×Rn) for i=1,2,

    (ii) for almost every t[0,+) and for all ϕ,φCc(Rn), we have

    ddtRnϕρdx=θRnρρϕdx+Rnρ(Sρρ+Kη)ϕdx,ddtRnφηdx=θRnηηφdx+Rnη(SηηαKρ)ϕdx.

    Theorem 2.1. Assume that (A1)-(A3) are satisfied. Let μ0=(ρ1,0,ρ2,0)P2(Rn)2 such that

    F[μ0](μ0)<+.

    Then, there exists a weak solution to (3) in the sense of Definition 2.1 with μ(0)=μ0.

    As already mentioned the proof of Theorem 2.1 is a special case of the results in [10, 20] and consists in the following main steps:

    1. Let τ>0 be a fixed time step and consider the initial datum μ0P2(Rn)2, such that F[μ0](μ0)<+. Define a sequence {μkτ}kN recursively: μ0τ=μ0 and, for a given μkτP2(Rn)2 with k0, we choose μk+1τ as follows:

    μk+1τargminμP2(Rn)2{12τW22(μkτ,μ)+F[μkτ](μ)}. (9)

    For a fixed T>0, set N:=[Tτ], and

    μτ(t)=(ρτ(t),ητ(t))=μkτt((k1)τ,kτ],kN,

    with μkτ defined in (9).

    2. There exists an absolutely continuous curve ˜μ:[0,T]P2(Rn)2 such that the piecewise constant interpolation μτ admits a sub-sequence μτh narrowly converging to ˜μ uniformly in t[0,T] as h+. This is a standard result coming from the minimising condition (9).

    3. There exist a constant C>0 such that

    T0[||ρτ(t,)||2H1(Rn)+||ητ(t,)||2H1(Rn)]dtC(T,μ0), (10)

    and the sequence μτh:[0,+)P2(Rn)2 converges to ˜μ strongly in

    L2((0,T)×Rn)×L2((0,T)×Rn).

    The estimate in (10) can be deduced by using the so called Flow-interchange Lemma introduced in [37], see also [25]. In order to deduce the strong convergence we use the extended version of the Aubin-Lions Lemma in [44].

    4. The approximating sequence μτh converges to a weak solution ˜μ to (3). This can be showed considering two consecutive steps in the semi-implicit JKO scheme (9), i.e. μkτ, μk+1τ, and perturbing in the following way

    μϵ=(ρϵ,ηϵ)=(Pϵ#ρk+1τ,ηkτ), (11)

    where Pϵ=id+ϵζ, for some ζCc(Rn;Rn) and ϵ0. From the minimizing property of μk+1τ we have

    012τ[W22(μk+1τ,μϵ)W22(μkτ,μk+1τ)]+F[μkτ](μϵ)F[μkτ](μϵ). (12)

    After some manipulations, sending first ϵ0 and then τ0 the inequality (12) leads to the first weak formulation in Definition 2.1. Perturbing now on η and repeating the same procedure we get the required convergence.

    The existence of weak solutions to the purely non-local systems, i.e.,

    {tρ=div(ρ(Sρρ+Kρη)),tη=div(η(Sηη+Kηρ)), (13)

    with generic cross-interaction kernels Kρ and Kη is investigated in [22], whereas studies on the shape of stationary states can be found in [17, 27]. Concerning the predator-prey modelling and patterns formation, in [15, 46] a minimal version of (1) has been considered with only one predator and arbitrarily many prey subject to (different) singular potentials. This model induces the formation of nontrivial patterns in some way to prevent the action of the predators. In [21] the authors study existence and stability of stationary states for the purely aggregative version of system (3), namely equation (3) with θ=0,

    {tρ=div(ρ(Sρρ+Kη)),tη=div(η(SηηαKρ)), (14)

    and its relation with the particle system

    {˙Xi(t)=Nk=1mkXSρ(Xi(t)Xk(t))Mk=1mkYK(Xi(t)Yk(t)),˙Yj(t)=Mk=1mkYSη(Yj(t)Yk(t))+αNk=1mkXK(Yj(t)Xk(t)). (15)

    It is proved that stationary states of system (14) are linear combinations of Dirac's deltas, namely ˉρ,ˉηP(Rn), with

    (ˉρ,ˉη)=(Nk=1mkXδˉXk(x),Mh=1mhYδˉYh(x)). (16)

    where {ˉXk}k, {ˉYh}h are stationary solutions of system (15), i.e.,

    {0=Nk=1Sρ(ˉXkˉXi)mkX+Mh=1K(ˉYhˉXi)mhY0=Mh=1Sη(ˉYhˉYj)mhYαNk=1K(ˉXkˉYj)mkX. (17)

    for i=1,...,N and j=1,...,M, see also [29, 30] for a symilar result in the one-species case. As pointed out in [21], system (17) is not enough to determine a unique steady state, since the linear combination of the first N equations, weighted with αmiX, and the final M equations weighted with coefficients mjY get the trivial identity 0=0. System (17) should be coupled with the quantity

    Cα=αNi=1miXXiMj=1mjYYj (18)

    that is a conserved quantities, and therefore one would like to produce a unique steady state once the quantity Cα is prescribed. Solutions to system (17) will play a crucial role in the proof of the main Theorem 1.1.

    As a preliminary result, we now prove the existence of one possible shape of steady state, that will be a prototype example for the general case. The steady state is what we can call a mixed steady state, that identifies the case in which the predators can catch the prey, see Figure 2.

    Figure 2. 

    Example of mixed stationary state. Note that by symmetry Lρ=Rρ and Lη=Rη

    .

    The proof of the existence of such steady state follows by using the strong version of the Krein-Rutman theorem, see [26].

    Theorem 2.2 (Krein-Rutman). Let X be a Banach space, KX be a solid cone, such that λKK for all λ0 and K has a nonempty interior Ko. Let T be a compact linear operator on X, which is strongly positive with respect to K, i.e. T[u]Ko if uK{0}. Then,

    (i) the spectral radius r(T) is strictly positive and r(T) is a simple eigenvalue with an eigenvector vKo. There is no other eigenvalue with a corresponding eigenvector vK.

    (ii) |λ|<r(T) for all other eigenvalues λr(T).

    As pointed out in [6], using this strategy we can only obtain existence of stationary states for a diffusion coefficient that depends on the support, without providing any one-to-one correspondence between the diffusion constant (eigenvalue) and the support. Even if non completely satisfactory, the following results give a useful insight on the possible conditions we can expect in order to get existence of steady states, see Remark 3.1.

    Let us first introduce a proper definition for mixed steady states as in Figure 2.

    Definition 2.2. Let 0<Rρ<Rη be fixed. We call a pair (ρ,η) a mixed steady state a solution to system (3) with ρ and η in L1L(R), non-negative, symmetric and radially decreasing functions with supports

    Iρ:=supp(ρ)=[Rρ,Rρ],andIη:=supp(η)=[Rη,Rη].

    Let us now assume that (ρ,η) is a steady state to system (3) as in Definition 2.2, then (4) can be rephrased as

    {θρ(x)Sρρ(x)Kη(x)=CρxIρθη(x)Sηη(x)+αKρ(x)=CηxIη. (19)

    where Cρ,Cη>0 are two constants. Differentiating the two equations in (19) w.r.t. xsupp(ρ) and xsupp(η) respectively, we obtain

    {θρx=RρRρSρ(xy)ρy(y)dy+RηRηK(xy)ηy(y)dyxIρθηx=RηRηSη(xy)ηy(y)dyαRρRρK(xy)ρy(y)dyxIη. (20)

    By symmetry properties of the kernels Sρ, Sη and K and the steady states ρ and η, for x>0, we get

    θρx=Rρ0(Sρ(xy)Sρ(x+y))ρy(y)dy+Rη0(K(xy)K(x+y))ηy(y)dy,θηx=Rη0(Sη(xy)Sη(x+y))ηy(y)dyαRρ0(K(xy)K(x+y))ρy(y)dy. (21)

    In order to simplify the notations, we set

    ˜G(x,y):=G(xy)G(x+y),forG=Sρ,Sη,K.

