Academic interest in understanding the role of financial technology (FinTech) in sustainable development has grown exponentially in recent years. Many studies have highlighted the context, yet no reviews have explored the integration of FinTech and sustainability through the lens of the banking aspect. Therefore, this study sheds light on the literature trends associated with FinTech and sustainable banking using an integrated bibliometric and systematic literature review (SLR). The bibliometric analysis explored publication trends, keyword analysis, top publisher, and author analysis. With the SLR approach, we pondered the theory-context-characteristics-methods (TCCM) framework with 44 articles published from 2002 to 2023. The findings presented a substantial nexus between FinTech and sustainable banking, showing an incremental interest among global scholars. We also provided a comprehensive finding regarding the dominant theories (i.e., technology acceptance model and autoregressive distributed lag model), specific contexts (i.e., industries and countries), characteristics (i.e., independent, dependent, moderating, and mediating variables), and methods (i.e., research approaches and tools). This review is the first to identify the less explored tie between FinTech and sustainable banking. The findings may help policymakers, banking service providers, and academicians understand the necessity of FinTech in sustainable banking. The future research agenda of this review will also facilitate future researchers to explore the research domain to find new insights.
Citation: Md. Shahinur Rahman, Iqbal Hossain Moral, Md. Abdul Kaium, Gertrude Arpa Sarker, Israt Zahan, Gazi Md. Shakhawat Hossain, Md Abdul Mannan Khan. FinTech in sustainable banking: An integrated systematic literature review and future research agenda with a TCCM framework[J]. Green Finance, 2024, 6(1): 92-116. doi: 10.3934/GF.2024005
[1] | Martin Bohner, Sabrina Streipert . Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences and Engineering, 2016, 13(4): 673-695. doi: 10.3934/mbe.2016014 |
[2] | Qianhong Zhang, Fubiao Lin, Xiaoying Zhong . On discrete time Beverton-Holt population model with fuzzy environment. Mathematical Biosciences and Engineering, 2019, 16(3): 1471-1488. doi: 10.3934/mbe.2019071 |
[3] | John E. Franke, Abdul-Aziz Yakubu . Periodically forced discrete-time SIS epidemic model with disease induced mortality. Mathematical Biosciences and Engineering, 2011, 8(2): 385-408. doi: 10.3934/mbe.2011.8.385 |
[4] | Yang Li, Jia Li . Stage-structured discrete-time models for interacting wild and sterile mosquitoes with beverton-holt survivability. Mathematical Biosciences and Engineering, 2019, 16(2): 572-602. doi: 10.3934/mbe.2019028 |
[5] | Shishi Wang, Yuting Ding, Hongfan Lu, Silin Gong . Stability and bifurcation analysis of SIQR for the COVID-19 epidemic model with time delay. Mathematical Biosciences and Engineering, 2021, 18(5): 5505-5524. doi: 10.3934/mbe.2021278 |
[6] | Jaqueline G. Mesquita, Urszula Ostaszewska, Ewa Schmeidel, Małgorzata Zdanowicz . Global attractors, extremal stability and periodicity for a delayed population model with survival rate on time scales. Mathematical Biosciences and Engineering, 2021, 18(5): 6819-6840. doi: 10.3934/mbe.2021339 |
[7] | Guilherme M Lopes, José F Fontanari . Influence of technological progress and renewability on the sustainability of ecosystem engineers populations. Mathematical Biosciences and Engineering, 2019, 16(5): 3450-3464. doi: 10.3934/mbe.2019173 |
[8] | Jordi Ripoll, Jordi Font . Numerical approach to an age-structured Lotka-Volterra model. Mathematical Biosciences and Engineering, 2023, 20(9): 15603-15622. doi: 10.3934/mbe.2023696 |
[9] | Michael Leguèbe . Cell scale modeling of electropermeabilization by periodic pulses. Mathematical Biosciences and Engineering, 2015, 12(3): 537-554. doi: 10.3934/mbe.2015.12.537 |
[10] | Islam A. Moneim, David Greenhalgh . Use Of A Periodic Vaccination Strategy To Control The Spread Of Epidemics With Seasonally Varying Contact Rate. Mathematical Biosciences and Engineering, 2005, 2(3): 591-611. doi: 10.3934/mbe.2005.2.591 |
Academic interest in understanding the role of financial technology (FinTech) in sustainable development has grown exponentially in recent years. Many studies have highlighted the context, yet no reviews have explored the integration of FinTech and sustainability through the lens of the banking aspect. Therefore, this study sheds light on the literature trends associated with FinTech and sustainable banking using an integrated bibliometric and systematic literature review (SLR). The bibliometric analysis explored publication trends, keyword analysis, top publisher, and author analysis. With the SLR approach, we pondered the theory-context-characteristics-methods (TCCM) framework with 44 articles published from 2002 to 2023. The findings presented a substantial nexus between FinTech and sustainable banking, showing an incremental interest among global scholars. We also provided a comprehensive finding regarding the dominant theories (i.e., technology acceptance model and autoregressive distributed lag model), specific contexts (i.e., industries and countries), characteristics (i.e., independent, dependent, moderating, and mediating variables), and methods (i.e., research approaches and tools). This review is the first to identify the less explored tie between FinTech and sustainable banking. The findings may help policymakers, banking service providers, and academicians understand the necessity of FinTech in sustainable banking. The future research agenda of this review will also facilitate future researchers to explore the research domain to find new insights.
The Beverton–Holt recurrence
xt+1=ηKxtK+(η−1)xt, | (1.1) |
where η is the proliferation rate and K is the carrying capacity, was derived in [1] in the context of fisheries. The solution of the logistic differential equation evaluated at time T+t0 is used to describe the adult fish population (new generation), and the solution at time t0 represents the juveniles (old generation). The derivation led to a proliferation rate η=erT>1 , where r>0 is the growth rate of the underlying continuous model and T is the time span until adulthood. Equation (1.1) for more general parameters is also known as the Pielou equation [2]. For related work on the Beverton–Holt equation, see [3,4,5,6,7,8,9,10,11,12], and for other related work, see [13,14,15,16].
In [17], Cushing and Henson investigated the effects of a periodically enforced carrying capacity onto flour beetles, which were modeled by the Beverton–Holt recurrence (1.1). Based on their observations, the authors conjectured that the introduction of a periodic environment on populations, modeled by (1.1), results in the existence of a unique periodic solution. Further, the authors predicted that a periodic environment is deleterious to the population, as the average of the unique periodic solution is bounded above by the mean of the periodic environment. These conclusions were formulated as the first and second Cushing–Henson conjectures.
In the case of a two-periodic Kt and constant η , the conjectures have been analytically verified in [18]. For higher-order periodic carrying capacities Kt , the conjectures have been the focus of the works [19,20]. A discussion assuming additionally time-dependent growth rates can be found in [21]. The extension of the conjectures to periodic time scales was addressed in [22]. Periodic time scales are time domains such that if t is in the time scale, then so is t+ω , hence requiring an additive time structure. The discrete time setting obeys this additive property and is a special case of a periodic time scale, in contrast to the quantum time setting qN0 , which is not periodic. Periodic time scales are a subset of arbitrary time scales, a theory developed by Stefan Hilger in 1988, that unifies the discrete and continuous theories. Studying the Beverton–Holt model on time scales provides a platform to consider time-dependent time spans until adulthood instead of a constant time span T , as assumed in the derivation of the Beverton–Holt model. Due to a lack of existing periodicity definitions for general time domains, the conjectures remained unsolved for this and other examples of nonperiodic time scales. In [23], in order to extend the study to the quantum time scale, the authors defined periodicity in the quantum setting and discussed the Cushing–Henson conjectures. In the quantum time setting, the Beverton–Holt model reads as
x(qt)=ηK(t)x(t)K(t)+(η−1)x(t) |
with the carrying capacity K:qN0→R+:=(0,∞) ( q>1 ) and proliferation rate η>1 . In [23], the authors proved the existence and global stability of a unique periodic solution for periodic carrying capacities, i.e., for K such that qωK(qωt)=K(t) , confirming the first Cushing–Henson conjecture for the q -Beverton–Holt model. In [24], the authors were able to extend this result to time-dependent proliferation rates. The second Cushing–Henson conjecture, however, only remained true in the quantum time scale with a slight modification as follows.
Theorem 1.1 (See [23, Theorem 5.6]). The average of the ω -periodic solution ˉx of the q -Beverton–Holt model is strictly lessthan the average of the ω -periodic carrying capacitytimes the constant q−λ1−λ , i.e.,
∫qω1ˉx(t)Δt≤q−λ1−λ∫qω1K(t)Δt, |
where λ=1−(q−1)A , η=11−(q−1)A , and A>0 .
The multiplicative constant in Theorem 1.1 can be expressed by
q−λ1−λ=(q−1)(1+A)(q−1)A=(q−1)η(q−1)+η−1η(q−1)η−1η=qη−1η−1. | (1.2) |
This indicates that the second conjecture does not necessarily hold, and it reveals that the time scale determines the proportionality constant linearly. Given this reformulation of the classical second Cushing–Henson conjecture in the special case of a quantum time scale, we aim to find a general formulation of this conjecture on arbitrary isolated time scales. On any isolated time scale T , we therefore consider the Beverton–Holt model
xσ=ηKxK+(η−1)x, | (1.3) |
where η,K:T→R+ , η>1 , and where σ(t) is the next time step following t . We can express the recurrence (1.3) as a dynamic equation
xσ−xμ=:xΔ=αxσ(1−xK)withα=η−1μη, | (1.4) |
where μ(t) is the distance to the time point following t , formally introduced in Section 2. Equation (1.4) is known as the logistic dynamic equation [25], and it can be transformed equivalently into a linear dynamic equation using the variable substitution u=1/x for x≠0 , namely
uΔ=−αu+αK. | (1.5) |
To study the effects of periodicity on (1.3) on arbitrary time scales, we utilize the recently introduced definition of periodicity in [26]. This new concept was already successfully applied in [27]. This and other useful definitions are stated in Section 2. Section 3 concerns the generalization of the first Cushing–Henson conjecture, discussing the existence of a unique globally asymptotically stable periodic solution. The generalization of the second Cushing–Henson conjecture is addressed in Section 4. The paper is completed in Section 5 with some concluding remarks.
In this section, we introduce some necessary time scales fundamentals. A time scale T is a closed nonempty subset of R .
Definition 2.1 (See [28, Definition 1.1]). For t∈T , the forward jump operator σ:T→T is defined by
σ(t)=inf{s∈T:s>t}. |
We adopt the convention that inf∅=supT . If σ(t)>t , then we say that t is right-scattered. If σ(t)=t , then we say that t is right-dense. Similarly, left-scattered and left-dense points are defined. The graininess μ:T→R+0 is defined by μ(t):=σ(t)−t . We define fσ:T→R by fσ:=f∘σ . If T has a left-scattered maximum M , then we define Tκ=T∖{M} ; otherwise, Tκ=T .
In this work, we focus on isolated time scales, i.e., all points are left-scattered and right-scattered. Hence, in what follows, throughout, T refers to an isolated time scale, and the following definitions and results are taking this special time structure already into account, as well as the entire remainder of this paper.
Definition 2.2 (See [25, Definition 2.25]). A function p:T→R is called regressive, denoted by R , provided
1+μ(t)p(t)≠0forallt∈T. |
Moreover, p is called positively regressive, denoted by R+ , provided
1+μ(t)p(t)>0forallt∈T. |
Remark 2.3. Assume f:T→R and t∈Tκ . Then the delta-derivative of f , denoted by fΔ (see [25, Definition 1.10]), is
fΔ(t)=f(σ(t))−f(t)μ(t). |
For f,g:T→R , the product-rule for t∈Tκ (see [25]) reads as
(fg)Δ=fΔgσ+fgΔ=fΔg+fσgΔ, | (2.1) |
and, if g≠0 , the quotient-rule (see [25]) reads as
(fg)Δ=fΔg−fgΔggσ. |
The delta integral is defined for a,b∈T with a<b by
∫baf(τ)Δτ=∑τ∈[a,b)∩Tμ(τ)f(τ), |
and consequently,
FΔ=f,ifF(t)=∫taf(τ)Δτ. |
Using the product rule, we get the integration by parts formula (see [25])
∫ba(fΔg)(t)Δt=(fg)(b)−(fg)(a)−∫ba(fσgΔ)(t)Δt. | (2.2) |
The following circle-plus addition turns (R,⊕) into an Abelian group.
Definition 2.4 (See [28, p. 13]). Define circle plus and circle minus by
p⊕q=p+q+μpq,p⊖q=p−q1+μqforq∈R. |
Theorem 2.5 (See [25, Theorem 2.33). Let p∈R and t0∈T . Then the initial value problem
yΔ=p(t)y,y(t0)=1 |
possesses a unique solution.
The unique solution from Theorem 2.5 is called the dynamic exponential function and is denoted by ep(⋅,t0) . On an isolated time scale, the dynamic exponential function for p∈R is
ep(t,t0)=∏s∈[t0,t)∩T(1+μ(s)p(s)),t>t0. |
Useful properties of the dynamic exponential function follow. Part 6 can easily be shown using parts 4 and 5. Part 8 is is the content of [25, Theorem 2.39]. Part 9 is in [25, Theorem 2.48(i)], and the remaining parts are from [25, Theorem 2.36].
Theorem 2.6. If p,q∈R and t,s,r∈T , then
1. e0(t,s)=1 and ep(t,t)=1 ,
2. ep⊕q(t,s)=ep(t,s)eq(t,s) ,
3. ep⊖q(t,s)=ep(t,s)eq(t,s) ,
4. e⊖p(t,s)=ep(s,t)=1ep(t,s) ,
5. ep(σ(t),s)=(1+μ(t)p(t))ep(t,s) ,
6. ep(t,σ(s))=ep(t,s)1+μ(s)p(s) ,
7. eΔp(⋅,s)=pep(⋅,s) ,
8. eΔp(s,⋅)=−peσp(s,⋅) ,
9. p∈R+ implies ep(t,s)>0 , and
10. the semigroup property holds: ep(t,r)ep(r,s)=ep(t,s) .
The following result was used in [26, Proof of Theorem 6.2]. A variant of it was also included in [29, Theorem 2.1], see also [30, First formula in the line two lines after (3)]). Here we state it explicitly and include its short proof.