    Notice that ˜G, under assumptions (A1)-(A3), is a nonnegative function for x,y>0. We also set p(x)=ρx(x) for x(Rρ,Rρ) and q(x)=ηx(x) for x(Rη,Rη). Hence, (21) is rewritten simply as

    {θp(x)=Rρ0˜Sρ(x,y)p(y)dy+Rη0˜K(x,y)q(y)dyθq(x)=Rη0˜Sη(x,y)q(y)dyαRρ0˜K(x,y)p(y)dy. (22)

    Proposition 2.1. Assume that Sρ,Sη,K satisfy (A1), (A2) and (A3) and fix 0<Rρ<Rη and 0<zρ, zη. Assume that Sη and K are strictly concave on [Rη,Rη] and [Rρ,Rρ] respectively. Assume that there exists a constant C such that

    C<Rη0˜Sη(x,y)h(y)dyRρ0˜K(x,y)k(y)dy, (23)

    for any (k,h)C1([0,Rρ])×C1([0,Rη]) with k(0)=h(0)=0 and k(0)>0, k(x)>0 for all x(0,Rρ), and h(0)>0, h(x)>0 for all x(0,Rη). Then, there exists a unique mixed steady state (ρ,η) in the sense of Definition 2.2 to system (3) with θ=θ(Rρ,Rη)>0, provided that

    α<min{C,Sη(Rη)zηR2ηK(0)zρ},

    where zρ and zη are masses of ρ and η respectively.

    Proof. Let us first introduce the following Banach space

    XRρ,Rη={(p,q)C1([0,Rρ])×C1([0,Rη]):p(0)=q(0)=0},

    endowed with the W1,-norm for the two components p and q. Define the operator TRρ,Rη[P] on the Banach space XRρ,Rη as

    TRρ,Rη[P]:=(f,g)C1([0,Rρ])×C1([0,Rη]),

    where P denotes the elements P=(p,q)XRρ,Rη, and (f,g) are given by

    f(x)=Rρ0˜Sρ(x,y)p(y)dy+Rη0˜K(x,y)q(y)dyforx[0,Rρ],g(x)=Rη0˜Sη(x,y)q(y)dyαRρ0˜K(x,y)p(y)dyforx[0,Rη].

    The assumptions on the kernels ensure that the operator TRρ,Rη is compact on the Banach space XRρ,Rη. Indeed, as the operator TRρ,Rη is defined on a space of C1 functions defined on compact intervals, and since the kernels Sρ,Sη,K are all in C2, which is from assumption (B1), defined in the operator on compact intervals, then using Arzelá's theorem, it is easy to prove that TRρ,Rη maps bounded sequences in XRρ,Rη into pre-compact ones. Now, consider the subset KRρ,RηXRρ,Rη defined as

    KRρ,Rη={PXRρ,Rη:p0,q0}.

    It can be shown that this set is indeed a solid cone in KRρ,Rη. Moreover, we have that

    HRρ,Rη={PKRρ,Rη:p(0)>0,p(x)>0forallx(0,Rρ),andq(0)>0,q(x)>0forallx(0,Rη)}KRρ,Rη,

    where KRρ,Rη denotes the interior of KRρ,Rη. We now show that the operator TRρ,Rη is strongly positive on the solid cone KRρ,Rη in the sense of Theorem 2.2. Let (p,q)KRρ,Rη with p,q0, then by the definition of the operator TRρ,Rη, it is easy to see that the first component is non-negative. Concerning the second component, we have

    Rη0˜Sη(x,y)q(y)dyαRρ0˜K(x,y)p(y)dy>0, (24)

    if and only if α<C with C as in (23). Next, it is easy to show that the derivative at x=0 of the first component is strictly positive. The derivative of the second component is given by

    ddx|x=0(Rη0˜Sη(x,y)q(y)dyαRρ0˜K(x,y)p(y)dy)=Rη0˜Sη,x(0,y)q(y)dyαRρ0˜Kx(0,y)p(y)dy=Rη0(Sη(y)Sη(y))q(y)dyαRρ0(K(y)K(y))p(y)dy=2Rη0Sη(y)q(y)dy+2αRρ0K(y)p(y)dy:=A.

    Now, we need to find the condition on α such that A>0. Chebyshev's inequality in the first integral of A and the concavity assumption of Sη on the interval [Rη,Rη] yields the bound

    2RηRη0Sη(y)q(y)dy=2RηRη0Sη(y)η(y)dy(1RηRη0Sη(y)dy)(2RηRη0η(y)dy)=Sη(Rη)zηR2η.

    Using the concavity assumption of K on the interval [Rρ,Rρ], the other integral can be easily bounded by

    2Rρ0K(y)p(y)dy=2Rρ0K(y)ρ(y)dy<K(0)zρ.

    Thus, A>0 holds under the condition

    α<Sη(Rη)zηR2ηK(0)zρ. (25)

    As a consequence, TRρ,Rη[P] belongs to HRρ,Rη, which implies that the operator TRρ,Rη is strongly positive on the solid cone KRρ,Rη. Then, the Krein-Rutman theorem applies and guarantees the existence of an eigenvalue θ=θ(Rρ,Rη) such that

    TRρ,Rη[P]=θP,

    with an eigenspace generated by one given nontrivial element (ˉp,ˉq) in the interior of the solid cone KRρ,Rη. Moreover, by (i) of Theorem 2.2, there exists no other eigenvalues to TRρ,Rη with corresponding eigenvectors in KRρ,Rη besides the one with eigenfunction (ˉp,ˉq), and by (ii) of Theorem 2.2 all other eigenvalues ˜θ with eigenfunctions in XRρ,Rη satisfy |˜θ|<θ.

    Remark 2.1. Motivated from the purely aggregative case (13), we actually expect a more rich behavior for the possible steady states configurations, such as the separated stationary state in Figure 3. This is expected as a possible winning strategy for the prey, since it corresponds to prey finding a safe distance from the predators. Unfortunately, the symmetrization argument used in the previous proof seems to fail, since in this case we need to require the symmetry around cm2η for the convolutions. In the next section we prove the existence and uniqueness for the multiple bumps stationary state in the sense of Definition 1.1, that includes the mixed and the separated one, using a completely different approach.

    Figure 3. 

    An example of a separated stationary state

    .

    In this Section we prove the existence and uniqueness of a multiple bumps steady state in the sense of Definition 1 fixing masses and a small diffusion coefficient. Following the approach in [5, 6], we first formulate the problem in terms of the pseudo-inverse functions and then we use the Implicit Function Theorem (cf. [19, Theorem 15.1]).

    We start rewriting our stationary system in terms of pseudo-inverse functions. Let (ρ,η) be a solution to the stationary system

    {0=(ρ(θρSρραρKη)x)x,0=(η(θηSηηαηKρ)x)x. (26)

    where αρ=1 and αη=α. Assume that (ρ,η) have masses zρ and zη respectively and denote by cml,l{ρ,η}, the centres of masses

    Rxρ(x)dx=cmρ,Rxη(x)dx=cmη.

    Remember that the only conserved quantity in the evolution, together with the masses, is the joint centre of mass

    CMα=αcmρcmη, (27)

    that we can consider fixed. Define the cumulative distribution functions of ρ and η as

    Fρ(x)=xρ(x)dx,Fη(x)=xη(x)dx.

    Let ul:[0,zl]R,l{ρ,η}, be the pseudo-inverse of Fl, namely

    ul(z)=inf{xR:Fl(x)z},l{ρ,η},

    supported on

    supp(ul)=[0,zl]:=Jl,l{ρ,η}.