Lemma 2.7. Let f:T→R∖{0} be delta-differentiable and s,t∈T . Then
efΔf(s,t)=f(s)f(t). | (2.3) |
Proof. For fixed t∈T , define w(s):=f(s)f(t) . Since f≠0 , fΔf∈R and
wΔ(s)=fΔ(s)f(t)=f(s)f(t)fΔ(s)f(s)=w(s)fΔ(s)f(s),w(t)=1. |
By Theorem 2.5, the claim follows.
Theorem 2.8 (See [31]). If f is nonnegative with −f∈R+ , then
1−∫tsf(τ)Δτ≤e−f(t,s)≤exp(−∫tsf(τ)Δτ). | (2.4) |
Theorem 2.9. (Variation of Constants, see [28, Theorem 2.77]). If p∈R , f:T→R , t0∈T , and y0∈R , then the unique solution of the IVP
yΔ=p(t)y+f(t),y(t0)=y0 |
is given by
y(t)=ep(t,t0)y0+∫tt0ep(t,σ(s))f(s)Δs. | (2.5) |
As mentioned in the introduction, the definition of periodicity is crucial in the discussion of effects of periodicity. We refer to our recent work [26], where we introduced periodicity on isolated time sales as follows.
Definition 2.10 (See [26, Definition 4.1]). Let ω∈N . A function f:T→R is called ω -periodic, denoted by f∈Pω , provided
νΔfν=f,whereν=σωandfν=f∘ν. | (2.6) |
Example 2.11. If T=Z , then ν(t)=t+ω , νΔ(t)=1 , and (2.6) reduces to the known definition of periodicity, f(t+ω)=f(t) .
Example 2.12. If T=qN0 , then ν(t)=qωt , νΔ(t)=qω , and (2.6) reduces to the known definition of periodicity, qωf(qωt)=f(t) , which was introduced in [23].
Now, from our recent paper [26], we collect some results, supplemented by some new tools (substitution rule and change of order of integration formula), that will be used in the remainder of this study.
Lemma 2.13. (See [26, Theorem 5.6 and Corollary 5.8]). If f,g∈Pω , then
f+g,f−g,⊖f,f⊕g∈Pω. |
Lemma 2.14. (See [26, Theorem 5.1]). f∈P1 iff μf is constant.
Lemma 2.15. (See [26, Lemma 4.6]). We have P1⊂Pω for all ω∈N .
Lemma 2.16. (See [26, Lemma 3.1]). We have the formula
μνΔ=μν. | (2.7) |
Theorem 2.17 (Chain Rule, Substitution Rule). For f:T→R , we have
FΔν=νΔfν−f,ifFν(t)=∫ν(t)tf(τ)Δτ. | (2.8) |
Moreover, if s,t∈T , then
∫ν(t)ν(s)f(τ)Δτ=∫tsνΔ(τ)f(ν(τ))Δτ. | (2.9) |
Proof. Equation (2.8) is the content of [26, Lemma 3.8]. Using (2.8), we get
∫ν(t)ν(s)f(τ)Δτ=Fν(t)−Fν(s)+∫tsf(τ)Δτ=∫ts(FΔν(τ)+f(τ))Δτ=∫tsνΔ(τ)fν(τ)Δτ, |
i.e., (2.9) holds.
Theorem 2.18. (See [26, Theorem 4.9]). If p∈Pω∩R and t,s∈T , then
ep(ν(t),t)=ep(ν(s),s)andep(ν(t),ν(s))=ep(t,s). | (2.10) |
To conclude this section, we include a result on how to change the order of integration in a double integral. The final formula in the following theorem will be needed in our proof of the second Cushing–Henson conjecture in Section 4, while the other formulas are included for future reference.
Theorem 2.19. Let f,g:T→R and a,b,c∈T . For Φ(t,s):=f(t)g(s) , we have
∫ba∫tcΦ(t,s)ΔsΔt=∫ba∫acΦ(t,s)ΔtΔs+∫ba∫bσ(s)Φ(t,s)ΔtΔs, | (2.11) |
∫ba∫ν(t)cΦ(t,s)ΔsΔt=∫ba∫ν(a)cΦ(t,s)ΔtΔs+∫ν(b)ν(a)∫bν−1(σ(s))Φ(t,s)ΔtΔs, | (2.12) |
∫ba∫ν(t)tΦ(t,s)ΔsΔt=∫ba∫ν(a)aΦ(t,s)ΔtΔs−∫ba∫bσ(s)Φ(t,s)ΔtΔs+∫ν(b)ν(a)∫bν−1(σ(s))Φ(t,s)ΔtΔs, | (2.13) |
∫ν(a)a∫ν(t)tΦ(t,s)ΔsΔt=∫ν(a)a∫σ(s)aΦ(t,s)ΔtΔs+∫ν(ν(a))ν(a)∫ν(a)ν−1(σ(s))Φ(t,s)ΔtΔs. | (2.14) |
Proof. In what follows, we use the notation
F(t):=−∫btf(s)Δs,G(t):=∫tcg(s)Δs,Gν(t):=∫ν(t)tg(s)Δs,ψ(t):=g(t)Fν−1(σ(t)), |
which imply
F(b)=0,FΔ=f,GΔ=g,GΔν(2.8)=νΔgν−g. |
First,
∫ba∫tcΦ(t,s)ΔsΔt(2.2)=∫baFΔ(t)G(t)Δt(2.2)=F(b)G(b)−F(a)G(a)−∫baF(σ(s))GΔ(s)Δs, |
so (2.11) holds. Next,
∫ba∫ν(t)tΦ(t,s)ΔsΔt(2.2)=∫baFΔ(t)Gν(t)Δt(2.2)=F(b)Gν(b)−F(a)Gν(a)−∫baF(σ(s))GΔν(s)Δs(2.2)=−F(a)Gν(a)−∫baνΔ(s)ψν(s)Δs−∫baF(σ(s))g(s)Δs(2.9)=−F(a)Gν(a)−∫ν(b)ν(a)ψ(s)Δs−∫baF(σ(s))g(s)Δs |
shows (2.13). Finally, (2.12) follows by adding (2.11) and (2.13), while (2.14) is the same as (2.13) with b=ν(a) .
Recall that throughout, ω∈N and ν=σω . Recall also that η>1 and
α:=η−1μη=−(⊖η−1μ),i.e.,η=1+μ(⊖(−α))=11−μα. | (3.1) |
Throughout the remainder of this paper, we assume
η:T→(1,∞),α,K:T→(0,∞),−α∈R+. |
We formulate some assumptions.
( A 1 ) (σΔη)ν=σΔη ,
( A 2 ) Kη−1∈Pω .
Theorem 3.1 (First Cushing–Henson Conjecture). Let t0∈T . Assume ( A 1 ) and ( A 2 ) . Define
λ:=νΔ(t0)e⊖(−α)(ν(t0),t0)−1. | (3.2) |
If λ≠0 , then (1.4) has a unique ω -periodic solution ˉx , given by
ˉx(t)=λ∫ν(t)te⊖(−α)(σ(s),t)α(s)K(s)Δs. | (3.3) |
If additionally, T is unbounded above, ∫∞t0α(s)Δs=∞ , and ˉx and K are bounded above, then ˉx is globally asymptotically stablefor solutions with positive initial conditions.
Given the structure of (1.3), we immediately obtain that solutions remain positive for positive initial conditions, i.e., for x0>0 , the solution x satisfies x(t)>0 for all t∈T , t≥t0 .
Before proving Theorem 3.1, we give a series of auxiliary results.
Lemma 3.2. Consider
( A 3 ) −α+1μσΔ∈Pω ,
( A 4 ) (⊖(−α)K)ν=⊖(−α)K .
( A 5 ) φ:=(−α)⊖μΔμ∈Pω .
Then ( A 1 ) holds iff ( A 3 ) holds iff ( A 5 ) holds, and ( A 2 ) holds iff ( A 4 ) holds.
Proof. The three calculations
μ{νΔ(−α+1μσΔ)ν−−α+1μσΔ}(2.7)=μ{νΔ(1−μαμσΔ)ν−1−μαμσΔ}(2.7)=(1−μασΔ)ν−1−μασΔ(3.1)=1(σΔη)ν−1σΔη, |
μ{νΔ(Kη−1)ν−Kη−1}(3.1)=μ{νΔ(Kμ(⊖(−α)))ν−Kμ(⊖(−α))}(2.7)=(K⊖(−α))ν−K⊖(−α), |
and (using 1+μΔ=σΔ and Lemmas 2.13, 2.14, and 2.15)
−α+1μσΔ−φ=1μ∈P1⊂Pω | (3.4) |
complete the proof.
Lemma 3.3. Let t,s∈T . For φ∈R defined in ( A 5 ) ,
eφ(t,s)=e−α(t,s)μ(s)μ(t) | (3.5) |
and
e⊖φ(ν(t),t)=νΔ(t)e⊖(−α)(ν(t),t). | (3.6) |
If ( A 5 ) holds, then
νΔ(t)e⊖(−α)(ν(t),t)=νΔ(s)e⊖(−α)(ν(s),s) | (3.7) |
and
νΔ(t)e⊖(−α)(ν(t),ν(s))=νΔ(s)e⊖(−α)(t,s). | (3.8) |
Proof. By Theorem 2.6 (part 3) and (2.3), we get (3.5). Using (3.5) with (2.7) yields (3.6). If φ∈Pω , then ⊖φ∈Pω by Lemma 2.13. Employing (3.5) and (3.6) together with (2.10) (applied to p=⊖φ ) implies (3.7). Theorem 2.6 and (3.7) result in (3.8).
Lemma 3.4. Assume ( A 3 ) and ( A 4 ) . Let t0∈T and define
Hν(t):=∫ν(t)th(s)Δswithh(t):=e⊖(−α)(σ(t),t0)α(t)K(t). | (3.9) |
Then, for λ defined in (3.2), we have
HΔν(t)=λh(t) | (3.10) |
and
Hνν(t)=(λ+1)Hν(t). | (3.11) |
Proof. First, by Theorem 2.6 (part 5), we have
h(t)=e⊖(−α)(t,t0)α(t)(1−μ(t)α(t))K(t)=(⊖(−α)K)(t)e⊖(−α)(t,t0), |
and hence
hν(t)(A4)=(⊖(−α)K)ν(t)e⊖(−α)(ν(t),t0)(A4)=(⊖(−α)K)(t)e⊖(−α)(ν(t),t0)=e⊖(−α)(ν(t),t)h(t). |
Thus,
HΔν(t)(2.8)=νΔ(t)hν(t)−h(t)(3.7)=(λ+1−1)h(t)=λh(t), |
which shows (3.10). Next,
Hνν(t)(3.10)=Hν(ν(t))=Hν(t)+∫ν(t)tHΔν(s)Δs(3.10)=Hν(t)+λ∫ν(t)th(s)Δs(3.10)=Hν(t)+λHν(t)=(λ+1)Hν(t) |
proves (3.11).
With Lemmas 3.3 and 3.4, we now have sufficient machinery to prove Theorem 3.1.
Proof of Theorem 3.1. In a first step, we show that ˉx given by (3.3) is an ω -periodic solution of (1.4). Note that
ˉx(t)(3.9)=λe−α(t,t0)Hν(t), |
and thus,
νΔ(t)ˉxν(t)(3.11)=λνΔ(t)e−α(ν(t),t0)Hνν(t)(3.11)=λνΔ(t)e−α(ν(t),t0)(λ+1)Hν(t)(3.11)=νΔ(t)e−α(ν(t),t)(λ+1)ˉx(t)(3.7)=ˉx(t) |
(use also (3.2) in the last equality), so ˉx is ω -periodic. With ˉu=1/ˉx , we get
ˉu(t)=1λe−α(t,t0)Hν(t), |
and thus,
ˉuΔ(t)(2.1)=1λ{e−α(σ(t),t0)HΔν(t)−α(t)e−α(t,t0)Hν(t)}(3.10)=e−α(σ(t),t0)h(t)−α(t)ˉu(t)(3.9)=α(t)K(t)−α(t)ˉu(t), |
so ˉu solves (1.5), and thus ˉx=1/ˉu solves (1.4). Altogether, ˉx is an ω -periodic solution of (1.4).
Conversely, we assume that ˜x is any ω -periodic solution of (1.4). Then ˜u=1/˜x satisfies (1.5), i.e., ˜uΔ(t)=−α(t)˜u(t)+α(t)K(t) . Hence,
νΔ(t)˜u(t)=νΔ(t)˜x(t)=1˜xν(t)=˜uν(t)=˜u(ν(t))(2.5)=e−α(ν(t),t)˜u(t)+∫ν(t)te−α(ν(t),σ(s))α(s)K(s)Δs=e−α(ν(t),t){˜u(t)+e−α(t,t0)∫ν(t)te⊖(−α)(σ(s),t0)α(s)K(s)Δs}=e−α(ν(t),t){˜u(t)+e−α(t,t0)Hν(t)} |
(note that (2.5) was applied with t0 replaced by t and t replaced by ν(t) ), so
(1+λ)˜u(t)(3.7)=νΔ(t)e⊖(−α)(ν(t),t)˜u(t)(3.7)=˜u(t)+e−α(t,t0)Hν(t), |
which, upon solving for ˜u(t) , results in
˜u(t)=e−α(t,t0)Hν(t)λ, |
i.e.,
˜x(t)=1˜u(t)=λe−α(t,t0)Hν(t)=ˉx(t). |
To prove the global asymptotic stability of ˉx , let x be the unique solution of (1.4) with initial condition x0>0 , and let ˉx0:=ˉx(t0) . Since 1/x solves (1.5), using (2.5), we get
x(t)=x0e−α(t,t0)(1+x0∫tt0h(s)Δs). |
Thus, we obtain
x(t)−ˉx(t)=x0e−α(t,t0)(1+x0∫tt0h(s)Δs)−ˉx0e−α(t,t0)(1+ˉx0∫tt0h(s)Δs)=x0−ˉx0e−α(t,t0)(1+x0∫tt0h(s)Δs)(1+ˉx0∫tt0h(s)Δs)=(x0−ˉx0)ˉx(t)ˉx0(1+x0∫tt0h(s)Δs), |
which tends to zero as t→∞ because α>0 and −α∈R+ so that
1+x0∫tt0h(s)Δs(2.4)≥1+x0‖K‖∞∫tt0e−α(t0,σ(s))α(s)Δs(2.4)=1+x0‖K‖∞(e−α(t0,t)−1)(2.4)=1−x0‖K‖∞+x0‖K‖∞e−α(t0,t)(2.4)≥1−x0‖K‖∞+x0‖K‖∞(1−∫t0tα(s)Δs)(2.4)=1+x0‖K‖∞∫tt0α(s)Δs→∞ast→∞, |
completing the proof.