    For ρ and η multiple bumps in the sense of Definition 1.1 we can denote the mass of each bump as

    ρi(x)dx=ziρ,ηj(x)dx=zjη,i=1,2,,Nρ,j=1,2,,Nη,

    and the centres of masses accordingly,

    xρi(x)dx=cmiρ,xηj(x)dx=cmjη,i=1,2,,Nρ,j=1,2,,Nη,

    and we can always assume that the centres of masses are ordered species by species, i.e. cmilcmjl if ij. Let us consider the case of centres of masses that are stationary solutions of the purely non-local particle system (17), that we recall for the reader convenience,

    {Nρj=1Sρ(cmiρcmjρ)zjρ+Nηj=1K(cmiρcmjη)zjη=0,i=1,,Nρ,Nηj=1Sη(cmiηcmjη)zjηαNρj=1K(cmiηcmjρ)zjρ=0,i=1,,Nη, (28)

    coupled with the conservation of the joint centre of mass CMα in (27), see the discussion in Section 2. For such a density the pseudo-inverse ul reads as

    ul(z)=Nli=1uil(z)1Jil(z),l{ρ,η},

    where

    supp(ul)=[0,zl]=Jl=Nli=1[ik=1zk1l,ik=1zkl]:=Nli=1[ˆzil,˜zil]:=Nli=1Jil,l{ρ,η},

    with z0l=0 and zl=Nlk=1zkl. We are now in the position of reformulating (26) in terms of the pseudo-inverse functions as follows:

    {θ2z((zuρ(z))2)=JρSρ(uρ(z)uρ(ξ))dξ+αρJηK(uρ(z)uη(ξ))dξ,zJρ,θ2z((zuη(z))2)=JηSη(uη(z)uη(ξ))dξ+αηJρK(uη(z)uρ(ξ))dξ,zJη. (29)

    The restriction to zJil, i=1,2,,Nl, and l{ρ,η}, allow us to rephrase (29) in the compact form

    θ2z((zuil(z))2)=Nlj=1JjlSl(uil(z)ujl(ξ))dξ+αlNhj=1JjhK(uil(z)ujh(ξ))dξ,zJil. (30)

    Similar to [5, 6], we suggest the linearization

    uil=cmil+δvili=1,2,,Nl,andl{ρ,η},

    with vil, being odd functions defined on Jil. Using this ansatz in (30), with the scaling θ=δ3 we have

    δ2z((zvil(z))2)=Nlj=1JjlSl(cmilcmjl+δ(vil(z)vjl(ξ)))dξ+αlNhj=1JjhK(cmilcmjh+δ(vil(z)vjh(ξ)))dξ. (31)

    Multiplying (31) by δzvil, and taking the primitives w.r.t. z, we obtain

    δ2zvil(z)=Nlj=1JjlSl(cmilcmjl+δ(vil(z)vjl(ξ)))dξ+αlNhj=1JjhK(cmilcmjh+δ(vil(z)vjh(ξ)))dξ+Ail,zJil, (32)

    where Ail are the integration constants. In order to recover the constants Ail, we substitute ˜zil into equation (32). Denoting by vil(˜zil)=λil, we obtain

    Ail=Nlj=1JjlSl(cmilcmjl+δ(λilvjl(ξ)))dξαlNhj=1JjhK(cmilcmjh+δ(λilvjh(ξ)))dξ. (33)

    Next, we set Gl and H such that Gl=Sl and H=K, with Gl,H to be odd and satisfy Gl(0)=H(0)=0. Then, multiplying (32) again by δzvil and taking the primitives w.r.t. zJil, we obtain

    δ3z=Nlj=1JjlGl(cmilcmjl+δ(vil(z)vjl(ξ)))dξ+αlNhj=1JjhH(cmilcmjh+δ(vil(z)vjh(ξ)))dξ+Ailδvil(z)+βil,zJil. (34)

    Let us denote with ˉzil the middle point of each interval Jil. Then, in order to recover the integration constants βil, we substitute ˉzil in (34) which yields

    βil=δ3ˉzilNlj=1JjlGl(cmilcmjlδvjl(ξ))dξαlNhj=1JjhH(cmilcmjhδvjh(ξ))dξ. (35)

    As a consequence of all above computations, we get the following relation for zJil,

    δ3(zˉzil)=Nlj=1JjlGl(cmilcmjl+δ(vil(z)vjl(ξ)))Gl(cmilcmjlδvjl(ξ))dξδvil(z)Nlj=1JjlSl(cmilcmjl+δ(λilvjl(ξ)))dξ+αlNhj=1JjhH(cmilcmjh+δ(vil(z)vjh(ξ)))H(cmilcmjhδvjh(ξ))dξδvil(z)αlNhj=1JjhK(cmilcmjh+δ(λilvjh(ξ)))dξ. (36)

    If we define, for p=(v1ρ,,vNρρ,v1η,,vNηη)

    Fil[p;δ](z)=ˉzilz+δ3[Nlj=1JjlGl(cmilcmjl+δ(vil(z)vjl(ξ)))Gl(cmilcmjlδvjl(ξ))dξδvil(z)Nlj=1JjlSl(cmilcmjl+δ(λilvjl(ξ)))dξ+αlNhj=1JjhH(cmilcmjh+δ(vil(z)vjh(ξ)))H(cmilcmjhδvjh(ξ))dξδvil(z)αlNhj=1JjhK(cmilcmjh+δ(λilvjh(ξ)))dξ],zJil, (37)

    we have that (30) reduces to the equation Fil[p;δ](z)=0. In the following we compute the Taylor expansion of Fil[p;δ](z) around δ=0. Let us begin with the first integral on the r.h.s. of (37), i.e.,

    JjlGl(cmilcmjl+δ(vil(z)vjl(ξ)))Gl(cmilcmjlδvjl(ξ))dξ=[Sl(cmilcmjl)δvil(z)+δ22Sl(cmilcmjl)(vil(z))2+δ36Sl(cmilcmjl)(vil(z))3]|Jjl|+Jjlδ32Sl(cmilcmjl)(vjl(ξ))2vil(z)dξ+R(Sl,δ4), (38)

    where we used the fact that Jilvil(ξ)dξ=0 and R(Sl,δ4) is a remainder term. For the second integral we have

    δvil(z)JjlSl(cmilcmjl+δ(λilvjl(ξ)))dξ=[Sl(cmilcmjl)δvil(z)δ2Sl(cmilcmjl)λilvil(z)δ32Sl(cmilcmjl)(λil)2vil(z)]|Jjl|Jjlδ32Sl(cmilcmjl)(vjl(ξ))2vil(z)dξ+R(Sl,δ4). (39)

    Summing up the contributions in (38) to (39), we get that the self-interaction part in (37) reduces to

    δ3[δ12Sl(cmilcmjl)vil(z)(vil(z)2λil)+16Sl(cmilcmjl)((vil(z))33vil(z)(λil)2)]|Jjl|+R(Sl,δ4). (40)

    Similarly, for the cross-interaction part we obtain

    δ3[δ12K(cmilcmjh)vil(z)(vil(z)2λil)+16K(cmilcmjh)((vil(z))33vil(z)(λil)2)]|Jjh|+R(K,δ4). (41)

    Putting together the contributions of (40) and (41) in the functional equation (37), we get

    Fil[p;δ](z)=(ˉzilz)+Dil6(3vil(z)(λil)2(vil(z))3)+δ1Bil2vil(z)(vil(z)2λil)+R(Sl,K,δ4), (42)

    where we used the notations introduced in (6) and (5), namely

    Dil=Nlj=1Sl(cmilcmjl)|Jjl|αlNhj=1K(cmilcmjh)|Jjh|,

    and

    Bil=Nlj=1Sl(cmilcmjl)|Jjl|+αlNhj=1K(cmilcmjh)|Jjh|.

    Note that since the values cmil satisfy (28) we have that Bil=0. After the manipulations above, equation Fil[p;0](z)=0 reads as

    (ˉzilz)+Dil6(3vil(z)(λil)2(vil(z))3)=0,zJil, (43)

    that gives a unique solution once the value of λil is determined. In order to do that, we evaluate the functional Fil in the end point ˜zil of the corresponding interval

    Λil[p;δ]=Fil[p;δ](˜zil). (44)

    Performing Taylor expansions similar to the ones in (38) and (39) we get that

    Λil[p;0]=(ˉzil˜zil)+Dil3(λil)3, (45)

    and we are now in the position to solve

    {(ˉzilz)+Dil6(3vil(z)(λil)2(vil(z))3)=0,(ˉzil˜zil)+Dil3(λil)3=0. (46)

    The second equation in (46) admits a solution once we have that Dil>0, and λil is uniquely determined by

    λil=(3(˜zilˉzil)Dil)1/3. (47)

    By construction λilλjl if ij, and this implies that equation (46) admits the unique solution ˉvil which can be recovered as the pseudo inverse of the following Barenblatt type profiles

    ˉρi(x)=Diρ2((λiρ)2(xcmiρ)2)1Iiρ(x),i=1,,Nρ,ˉηh(x)=Dhη2((λhη)2(xcmhη)2)1Ihη(x),h=1,,Nη, (48)

    where the intervals Iiρ=[liρ,riρ] and Ihη=[lhη,rhη] are determined imposing

    likk=cmikkλikk,rikk=cmikk+λikk,ik=1,,Nk,k=ρ,η.

    We are now ready to reformulate (36) as a functional equation on a proper Banach space. Consider the spaces

    Ωil={vL([ˉzil,˜zil))|vincreasing,v(ˉzil)=0},i=1,,Nl,l{ρ,η}, (49)

    endowed with the L norm and take the product spaces

    Ωl=Nl×i=1Ωil,forl{ρ,η}.

    We now introduce the space Ω defined by

    Ω=Ωρ×RNρ×Ωη×RNη, (50)

    with elements ω=(v1ρ,,vNρρ,λ1ρ,,λNρρ,v1η,,vNηη,λ1η,,λNηη) endowed with the norm

    |||ω|||=Nρi=1(viρL+|λiρ|)+Nηi=1(viηL+|λiη|). (51)

    For γ>0, calling ˜Jil=[ˉzil,˜zil), we consider the norm

    |||ω|||γ=|||ω|||+l{ρ,η}Nli=1supz˜Jil|λilvil(z)|(˜zilz)γ, (52)

    and set Ωγ:={ωΩ:|||ω|||γ<+}. For a given ωΩ12, we define the operator T:Ω12Ω1

    T[ω;δ](z):=(Fρ[ω;δ](z)Λρ[ω;δ]Fη[ω;δ](z)Λη[ω;δ]), (53)

    where for l{ρ,η} we have shorted the notation introducing

    Fl[ω;δ](z)=(F1l[ω;δ](z),,FNll[ω;δ](z)),Λl[ω;δ](z):=(Λ1l[ω;δ],,ΛNll[ω;δ]). (54)

    The operator T is a bounded operator for any fixed δ0 and can be continuously extended at δ=0 to (43) and (45). In order to prove existence of stationary solutions for small δ>0 using the Implicit Function Theorem, we need to prove that the Jacobian matrix of T is a bounded linear operator form Ω1/2 to Ω1 with bounded inverse. The Jacobian of T has the following structure

    DT[ω;δ]=(DvρFρ(δ)DλρFρ(δ)DvηFρ(δ)DληFρ(δ)DvρΛρ(δ)DλρΛρ(δ)DvηΛρ(δ)DληΛρ(δ)DvρFη(δ)DλρFη(δ)DvηFη(δ)DληFη(δ)DvρΛη(δ)DλρΛη(δ)DvηΛη(δ)DληΛη(δ)), (55)

    where the components are actually matrices defined by

    DvhFl(δ)=(Fil[ω;δ]vjh(νjh))Nl,Nhi,j=1,DλhFl(δ)=(Fil[ω;δ]λjh(ajh))Nl,Nhi,j=1DvhΛl(δ)=(Λil[ω;δ]vjh(νjh))Nl,Nhi,j=1,DλhΛl(δ)=(Λil[ω;δ]λjh(ajh))Nl,Nhi,j=1,

    where νjh and ajh are generic directions. We first compute the diagonal terms in the matrix DvlFl(δ). We have

    Fil[ω;δ]vil(νil)=δ2Jilνil(ξ)[Sl(δ(vil(z)vil(ξ)))δvil(z)Sl(δ(λilvil(ξ)))Sl(δvil(ξ))]dξ+δ2νil(z)Nlj=1Jjl[Sl(cmilcmjl+δ(vil(z)vjl(ξ)))Sl(cmilcmjl+δ(λilvjl(ξ)))]dξ+δ2νil(z)αlNhj=1Jjh[K(cmilcmjh+δ(vil(z)vjh(ξ)))K(cmilcmjh+δ(λilvjh(ξ)))]dξ.

    A Taylor expansion around δ=0 similar to the ones in (38) - (41) easily gives that in the limit δ0 we obtain

    Fil[ω;0]vil(νil)=Dil2((λil)2(vil(z))2)νil(z).

    Concerning the other terms in DvlFl(δ) we get

    Fil[ω;δ]vjl(νjl)=δ2Nlj=1Jjlνjl(ξ)[Sl(cmilcmjl+δ(vil(z)vjl(ξ)))Sl((cmilcmjlδvjl(ξ))δvil(z)Sl(cmilcmjl+δ(λilvjl(ξ)))]dξ,

    that all vanish in the limit δ0. Let us now focus on the matrix DλlFl(δ). By (37) it is easy to see that the only non-zero terms in DλlFl(δ) are the diagonal ones that are given by

    Fil[ω;δ]λil(ail)=δ1vil(z)ail[Nlj=1JjlSl(cmilcmjl+δ(λilvjl(ξ)))dξ+αlNhj=1JjhK(cmilcmjh+δ(λilvjh(ξ)))dξ].

    Then, Taylor expansion w.r.t. δ yields

    Fil[ω;0]λil(ail)=Dilλilvil(z)ail.

    Since all the entrances in the matrix DλhFl(δ) are zero, the last matrix that concerns Fil is DvhFl(δ). The elements of this matrix are given by

    Fil[ω;δ]vjh(νjh)=δ2αlNhj=1Jjhνjh(ξ)[K(cmilcmjh+δ(vil(z)vjh(ξ)))K(cmilcmjhδvjh(ξ))δvil(z)K(cmilcmjh+δ(λilvjh(ξ)))]dξ,

    that vanish in the limit δ0. We now start in computing the functional derivatives for Λil in (44). Again we should consider the four matrices in (55), and we start from DvlΛl(δ). Note that the terms in the diagonal are zero in this case and the others are given by

    Λil[ω;δ]vjl(νjl)=δ2Nlj=1Jjlνjl(ξ)[Sl(cmilcmjl+δ(λil(z)vjl(ξ)))Sl((cmilcmjlδvjl(ξ))δλilSl(cmilcmjl+δ(λilvjl(ξ)))]dξ.

    The terms in DvhΛl(δ) are

    Λil[ω;δ]vjh(νjh)=δ2αlNhj=1Jjhνjh(ξ)[K(cmilcmjh+δ(λilvjh(ξ)))K(cmilcmjhδvjh(ξ))δλilK(cmilcmjh+δ(λilvjh(ξ)))]dξ,

    and the usual Taylor expansions around δ=0, show that both the matrices DvlΛl(0)=DvhΛl(0)=0. Since DλhΛl(δ) is trivially a zero matrix, only remains to compute the diagonal terms in DλlΛl(δ). We have

    Λil[ω;δ]λil(ail)=δ1ailNlj=1JjlλilSl(cmilcmjl+δ(λilvjl(ξ)))dξδ1ailαlNhj=1JjhλilKl(cmilcmjh+δ(λilvjh(ξ)))dξ.

    The last Taylor expansion gives

    Λil[ω;0]λil(ail)=Dil(λil)2ail.