Example 3.5. If T=Z , then
σ(t)=t+1,σΔ(t)=1,ν(t)=t+ω,νΔ(t)=1,μ(t)=1,μΔ(t)=0 |
and, as noted in Example 2.11, periodicity defined in (2.6) is consistent with the classical periodicity definition, i.e., f is ω -periodic if f(t+ω)=f(t) for all t∈Z . In this case, ( A 1 ) states that
η(t+ω)=η(t), |
and ( A 2 ) says that
K(t+ω)η(t+ω)−1=K(t)η(t)−1. |
Together, ( A 1 ) and ( A 2 ) are equivalent to
η(t+ω)=η(t)andK(t+ω)=K(t), |
i.e., both η and K are ω -periodic. Next, ( A 3 ) says that
−α(t+ω)+1=−α(t)+1, |
and ( A 4 ) states that
α(t+ω)1−α(t+ω)⋅1K(t+ω)=α(t)1−α(t)⋅1K(t). |
Together, ( A 3 ) and ( A 4 ) are equivalent to
α(t+ω)=α(t)andK(t+ω)=K(t), |
i.e., both α and K are ω -periodic. We also note that φ=(−α)⊖0=−α , and so φ is ω -periodic if and only if α is ω -periodic. If α>1 is constant and K is ω -periodic, then ( A 3 ) and ( A 4 ) are satisfied, and ˉx from (3.3) is consistent with the unique ω -periodic solution derived in [22]. In that case, ˉx and K are bounded, as any periodic function on Z is bounded, and
∫tt0αΔs=α(t−t0)→∞ast→∞. |
Hence, the ω -periodic solution is globally asymptotically stable for solutions with positive initial conditions. The classical first Cushing–Henson conjecture is therefore a special case of Theorem 3.1. Further, Theorem 3.1 for T=Z also contains an extension of the classical first Cushing–Henson conjecture as presented in [21], where the authors considered both K and α to be ω -periodic. Again, in this case, all assumptions of Theorem 3.1 are satisfied, and the unique ω -periodic solution is globally asymptotically stable.
Example 3.6. If T=qN0 , then
σ(t)=qt,σΔ(t)=q,ν(t)=qωt,νΔ(t)=qω,μ(t)=(q−1)t,μΔ(t)=q−1 |
and, as noted in Example 2.12, periodicity defined in (2.6) is consistent with the periodicity definition from [23], i.e., f is ω -periodic if qωf(qωt)=f(t) for all t∈qN0 . In this case, ( A 1 ) states that
qη(qωt)=qη(t), |
and ( A 2 ) says that
qωK(qωt)η(qωt)−1=K(t)η(t)−1. |
Together, ( A 1 ) and ( A 2 ) are equivalent to
η(qωt)=η(t)andqωK(qωt)=K(t), |
i.e., both η/μ and K are ω -periodic. Next, ( A 3 ) says that
qω−α(qωt)+1(q−1)qωtq=−α(t)+1(q−1)tq, |
and ( A 4 ) states that
α(qωt)1−(q−1)qωtα(qωt)⋅1K(qωt)=α(t)1−(q−1)tα(t)⋅1K(t). |
Together, ( A 3 ) and ( A 4 ) are equivalent to
qωα(qωt)=α(t)andqωK(qω)=K(t), |
i.e., both α and K are ω -periodic. We also note that
φ(t)=−α(t)−1t1+(q−1)tt=−α(t)+1tq, |
and so φ is ω -periodic if and only if α is ω -periodic. Since these assumptions coincide with the assumptions in [24], [24, Conjecture 1] is the same as Theorem 3.1 if T=qN0 . We would like to remind the reader that Theorem 3.1 is therefore also a generalization of [23, Conjecture 1] that assumes α to be 1 -periodic.
Examples 3.5 and 3.6 show that assumptions ( A 4 ) and ( A 5 ) are equivalent to α and K being ω -periodic, in the sense of (2.6), if T=Z or T=qN0 . One might wonder for which other time scales this observation is true.
Theorem 3.7. Assume T is such that
μΔμ∈Pω,i.e.,μΔν=μΔ. | (3.12) |
Then ( A 4 ) and ( A 5 ) hold if and only ifboth α and K are ω -periodic.
Proof. Assume (3.12). First, assuming ( A 4 ) and ( A 5 ) hold, we get from ( A 5 ) that
−α=φ⊕μΔμ∈Pω |
due to (3.12) and Lemma 2.13. Hence, using again Lemma 2.13, we obtain α∈Pω . Then, by ( A 4 ) ,
⊖(−α)K=(⊖(−α)K)ν=νΔνΔ(⊖(−α)K)ν=νΔ(⊖(−α))ννΔKν=⊖(−α)νΔKν, |
so K is ω -periodic. Conversely, assuming both α and K are ω -periodic, we get by Lemma 2.13 that −α∈Pω , and hence
φ=(−α)⊖μΔμ∈Pω |
due to (3.12) and Lemma 2.13. Hence, ( A 5 ) holds. Moreover,
(⊖(−α)K)ν=νΔνΔ(⊖(−α)K)ν=νΔ(⊖(−α))ννΔKν=⊖(−α)K, |
showing ( A 4 ) .
However, if (3.12) does not hold, then, assuming α and K are ω -periodic instead of ( A 4 ) and ( A 5 ) , there does not even exist an ω -periodic solution of (1.4) in general. This can be verified easily with ω=1 according to the following example.
Example 3.8. Let ω=1 and assume α and K are ω -periodic. Let ˜x be an ω -periodic solution of (1.4). Let ˜u=1/˜x . By Lemma 2.14, ˜c:=α/K is constant. Moreover, since ˜x=1/˜u satisfies (2.6), we get
σΔ˜u=˜uσ. | (3.13) |
Thus,
˜uΔ=˜uσ−˜uμ(3.13)=σΔ˜u−˜uμ=μΔμ˜u, |
and hence, due to
0(1.5)=˜uΔ+α˜u−˜c=(μΔμ+α)˜u−˜c, |
we obtain
˜u=μ˜cμΔ+μα. | (3.14) |
Hence,
˜uσ(3.14)=μσ˜cμΔσ+μσασ(2.7)=μσΔ˜cμΔσ+μσΔασ(2.6)=μσΔ˜cμΔσ+μα |
and
σΔ˜u(3.14)=μσΔ˜cμΔ+μα. |
Thus, with (3.13), we obtain (3.12).
Example 3.9. Let ω=4 . For q>0 , consider
T={tm:m∈N0},wheretm=m−1∑i=0q(−1)iform∈N0, |
where the "empty sum" is by convention zero, i.e., t0=0 . Then
σ(tm)=tm+1,μ(tm)=tm+1−tm=q(−1)m,σΔ(tm)=σ(tm+1)−σ(tm)μ(tm)=μ(tm+1)μ(tm)=q2(−1)m+1,μΔ(tm)=σΔ(tm)−1=q2(−1)m+1−1. |
Hence (3.12) holds. Let K0,K1,K2,K3>0 , ˉKi=Kimod4 , a≠1 and define
η(tm)=aq2(−1)m,K(tm)=q(−1)m+1ˉKm. |
Clearly, K∈Pω by design, and since ην=η , we have α∈Pω . By Theorem 3.7, ( A 3 ) and ( A 5 ) hold, so that by Theorem 3.1, the unique 4 -periodic solution is given by
ˉx(tm)=λ(η(tm)−1)ˉKm+ˉKm+2ˉKmˉKm+2+η(tm)(η(σ(tm))−1)ˉKm+1+ˉKm+3ˉKm+1ˉKm+3, |
where λ=a4−1≠0 .