    We have proved that

    DT[ω;0]=(El1dg(Diρλiρviρaiρ)000dg(Diρ(λiρ)2aiρ)0000El2dg(Diηλiηviηaiη)000dg(Diη(λiη)2aiη)), (56)

    where

    El1=dg(Diρ2((λiρ)2(viρ)2)νiρ),El2=dg(Diη2((λiη)2(viη)2)νiη),

    and dg(Ai) is a diagonal matrix with elements Ai. Let us denote by ω0 the unique solution to (46), we have the following lemma:

    Lemma 3.1. For δ>0 small enough, the operator DT[ω0;δ] is a bounded linear operator from Ω1/2 to Ω1.

    Proof. Thanks to the previous computations it is easy to see that DT is a bounded linear operator from Ω into itself and it is continuous at δ=0. The definition of the norm in (52) implies that for z˜Jil we need to control only that

    sup|||ω|||1/211(˜zilz)|Filvil[,δ](νil)Λilvil[;δ](νil)(Filvil[;0](νil)Λilvil[;0](νil))+Filλil[,δ](ail)Λilλil[;δ](ail)(Filλil[;0](ail)Λilλil[;0](ail))+Filvjh[,δ](νjh)Λilvjh[;δ](νjh)(Filvjh[;0](νjh)Λilvjh[;0](νjh))+Filλjh[,δ](ajh)Λilλjh[;δ](ajh)(Filλjh[;0](ajh)Λilλjh[;0](ajh))|0, (57)

    as δ0. We start estimating the third row in (57),

    1(˜zilz)[Filvjh[,δ](νjh)Λilvjh[;δ](νjh)(Filvjh[;0](νjh)Λilvjh[;0](νjh))]=αl(vil(z)λil)2˜zilzNhj=1JjhK(cmilcmjh+δ(λilvjh(ξ)))νjh(ξ)2dξδαl(vil(z)λil)3˜zilzNhj=1JjhK(˜x(ξ))νjh(ξ)6dξ=δαl(vil(z)λil)2˜zilzNhj=1JjhK(ˉx(ξ))(λilvjh(ξ))νjh(ξ)2dξδαl(vil(z)λil)3˜zilzNhj=1JjhK(˜x(ξ))νjh(ξ)6dξ.=αl((vil(z)λil)2˜zilz+(vil(z)λil)3˜zilz)O(δ),

    where in the first equality we did a Taylor expansion around the point x0=cmilcmjh+δ(λilvjh(ξ)) for the kernel K(cmilcmjh+δ(vil(z)vjh(ξ))), while in the second equality we did a Taylor expansion around the point x0=cmilcmjh for the kernel K(cmilcmjh+δ(λilvjh(ξ))). Similarly, we can show that

    1(˜zilz)[Filλjh[,δ](ajh)Λilλjh[;δ](ajh)(Filλjh[;0](ajh)Λilλjh[;0](ajh))]=0.

    The first two rows in (57) can be treated as follows,

    1(˜zilz)[Filvil[,δ](νil)Λilvil[;δ](νil)(Filvil[;0](νil)Λilvil[;0](νil))+Filλil[,δ](ail)Λilλil[;δ](ail)(Filλil[;0](ail)Λilλil[;0](ail))]=δ(˜zilz)[Nlj=1Jjlδ2(vil(z)λil)(νil(z)ail)Sl(cmilcmjl+δ(λilvjl(ξ)))+δ112(vil(z)λil)2(νil(z)νjl(ξ))Sl(cmilcmjl+δ(λilvjl(ξ)))+16(vil(z)λil)3(νil(z)νjl(ξ))Sl(˜x1(ξ))dξ+αlNhj=1Jjhδ2(vil(z)λil)(νil(z)ail)K(cmilcmjh+δ(λilvjh(ξ)))+δ112(vil(z)λil)2(νil(z)νjl(ξ))K(cmilcmjh+δ(λilvjh(ξ)))+16(vil(z)λil)3(νil(z)νjl(ξ))K(˜x2(ξ))dξ]=(2(νil(z)ail)(vil(z)λil)(˜zilz)+(2νil(z)1)((vil(z)λil)3(˜zilz)+(vil(z)λil)2(˜zilz)))O(δ).

    Since the functions v_l^i are components of a vector \omega belonging to \Omega_{1/2}, the quantities

    \frac{\lambda_l^i-v_l^i(z)}{(\tilde{z}_l^i - z)^{1/2}},

    are uniformly bounded in \tilde{J}_l^i , that gives (57).

    Lemma 3.2. For any \delta>0 small enough, D\mathcal{T}[\omega_0; 0]:\Omega_{1/2}\rightarrow \Omega_{1} is a linear isomorphism.

    Proof. Given w\in\Omega_1 , we have to prove that

    \begin{equation} D\mathcal{T}[\omega_0;0] \omega = w, \end{equation} (58)

    admits a unique solution \omega\in\Omega_{1/2} with the property

    \begin{equation*} ||\omega||_{1/2}\leq C ||w||_{1}. \end{equation*}

    The determinant of the matrix in (56) is given by

    \begin{equation} \det D\mathcal{T} = \left(\prod\limits_{i = 1}^{N_{\rho}}\frac{(D_{\rho}^i)^2}{2} \Big( (\lambda_{\rho}^i)^2 - (v_{\rho}^i)^2 \Big) (\lambda_{\rho}^i)^2 \right) \cdot \left(\prod\limits_{i = 1}^{N_{\eta}}\frac{(D_{\eta}^i)^2}{2} \Big( (\lambda_{\eta}^i)^2 - (v_{\eta}^i)^2 \Big) (\lambda_{\eta}^i)^2 \right) , \end{equation} (59)

    that is always different from zero under the condition D_l^i > 0 and since \big( v_l^i(z)-\lambda_l^i\big) < 0 on z\in[\bar{z}_l^i, \tilde{z}_l^i) . Thanks to the structure in (56) and denoting with \nu_l^i,a_l^i and \sigma_l^i,k_l^i the generic entrances in \omega and w respectively, we easily get that

    \begin{equation*} \begin{aligned} \nu_l^i(z) & = \frac{- 2 \sigma_l^i(z)}{D_l^i \big( (v_l^i(z))^2 - (\lambda_l^i)^2 \big)} + \frac{2 \lambda_l^i v_l^i(z) a_l^i}{ (v_l^i(z))^2 - (\lambda_l^i)^2 }, \\ a_l^i & = \frac{k_l^i}{D_l^i(\lambda_l^i)^2}, \end{aligned} \end{equation*}

    that implies ||\nu_l^i||_{1/2}\leq C ||\sigma_l^i||_{\infty} for i = 1,2,\cdots,N_l, and l \in \{\rho,\eta\} . In order to close the argument, it is enough to note that the ratio

    \begin{equation*} \frac{a_l^i-\nu_l^i(z)}{(\tilde{z}_l^i - z)^{1/2}} , \end{equation*}

    is uniformly bounded since (\lambda_l^i - v_l^i(z))/(\tilde{z}_l^i-z) is uniformly bounded, see [5, Lemma 4.4].

    We are now in the position of proving the main result of the paper, namely Theorem 1.1, that we recall below for convenience.