We now bring our attention to the second Cushing–Henson conjecture, which reads for the Beverton–Holt difference equation as follows. If η>1 is constant, K:Z→R+ is ω -periodic, i.e., K(t+ω)=K(t) for all t∈Z , then the average of the unique periodic solution ˉx of (1.1) is less than or equal (equal iff K is constant) the average of the periodic carrying capacity, i.e.,
1ωω−1∑t=0ˉx(t)≤1ωω−1∑t=0K(t). |
Biologically, this inequality is interpreted as deleterious effect of a periodic environment to the population. In order to extend this result to isolated time scales, we aim to find an upper bound for the average of the unique periodic solution. Similar to the discrete case, where the second Cushing–Henson conjecture assumed a constant proliferation rate, we adjust ( A 1 ) accordingly for ω=1 . More specifically, we consider the assumptions
( A 6 ) (σΔη)σ=σΔη , ( A 7 ) −α+1μσΔ∈P1 , ( A 8 ) φ=(−α)⊖μΔμ∈P1 .
Remark 4.1. According to Lemma 3.2, we have
(A6) holds iff (A7) holds iff (A8) holds, |
and, by Lemma 2.15,
any of (A6),(A7),(A8) implies any of (A1),(A3),(A5). |
Assume now any of the conditions ( A 6 ) , ( A 7 ) , and ( A 8 ) . By Remark 2.14,
C:=μ−α+1μσΔ=1−μασΔisconstant. | (4.1) |
Moreover, due to (3.4), we have
μφ=C−1andthusμ(⊖φ)=−μφ1+μφ=1−CC=:D. | (4.2) |
Because of
e⊖φ(ν(t0),t0)=∏τ∈[t0,ν(t0))∩T(1+μ(τ)(⊖φ)(τ))=(1+1−CC)ω=1Cω, |
and thus, we get that
λ(3.2)=νΔ(t0)e⊖(−α)(ν(t0),t0)−1(3.6)=e⊖φ(ν(t0),t0)−1=1Cω−1. |
Theorem 4.2 (Second Cushing–Henson Conjecture). Assume ( A 4 ) , ( A 8 ) , and C<1 , where C is defined in (4.1). Then the average of the unique ω -periodic solution ˉx of (1.4) is bounded above by
1ω∫ν(t0)t0ˉx(t)Δt≤1ω∫ν(t0)t01−C1σΔ(t)−CK(t)Δt, | (4.3) |
and equality holds iff K⊖(−α) is constant.
The central tool in the proof of Theorem 4.2 is the following generalized Jensen inequality from [32, Theorem 2.2], which reads for the strictly convex function 1/z as follows:
∫baw(s)Δs∫baw(s)v(s)Δs≤∫baw(s)v(s)Δs∫baw(s)Δswithw>0. | (4.4) |
We apply (4.4) with
wt(s):=−φ(s)eφ(t,σ(s))>0andvt(s):=μ(t)˜β(s)(⊖φ)(s), |
where we also put
˜β:=⊖(−α)μKandβ:=1μ2˜β=Kμ(⊖(−α)). |
Note that ( A 4 ) implies (use (2.7))
β,˜β∈Pω. | (4.5) |
Before proving Theorem 4.2, we offer the following auxiliary result.
Lemma 4.3. Assume ( A 4 ) and ( A 5 ) . Define λ by (3.2). We have
∫ν(t)twt(s)Δs=λ | (4.6) |
and
wt(s)vt(s)=e⊖(−α)(σ(s),t)α(s)K(s). | (4.7) |
Moreover, if ( A 8 ) holds, then
wt(s)vt(s)=−Dβ(s)φ(t)eφ(t,σ(s)), | (4.8) |
where D is defined in (4.2).
Proof. First, we use Theorem 2.6 (part 8) to integrate
∫ν(t)twt(s)Δs=−∫ν(t)tφ(s)eφ(t,σ(s))Δs=eφ(t,ν(t))−1=λ, |
where we also used (3.6), (3.7), and (3.2). This proves (4.6). Next, using (3.5), we get
wt(s)vt(s)=−φ(s)eφ(t,σ(s))μ(t)˜β(s)(⊖φ)(s)=eφ(t,s)μ(t)˜β(s)=e−α(t,s)μ(s)˜β(s)=e−α(t,s)(⊖(−α))(s)K(s)=e⊖(−α)(σ(s),t)α(s)K(s), |
which shows (4.7). Finally, assuming ( A 8 ) , we have (4.1) and (4.2). Then
wt(s)vt(s)=−φ(s)eφ(t,σ(s))μ(t)˜β(s)(⊖φ)(s)=−φ(s)(⊖φ)(s)μ(t)μ2(s)β(s)eφ(t,σ(s))=−μ(s)φ(s)μ(s)(⊖φ)(s)β(s)eφ(t,σ(s))μ(t)=−μ(t)φ(t)μ(s)(⊖φ)(s)β(s)eφ(t,σ(s))μ(t)=−μ(s)(⊖φ)(s)β(s)φ(t)eφ(t,σ(s))=−Dβ(s)φ(t)eφ(t,σ(s)) |
shows (4.8).
We can now bring our attention to the proof of the second Cushing–Henson conjecture on isolated time scales.
Proof of Theorem 4.2. We apply the generalized Jensen inequality (4.4) on time scales in the single forthcoming calculation to estimate
∫ν(t0)t0ˉx(t)Δt(3.3)=∫ν(t0)t0λ∫ν(t)te⊖(−α)(σ(s),t)α(s)K(s)ΔsΔt(4.6)=∫ν(t0)t0∫ν(t)twt(s)Δs∫ν(t)te⊖(−α)(σ(s),t)α(s)K(s)ΔsΔt(4.7)=∫ν(t0)t0∫ν(t)twt(s)Δs∫ν(t)twt(s)vt(s)ΔsΔt(4.4)≤∫ν(t0)t0∫ν(t)twt(s)vt(s)Δs∫ν(t)twt(s)ΔsΔt(4.6)=1λ∫ν(t0)t0∫ν(t)twt(s)vt(s)ΔsΔt(2.14)=1λ{∫ν(t0)t0∫σ(s)t0wt(s)vt(s)ΔtΔs+∫ν(ν(t0))ν(t0)∫ν(t0)ν−1(σ(s))wt(s)vt(s)ΔtΔs}(4.8)=−Dλ{∫ν(t0)t0β(s)∫σ(s)t0φ(t)eφ(t,σ(s))ΔtΔs+∫ν(ν(t0))ν(t0)β(s)∫ν(t0)ν−1(σ(s))φ(t)eφ(t,σ(s))ΔtΔs}(4.8)=Dλ{∫ν(t0)t0β(s)(eφ(t0,σ(s))−1)Δs+∫ν(ν(t0))ν(t0)β(s)(eφ(ν−1(σ(s)),σ(s))−eφ(ν(t0),σ(s)))Δs}(2.9)=Dλ{∫ν(t0)t0β(s)(eφ(t0,σ(s))−1)Δs+∫ν(t0)t0νΔ(s)βν(s)(eφ(σ(s),σ(ν(s)))−eφ(ν(t0),σ(ν(s))))Δs}(2.10)=Dλ{∫ν(t0)t0β(s)(eφ(t0,σ(s))−1)Δs+∫ν(t0)t0νΔ(s)βν(s)(eφ(t0,ν(t0))−eφ(t0,σ(s)))Δs}(4.5)=Dλ{∫ν(t0)t0β(s)(eφ(t0,σ(s))−1)Δs+∫ν(t0)t0β(s)(eφ(t0,ν(t0))−eφ(t0,σ(s)))Δs}(4.8)=Dλ∫ν(t0)t0β(s)(eφ(t0,ν(t0))−1)Δs(3.6)=D∫ν(t0)t0β(s)Δs=∫ν(t0)t01−C1σΔ(s)−CK(s)Δs, |
where the last equality holds because
Dβ(s)(4.2)=(1−C)(1−μ(s)α(s))K(s)Cμ(s)α(s)(4.1)=(1−C)σΔCC(1−σΔC). |
Note also, that due to the strict convexity of 1/z (see [32, Theorem 2.2]), equality holds in (4.4) if and only if vt(s) is independent of s , which means β∈P1 .
Remark 4.4. If ( A 4 ) and β∈P1 hold, then the unique periodic solution of (1.4) is Dβ , where D is given by (4.2). In detail, the unique periodic solution is
cDμ(t),wherec:=(K⊖(−α))(t)isconstant. | (4.9) |
We can check (4.9) in two simple ways, namely by calculating it from (3.3), i.e.,
ˉx(t)(3.3)=λ∫ν(t)te⊖(−α)(σ(s),t)α(s)K(s)Δs(3.3)=λ∫ν(t)te−α(t,s)(⊖(−α)K)(s)Δs=λc∫ν(t)te−α(t,s)Δs(3.5)=λc∫ν(t)teφ(t,s)μ(t)μ(s)Δs(3.5)=λc∫ν(t)teφ(t,s)μ(t)(⊖φ)(s)μ(s)(⊖φ)(s)Δs(4.2)=λcDμ(t)∫ν(t)te⊖φ(s,t)(⊖φ)(s)Δs(3.6)=cDμ(t), |
or by directly checking that it is 1 -periodic (this is clear from Lemma 2.14) and verifying that it solves (1.4), i.e.,
α(cDμ)σ(1−cDμK)−(cDμ)Δ=αcDμσ(1−D(1−μα)μα)+cDμΔμμσ=cDμμσ(μα−D(1−μα)+σΔ−1)=cDμμσ((μα−1)(1+D)+σΔ)=cDμμσ(−C(1+D)+1)σΔ(4.2)=0. |
It can also be verified easily that for (4.9), the inequality (4.3) becomes an equality with cD on both sides:
1ω∫ν(t0)t0cDμ(t)Δt=cD |
and
1ω∫ν(t0)t01−C1σΔ(t)−CK(t)Δt=1ω∫ν(t0)t0(1−C)cα(t)(1σΔ(t)−C)(1−μ(t)α(t))Δt=1ω∫ν(t0)t0(1−C)cα(t)(1σΔ(t)−C)CσΔΔt=1ω∫ν(t0)t0(1−C)cα(t)C−C2σΔ(t)Δt=1ω∫ν(t0)t0(1−C)cα(t)Cμ(t)α(t)Δt=cDω∫ν(t0)t0Δtμ(t)=cD. |
Remark 4.5. For all isolated time scales such that σΔ=c is constant for some c∈R , Theorem 4.2 implies that the average of the unique periodic solution is less than or equal to the average of the carrying capacity multiplied by the constant ηc−1η−1 . If T=Z , then c=1 , and the classical second Cushing–Henson conjecture is retrieved. If T=qN0 , then c=q , and Theorem 4.2 is consistent with the second Cushing–Henson conjecture formulated in [23], see also (1.2). The inequality (4.3) reveals that the upper bound is increasing in σΔ . Since
1−C1σΔ−C=1+1−1σΔ1σΔ−C=1+σΔ−11−CσΔ(4.1)=1+μΔμα, |
we can write (4.3) also as
∫ν(t0)t0ˉx(t)Δt≤∫ν(t0)t0(1+μΔ(t)μ(t)α(s))K(t)Δt. |
Hence if μΔ(s)<0 for all s∈[t0,ν(t0)) , then the upper bound for the average periodic solution is smaller than the average of the carrying capacities, suggesting an even stronger negative effect of periodicity onto the population compared to the classical case. If μΔ(s)>0 for all s∈[t0,ν(t0)) , then the second Cushing–Henson conjecture does not necessarily hold as the multiplicative factor exceeds one. The inequality (4.3) exposes the effects of the time structure onto the upper bound of the mean periodic population.
In this work, we studied the Beverton–Holt model on arbitrary isolated time scales with time-dependent coefficients. Using the recently formulated periodicity concept for isolated time scales allows to address the Cushing–Henson conjectures for nonperiodic time scales. After an introduction in Section 1 and some preliminaries in Section 2, in Section 3, we provided conditions for the existence and uniqueness of a globally asymptotically stable periodic solution. This generalizes the first Cushing–Henson conjecture to an arbitrary isolated time scale. The provided theorem, when applied to the special case of the discrete time domain Z , coincides with results in existing literature. The presented conditions for existence and uniqueness of the periodic solution and its global asymptotic stability are equivalent to the conditions in the first conjecture presented in [21]. It therefore generalizes the classical formulation of the first Cushing–Henson conjecture. We also showed that our result is consistent [24, Conjecture 1] in the special case of a quantum time scale. A special subcase in this time scale was discussed in [23]. As we outlined, Theorem 3.1 contains these works as special cases. In Section 4, we focused on the discussion of the second Cushing–Henson conjecture on arbitrary isolated time scales. In the classical case, when T=Z , the conjecture concerns the effects of a periodic environment under constant proliferation rate, mathematically formulated by an upper bound of the average periodic solution. The derived upper bound in Theorem 4.2 is, in contrast to the classical case, a weighted average dependent on changes of the time scale. This highlights that the second Cushing–Henson conjecture does not necessarily hold in general, and its statement depends on the change of the time scale. If the time scale changes with a constant rate, that is, σΔ is constant, then the average of the periodic solution is bounded by a factor times the average of the carrying capacity. Examples of this special case contain the discrete and the quantum time scale. For both of these time scales, Theorem 4.2 is consistent with existing formulations of the second Cushing–Henson conjecture in [18,21,23]. Our results complement work in [22], where the authors consider the Cushing–Henson conjectures for the Beverton–Holt model on periodic time scales. In contrast to isolated time scales, ω -periodic time scales assume t+ω∈T for all t∈T . Since periodic and isolated time scales intersect but neither is a subset of the other, modifications of the conjectures remain unknown on arbitrary time scales. We highlight that this work is an application of the new definition of periodicity on isolated time scales, defined in [26]. The introduced method of the application of periodicity on isolated time scales can now be extended to other models, such as delay Beverton–Holt models. In fact, in [33,34,35], the Cushing–Henson conjectures for different delay Beverton–Holt models in the discrete case are discussed. The tools used in this paper can be used to establish these results on an arbitrary isolated time scale.