    Theorem 3.1. Assume that S_\rho , S_\eta and K are interaction kernels are under the assumptions (A1), (A2) and (A3). Consider N_\rho,N_\eta\in\mathbb{N} and let z_l^i be fixed positive numbers for i = 1,2,\cdots,N_l, and l \in \{\rho,\eta\} . Consider two families of real numbers \{cm_\rho^i\}_{i = 1}^{N_\rho} and \{cm_\eta^i\}_{i = 1}^{N_\eta} such that

    (i) \{cm_\rho^i\}_{i = 1}^{N_\rho} and \{cm_\eta^i\}_{i = 1}^{N_\eta} are stationary solutions of the purely non-local particle system, that is, for i = 1,2,\cdots,N_l, for l,h \in \{\rho,\eta\} and l\neq h ,

    \begin{equation} B_{l}^i = \sum\limits_{j = 1}^{N_l} S'_l( cm_l^i - cm_l^j ) z_l^j+ \alpha_l\sum\limits_{j = 1}^{N_h} K'( cm_l^i - cm_h^j ) z_h^j = 0, \end{equation} (60)

    (ii) the following quantities

    \begin{equation} D_{l}^i = - \sum\limits_{j = 1}^{N_l} S''_l(cm_l^i - cm_l^j) z_l^j -\alpha_l \sum\limits_{j = 1}^{N_h} K''(cm_l^i - cm_h^j ) z_h^j, \end{equation} (61)

    are strictly positive, for all i = 1,2,\cdots,N_l , l,h \in \{\rho,\eta\} and l\neq h .

    where \alpha_{\rho} = 1 and \alpha_{\eta} = -\alpha . Then, there exists a constant \theta_0 such that for all \theta\in (0, \theta_0) the stationary equation (4) admits a unique solution in the sense of Definition 1.1 of the form

    where

    each interval I_l^i is symmetric around cm_l^i for all i = 1,2,\cdots,N_l , l \in \{\rho,\eta\} ,

    \rho^i and \eta^j are C^1 , non-negative and even w.r.t the centres of I_\rho^i and I_\eta^j respectively, with masses z_\rho^i and z_\eta^j , for i = 1,...,N_\rho and j = 1,...,N_\eta ,

    the solutions \rho and \eta have fixed masses

    z_\rho = \sum\limits_{i = 1}^{N_\rho} z_\rho^i \;\mathit{and}\; z_\eta = \sum\limits_{i = 1}^{N_\eta} z_\eta^i,

    respectively.

    Proof. Consider z_\rho and z_\eta fixed masses and a set of points cm_l^i for i = 1,2,\cdots,N_l, and l \in \{\rho,\eta\} that satisfy (i) and (ii). The results in Lemma 3.1 and Lemma 3.2 imply that given \mathcal{T} defined in (53), the functional equation

    \mathcal{T}[\omega;\delta](z) = 0,

    admits a unique solution

    \omega = ( v_\rho^1(z),\ldots,v_\rho^{N_\rho}(z),\lambda_\rho^1,\ldots,\lambda_\rho^{N_\rho},v_\eta^1(z),\ldots,v_\eta^{N_\eta}(z),\lambda_\eta^1,\ldots,\lambda_\eta^{N_\eta} )\,,

    for \delta >0 small enough. The entrances v_l^i(z) are solutions to (36) for z\in J_l^i . Consider now u_l^i defined for z\in J_l^i as u_l^i (z) = cm_l^i + \delta v_l^i(z) . Differentiating (36) twice w.r.t z we get that v_l^i is differentiable and strictly increasing for z\in J_l^i and that u_l^i is a solution to (30). Moreover u_l^i is also strictly increasing and we can define the inverse F_l^i that and its spatial derivative \rho_l^i = \partial_x F_l^i is a solution to (4).

    Remark 3.1 Note that conditions (ii) in Theorem 3.1 turn to be conditions on the positions of the centres of masses and on the value of \alpha . Indeed, as a sufficient condition for D_\rho^i> 0 we can assume that all the differences between the centres of masses are in the range of concavity of the kernels. Moreover, D_\eta^i> 0 is satisfied if

    \alpha < \min\limits_{i = 1,\ldots,N_\eta}\frac{ \sum\limits_{j = 1}^{N_\eta} S''_\eta(cm_\eta^i - cm_\eta^j) |J_\eta^j|}{ \sum\limits_{j = 1}^{N_\rho} K''(cm_\eta^i - cm_\rho^j ) |J_\rho^j| }.

    Note that the above conditions are comparable to the one we got in the proof of Section 2 using the Krein-Rutman approach.

    In this section, we study numerically solutions to system (3) using two different methods, the finite volume method introduced in [12, 13] and the particles method studied in [23, 28]. We validate the results about the existence of the mixed steady state and the multiple bumps steady states. Moreover, we perform some examples to show the variation in the behaviour of the solution to system (3) under different choices of initial data and the parameter \alpha , which, in turn, suggests future work on the stability of the solutions to system (3). Finally, travelling waves are detected under a special choice of initial data and value for the parameter \alpha . We begin by sketching the particles method. This method essentially consists in a finite difference discretization in space to the pseudo inverse version of system (3)

    \begin{equation} \begin{cases} \partial_t u_{\rho}(z) = &-\frac{\theta}{2}\partial_{z}\Big(\big(\partial_{z}u_{\rho}(z)\big)^{-2}\Big) + \int_{J_{\rho}}S_{\rho}^{\prime}\big(u_{\rho}(z)-u_{\rho}(\xi)\big)d\xi \\ &+ \alpha_{\rho}\int_{J_{\eta}}K^{\prime}\big(u_{\rho}(z)-u_{\eta}(\xi)\big)d\xi,\; \; z\in J_{\rho} \\ \partial_t u_{\eta}(z) = &-\frac{\theta}{2}\partial_{z}\Big(\big(\partial_{z}u_{\eta}(z)\big)^{-2}\Big) + \int_{J_{\eta}}S_{\eta}^{\prime}\big(u_{\eta}(z)-u_{\eta}(\xi)\big)d\xi \\ &+ \alpha_{\eta}\int_{J_{\rho}}K^{\prime}\big(u_{\eta}(z)-u_{\rho}(\xi)\big)d\xi,\; \; z\in J_{\eta}, \end{cases} \end{equation} (62)

    as the following: Let N\in\mathbb{N} , let \{z^i\}_{i = 1}^N be a sequence of points that partition the interval [0,1] uniformly. Denote by X_l^i(t): = u_l(t,z^i) the approximating particles of the pseudo inverse at each point z^i of the partition. Assuming that the densities \rho and \eta are of unit masses, then we have the following approximating system of ODEs

    \begin{equation} \begin{cases} \partial_t X_{\rho}^i(t) = & \frac{\theta}{2N} \Big(\big(\rho^{i-1}(t)\big)^{-2} - \big(\rho^{i}(t)\big)^{-2} \Big) + \frac{1}{N}\sum\limits_{j = 1}^N S_{\rho}^{\prime}\big(X_{\rho}^i(t)-X_{\rho}^j(t)\big)d\xi \\ & + \frac{\alpha_{\rho}}{N}\sum\limits_{j = 1}^N K^{\prime}\big(X_{\rho}^i(t)-X_{\eta}^j(t)\big)d\xi, \qquad i = 1,\cdots N, \\ \partial_t X_{\eta}^i(t) = & \frac{\theta}{2N} \Big(\big(\eta^{i-1}(t)\big)^{-2} - \big(\eta^{i}(t)\big)^{-2} \Big) + \frac{1}{N}\sum\limits_{j = 1}^N S_{\eta}^{\prime}\big(X_{\eta}^i(t)-X_{\eta}^j(t)\big)d\xi \\ & + \frac{\alpha_{\eta}}{N}\sum\limits_{j = 1}^N K^{\prime}\big(X_{\eta}^i(t)-X_{\rho}^j(t)\big)d\xi, \qquad i = 1,\cdots N, \end{cases} \end{equation} (63)

    where the densities are reconstructed as

    \begin{equation} \begin{cases} \rho^i(t) = \frac{1}{N ( X_l^{i+1}(t) - X_l^i(t) )}, \qquad i = 1,\cdots, N-1, \\ \rho^0(t) = 0, \\ \rho^N(t) = 0, \end{cases} \end{equation} (64)

    and the same for the \eta^i(t) . To this end, we are ready to solve this particle system by applying the Runge-Kutta MATLAB solver ODE23, with initial positions X_l(0) = X_{l,0} = \{ X_{l,0}^i \}_{i = 1}^N,\; l = \rho,\eta determined by solving

    \int_{X_{\rho,0}^i}^{X_{\rho,0}^{i+1}} \rho(t = 0) dX = \frac{1}{N-1}, \qquad i = 1,\cdots,N-1.

    The second method we use is the finite volume method which introduced in [12] and extended to systems in [13], that consists in a 1 D positive preserving finite-volume method for system (3). To do so, we first partition the computational domain into finite-volume cells U_{i} = [x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}] of a uniform size \Delta x with x_{i} = i\Delta x , i \in\{-s,\; \dots,s \} . Define

    \begin{equation*} \widetilde{\rho}_{i}(t) : = \frac{1}{\Delta x}\int_{U_{i}}\rho(x,t)dx,\qquad \widetilde{\eta}_{i}(t) : = \frac{1}{\Delta x}\int_{U_{i}}\eta(x,t)dx, \end{equation*}

    the averages of the solutions \rho,\eta computed at each cell U_{i} . Then we integrate each equation in system (3) over each cell U_{i} , and so we obtain a semi-discrete finite-volume scheme described by the following system of ODEs for \overline{\rho}^{i} and \overline{\eta}^{i}

    \begin{equation} \begin{cases} \frac{d\widetilde{\rho}_{i}(t)}{dt} = -\frac{F_{i+\frac{1}{2}}^{\rho}(t) - F_{i-\frac{1}{2}}^{\rho}(t) }{\Delta x}, \\ \frac{d\widetilde{\eta}_{i}(t)}{dt} = -\frac{F_{i+\frac{1}{2}}^{\eta}(t) - F_{i-\frac{1}{2}}^{\eta}(t) }{\Delta x}, \end{cases} \end{equation} (65)

    where the numerical flux F_{i+\frac{1}{2}}^l , l = \rho,\eta , is considered as an approximation for the continuous fluxes -\rho(\theta\rho + S_{\rho}*\rho + \alpha_{\rho} K*\eta )_{x} and -\eta(\theta\eta + S_{\eta}*\eta + \alpha_{\eta} K*\rho )_{x} respectively. More precisely, the expression for F_{i+\frac{1}{2}}^{\rho} is given by

    \begin{equation} F_{i+\frac{1}{2}}^{\rho} = \max (\vartheta_{\rho}^{i+1},0)\Big[ \widetilde{\rho}_{i} + \frac{\Delta x}{2}(\rho_{x})_{i} \Big] + \min (\vartheta_{\rho}^{i+1},0)\Big[ \widetilde{\rho}_{i} - \frac{\Delta x}{2}(\rho_{x})^{i} \Big], \end{equation} (66)

    where

    \begin{equation} \begin{aligned} \vartheta_{\rho}^{i+1} = &- \frac{\theta}{\Delta x} \big( \widetilde{\rho}_{i+1} - \widetilde{\rho}_{i} \big) - \sum\limits_{j}\widetilde{\rho}_{j}\big( S_{\rho}(x_{i+1} - x_{j}) - S_{\rho}(x_{i} - x_{j}) \big) \\ &- \alpha_{\rho} \sum\limits_{j}\widetilde{\eta}_{j}\big( K(x_{i+1} - x_{j}) - K(x_{i} - x_{j}) \big), \end{aligned} \end{equation} (67)

    and

    \begin{equation} (\rho_{x})^{i} = {\rm{minmod}} \Bigg( 2\frac{\widetilde{\rho}_{i+1} - \widetilde{\rho}_{i}}{\Delta x},\; \frac{\widetilde{\rho}_{i+1} - \widetilde{\rho}_{i-1}}{2\Delta x},\; 2\frac{\widetilde{\rho}_{i} - \widetilde{\rho}_{i-1}}{\Delta x} \Bigg). \end{equation} (68)

    The minmod limiter in (68) has the following definition

    \begin{equation} {\rm{minmod}}(a_{1}, a_{2},\; \dots) : = \begin{cases} \min (a_{1}, a_{2},\; \dots), \quad {\rm{if}} \; a_{i} > 0\quad \forall i \\ \max (a_{1}, a_{2},\; \dots), \quad {\rm{if}} \; a_{i} < 0\quad \forall i \\ 0, \qquad\qquad\qquad{\rm{otherwise.}} \end{cases} \end{equation} (69)

    We have the same as the above expresions for \eta . Finally, we integrate the semi-discrete scheme (65) numerically using the third-order strong preserving Runge-Kutta (SSP-RK) ODE solver used in [31].

    In all the simulations below, we will fix the kernels as a normalised Gaussian potentials

    S_{\rho}(x) = S_{\eta}(x) = K(x) = \frac{1}{\sqrt{\pi}} e^{-x^2},

    that are under the assumptions on the kernels (A1), (A2) and (A3). This choice helps us in better understanding the variation in the behavior of the solutions w.r.t. the change in the initial data and the parameter \alpha . In the first five examples, see Figures 4, 5, 6, 7 and 8 we show:

    Figure 4. 

    In this figure, a mixed steady state is plotted by using initial data given by (70), \alpha = 0.1 , \theta = 0.4 . Number of particles are chosen equal to number of cells in the finite volume method, which is N = 71

    .
    Figure 5. 

    A separated steady state is presented in this figure. Initial data are given by (71). The parameters are \alpha = 0.2 and \theta = 0.4 with N = 91

    .
    Figure 6. 

    This figure shows how from the initial densities \rho_0,\eta_0 and \theta , as in Figure 4, a transition between mixed and a sort of separated steady state appears by choosing the value of \alpha = 6 . This large value of \alpha suggests an unstable behavior in the profile, see Remark 3.1

    .
    Figure 7. 

    A steady state of four bumps is showed in this figure starting from initial data as in (72) with \alpha = 0.05 and \theta = 0.3 . The number of particles N = 181 , which is the same as number of cells

    .
    Figure 8. 

    Starting from initial data as in (73) with \alpha = 1 and \theta = 0.3 , we get a five bumps steady state

    .

    ● in the first row steady states are plotted at the level of density, on the l.h.s. we compare the two methods illustrated above, while on the r.h.s. we show the evolution by the finite volume method,

    ● in the second row we plot the particles paths for both species obtained with the particles method,

    ● in the last row we show the pseudo inverse functions corresponding to the steady state densities.

    The last example we present shows an interesting traveling waves-type evolution.

    The first example is devoted to validating existence of mixed steady state and separated steady state. By choosing the initial data (\rho_0,\eta_0) as

    (70)

    and fixing \alpha = 0.1 and \theta = 0.4 , we obtain a mixed steady state as plotted in Figure 4. Note that, the small value of \alpha allows the predators to dominate the prey which results in the shape shown in Figure 4. Next, we take the initial data as

    (71)

    with the same \theta as above and \alpha = 0.2 . This choice of the initial data, actually, introduce two equal attractive forces on the right and left hand sides of the predators which, in turn, fix the predators at the centre and gives the required shape of the separated steady state as shown in Figure 5. Finally, a sort of separated steady state can also be obtained starting from the same initial data as in (70) and same diffusion parameter \theta . If we choose \alpha = 6 , the prey \eta will have enough speed to get out of the predators region, producing a transition between the mixed state to the separated one, this is illustrated in Figure 6.

    We then, test two cases where we validate the existence of steady states of more bumps. Let us fix \theta = 0.3 . Then, a four-bumps steady state is performed and plotted in Figure 7, where we consider

    (72)

    as initial data and we take \alpha = 0.05 . This way of choosing initial data produces a balanced attractive forces which in turn form the required steady state. Similarly, we can obtain a steady state of five bumps by using initial data

    (73)

    and \alpha = 1 , see Figure 8.

    Finally, in this example, we detect existence of traveling waves. Indeed, by choosing initial data as

    (74)

    \alpha = 1 and \theta = 0.2 , we obtain a traveling wave that is shown in Figure 9. Once the initial data and the value of \theta are fixed, if \alpha is taken as a small value, then we will come out with a mixed steady state. If we take a larger value for \alpha then the prey will be fast escaping from the predators. Therefore, the proper value for \alpha will produce a situation where the attack speed of the predators is equal to the escape speed of the prey, which results in a traveling wave of both the densities \rho and \eta . All the simulations above motivate further studies on stability vs. instability for such a system, together with the possible existence on traveling wave type solutions.

    Figure 9. 

    This last figure shows a possible existence of traveling waves by choosing initial data as in (74), \alpha = 1 and \theta = 0.2 . The first two plots are performed by particles method, while the third one is done by finite volume method. Here we fix N = 101

    .


    [1] M. Abbas, B. Damjanović, R. Lazović, Fuzzy common fixed point theorems for generalized contractive mappings, Appl. Math. Lett., 23 (2010), 1326–1330. https://doi.org/10.1016/j.aml.2010.06.023 doi: 10.1016/j.aml.2010.06.023
    [2] M. A. Alghamdi, N. Hussain, P. Salimi, Fixed point and couplete fixed point theorems on b-metric-like spaces, J. Inequal. Appl., 2013 (2013), 402. https://doi.org/10.1186/1029-242X-2013-402 doi: 10.1186/1029-242X-2013-402
    [3] A. Ali, F. Uddin, M. Arshad, M. Rashid, Hybrid fixed point results via generalized dynamic process for F-HRS type contractions with application, Phys. A: Stat. Mech. Appl., 538 (2020), 122669. https://doi.org/10.1016/j.physa.2019.122669 doi: 10.1016/j.physa.2019.122669
    [4] S. Atailia, N. Redjel, A. Dehici, Some fixed point results for (c)-mappings in Banach spaces, J. Fixed Point Theory Appl., 22 (2020), 51. https://doi.org/10.1007/s11784-020-00787-4 doi: 10.1007/s11784-020-00787-4
    [5] A. Azam, Fuzzy fixed points of fuzzy mappings via a rational inequality, Hacet. J. Math. Stat., 40 (2011), 421–431. https://doi.org/10.1016/j.physa.2019.122669 doi: 10.1016/j.physa.2019.122669
    [6] A. Azam, M. Arshad, P. Vetro, On a pair of fuzzy \phi-contractive mappings, Math. Comput. Model., 52 (2010), 207–214. https://doi.org/10.1016/j.mcm.2010.02.010 doi: 10.1016/j.mcm.2010.02.010
    [7] J. P. Aubin, I. Ekeland, Applied nonlinear analysis, John Wiley Sons Wiley, 1984.
    [8] J. P. Aubin, J. Siegel, Fixed points and stationary points of dissipative multivalued maps, Proc. Am. Math. Soc., 78 (1980), 391–398. https://doi.org/10.2307/2042331 doi: 10.2307/2042331
    [9] A. Ali, A. Hussain, M. Arshad, H. A. Sulami, M. Tariq, Certain new development to the orthogonal binary relations, Symmetry, 14 (2022), 1954. https://doi.org/10.3390/sym14101954 doi: 10.3390/sym14101954
    [10] A. Ali, A. Muhammad, A. Hussain, N. Hussain, S. M. Alsulami, On new generalized \theta_{b}-contractions and related fixed point theorems, J. Inequal. Appl., 2022 (2022), 37. https://doi.org/10.1186/s13660-022-02770-8 doi: 10.1186/s13660-022-02770-8
    [11] R. Bellman, E. S. Lee, Functional equations in dynamic programming, Aequat. Math., 17 (1978), 1–18. https://doi.org/10.1007/BF01818535 doi: 10.1007/BF01818535
    [12] M. Cosentino, P. Vetro, Fixed point results for F-contractive mappings of Hardy-Rogers-type, Filomat, 28 (2014), 715–722. https://doi.org/10.2298/FIL1404715C doi: 10.2298/FIL1404715C
    [13] Y. J. Cho, M. Jleli, M. Mursaleen, B. Samet, C. Vetro, Advances in metric fixed point theory and applications, Singapore Springer Nature, 2021. https://doi.org/10.1007/978-981-33-6647-3
    [14] D. Derouiche, H. Ramoul, New fixed point results for F-contractions of Hardy-Rogers type in b-metric spaces with applications, J. Fixed Point Theory Appl., 22 (2020), 1–44. https://doi.org/10.1007/s11784-020-00822-4 doi: 10.1007/s11784-020-00822-4
    [15] P. Debnath, N. Konwar, S. Radenović, Metric fixed point theory: Applications in science, engineering and behavioural sciences, Springer Verlag, 2021. https://doi.org/10.1007/978-981-16-4896-0
    [16] N. Hussain, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Fixed points of contractive mappings in b-metric-like spaces, Sci. World J., 2014 (2014), 471827. https://doi.org/10.1155/2014/471827 doi: 10.1155/2014/471827
    [17] M. Kikkawa, T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal. Theory Methods Appl., 69 (2008), 2942–2949. https://doi.org/10.1016/j.na.2007.08.064 doi: 10.1016/j.na.2007.08.064
    [18] D. Klim, D. Wardowski, Fixed points of dynamic processes of set-valued F-contractions and application to functional equations, Fixed Point Theory Appl., 2015 (2015), 22. https://doi.org/10.1186/s13663-015-0272-y doi: 10.1186/s13663-015-0272-y
    [19] R. Kruse, K. D. Meyer, Statistics with vague data, Reidel, 1987. https://doi.org/10.1007/978-94-009-3943-1
    [20] V. Lakshmikantham, R. Mohapatra, Theory of fuzzy differential equations and inclusions, CRC Press, 2003. https://doi.org/10.1201/9780203011386
    [21] S. G. Matthews, Partial metric topology, in proceedings of the 11th summer conference on general topology an applications, Ann. New York Acad. Sci., 728 (1994), 183–197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x doi: 10.1111/j.1749-6632.1994.tb44144.x
    [22] S. B. J. Nadler, Multi-valued contraction mappings, Pac. J. Math., 30 (1969), 475–488. https://doi.org/10.2140/PJM.1969.30.475 doi: 10.2140/PJM.1969.30.475
    [23] H. K. Nashine, C. Vetro, W. Kumam, P. Kumam, Fixed point theorems for fuzzy mappings and applications to ordinary fuzzy differential equations, Adv. Differ. Equ., 2014 (2014), 232. https://doi.org/10.1186/1687-1847-2014-232 doi: 10.1186/1687-1847-2014-232
    [24] N. Onjai-Uea, P. Kumam, A generalized nonlinear random equations with random fuzzy mappings in uniformly smooth Banach spaces, J. Inequal. Appl., 2010 (2010), 728452. https://doi.org/10.1155/2010/728452 doi: 10.1155/2010/728452
    [25] R. A. Rashwan, M. A. Ahmed, Common fixed point theorems for fuzzy mappings, Arch. Math., 38 (2002), 219–226.
    [26] M. Sgroi, C. Vetro, Multi-valued F-contractions and the solution of certain functional and integral equations, Filomat, 27 (2013), 1259–1268. https://doi.org/10.2298/FIL1307259S doi: 10.2298/FIL1307259S
    [27] H. M. Srivastava, A. Ali, A. Hussain, M. Arshad, H. A. Sulami, A certain class of \theta_{L}-type non-linear operatorsand some related fixed point results, J. Nonlinear Var. Anal., 6 (2022), 69–87. https://doi.org/10.23952/jnva.6.2022.1.05 doi: 10.23952/jnva.6.2022.1.05
    [28] T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal. Theory Methods Appl., 71 (2009), 5313–5317. https://doi.org/10.1016/j.na.2009.04.017 doi: 10.1016/j.na.2009.04.017
    [29] D. Turkoglu, B. E. Rhoades, A fixed fuzzy point for fuzzy mapping in complete metric spaces, Math. Commun., 10 (2005), 115–121.
    [30] V. Todorcevic, Harmonic quasiconformal mappings and hyperbolic type metrics, Cham Springer International, 2019. https://doi.org/10.1007/978-3-030-22591-9
    [31] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 94. https://doi.org/10.1186/1687-1812-2012-94 doi: 10.1186/1687-1812-2012-94
    [32] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
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