The authors would like to thank the three anonymous referees and the handling editor for many useful comments and suggestions, leading to a substantial improvement of the presentation of this article. The second author acknowledges partial support by the project UnB DPI/DPG - 03/2020 and CNPq grant 307582/2018-3.
The authors declare there is no conflict of interest.
[1] |
Abdul-Rahim R, Bohari SA, Aman A, et al. (2022) Benefit–risk perceptions of FinTech adoption for sustainability from bank consumers' perspective: The moderating role of fear of COVID-19. Sustainability 14: 8357. https://doi.org/10.3390/su14148357 doi: 10.3390/su14148357
![]() |
[2] |
Aduba JJ (2021) On the determinants, gains and challenges of electronic banking adoption in Nigeria. Int J Soc Econ 48: 1021–1043. https://doi.org/10.1108/IJSE-07-2020-0452 doi: 10.1108/IJSE-07-2020-0452
![]() |
[3] |
Alaabed A, Masih M, Mirakhor A (2016) Investigating risk shifting in Islamic banks in the dual banking systems of OIC member countries: an application of two-step dynamic GMM. Risk Manage 18: 236–263. https://doi.org/10.1057/s41283-016-0007-3 doi: 10.1057/s41283-016-0007-3
![]() |
[4] |
Aracil E, Nájera-Sánchez JJ, Forcadell FJ (2021) Sustainable banking: A literature review and integrative framework. Financ Res Lett 42: 101932. https://doi.org/10.1016/j.frl.2021.101932 doi: 10.1016/j.frl.2021.101932
![]() |
[5] | Ashrafi DM, Dovash RH, Kabir MR (2022) Determinants of fintech service continuance behavior: moderating role of transaction security and trust. J Global Bus Technol 18. Available from: https://www.proquest.com/docview/2766511260?pq-origsite = gscholar & fromopenview = true & sourcetype = Scholarly%20Journals. |
[6] |
Ashta A, Herrmann H (2021) Artificial intelligence and fintech: An overview of opportunities and risks for banking, investments, and microfinance. Strateg Change 30: 211–222. https://doi.org/10.1002/jsc.2404 doi: 10.1002/jsc.2404
![]() |
[7] |
Banna H, Hassan MK, Ahmad R, et al. (2022) Islamic banking stability amidst the COVID-19 pandemic: the role of digital financial inclusion. Int J Islamic Middle 15: 310–330. https://doi.org/10.1108/IMEFM-08-2020-0389 doi: 10.1108/IMEFM-08-2020-0389
![]() |
[8] |
Boratyńska K (2019) Impact of digital transformation on value creation in Fintech services: an innovative approach. J Promot Manage 25: 631–639. https://doi.org/10.1080/10496491.2019.1585543 doi: 10.1080/10496491.2019.1585543
![]() |
[9] |
Bose S, Khan HZ, Rashid A, et al. (2018) What drives green banking disclosure? An institutional and corporate governance perspective. Asia Pac J Manage 35: 501–527. https://doi.org/10.1007/s10490-017-9528-x doi: 10.1007/s10490-017-9528-x
![]() |
[10] |
Brahmi M, Esposito L, Parziale A, et al. (2023) The role of greener innovations in promoting financial inclusion to achieve carbon neutrality: an integrative review. Economies 11: 194. https://doi.org/10.3390/economies11070194 doi: 10.3390/economies11070194
![]() |
[11] |
Çera G, Phan QPT, Androniceanu A, et al. (2020) Financial capability and technology implications for online shopping. EaM Ekon Manag. https://doi.org/10.15240/tul/001/2020-2-011 doi: 10.15240/tul/001/2020-2-011
![]() |
[12] |
Chang HY, Liang LW, Liu YL (2021) Using environmental, social, governance (ESG) and financial indicators to measure bank cost efficiency in Asia. Sustainability 13: 11139. https://doi.org/10.3390/su132011139 doi: 10.3390/su132011139
![]() |
[13] |
Coffie CPK, Zhao H, Adjei Mensah I (2020) Panel econometric analysis on mobile payment transactions and traditional banks effort toward financial accessibility in Sub-Sahara Africa. Sustainability 12: 895. https://doi.org/10.3390/su12030895 doi: 10.3390/su12030895
![]() |
[14] |
Cumming D, Johan S, Reardon R (2023) Global fintech trends and their impact on international business: a review. Multinatl Bus Rev 31: 413–436. https://doi.org/10.1108/MBR-05-2023-0077 doi: 10.1108/MBR-05-2023-0077
![]() |
[15] |
Danladi S, Prasad M, Modibbo UM, et al. (2023) Attaining Sustainable Development Goals through Financial Inclusion: Exploring Collaborative Approaches to Fintech Adoption in Developing Economies. Sustainability 15: 13039. https://doi.org/10.3390/su151713039 doi: 10.3390/su151713039
![]() |
[16] |
Davis FD, Bagozzi RP, Warshaw PR (1989) User acceptance of computer technology: A comparison of two theoretical models. Manage Sci 35: 982–1003. https://doi.org/10.1287/mnsc.35.8.982 doi: 10.1287/mnsc.35.8.982
![]() |
[17] |
Dewi IGAAO, Dewi IGAAP (2017) Corporate social responsibility, green banking, and going concern on banking company in Indonesia stock exchange. Int J Soc Sci Hum 1: 118–134. https://doi.org/10.29332/ijssh.v1n3.65 doi: 10.29332/ijssh.v1n3.65
![]() |
[18] |
Diep NTN, Canh TQ (2022) Impact analysis of peer-to-peer Fintech in Vietnam's banking industry. J Int Stud 15. https://doi.org/10.14254/2071-8330.2022/15-3/12 doi: 10.14254/2071-8330.2022/15-3/12
![]() |
[19] |
Dong Y, Chung M, Zhou C, et al. (2018) Banking on "mobile money": The implications of mobile money services on the value chain. Manuf Serv Oper Manag. https://doi.org/10.1287/msom.2018.0717 doi: 10.1287/msom.2018.0717
![]() |
[20] |
Eisingerich AB, Bell SJ (2008) Managing networks of interorganizational linkages and sustainable firm performance in business‐to‐business service contexts. J Serv Mark 22: 494–504. https://doi.org/10.1108/08876040810909631 doi: 10.1108/08876040810909631
![]() |
[21] |
Ellili NOD (2022) Is there any association between FinTech and sustainability? Evidence from bibliometric review and content analysis. J Financ Serv Mark 28: 748–762. https://doi.org/10.1057/s41264-022-00200-w doi: 10.1057/s41264-022-00200-w
![]() |
[22] |
Fenwick M, Vermeulen EP (2020) Banking and regulatory responses to FinTech revisited-building the sustainable financial service'ecosystems' of tomorrow. Singap J Legal Stud 2020: 165–189. https://doi.org/10.2139/ssrn.3446273 doi: 10.2139/ssrn.3446273
![]() |
[23] |
Gangi F, Meles A, Daniele LM, et al. (2021) Socially responsible investment (SRI): from niche to mainstream. The Evolution of Sustainable Investments and Finance: Theoretical Perspectives and New Challenges, 1–58. https://doi.org/10.1007/978-3-030-70350-9_1 doi: 10.1007/978-3-030-70350-9_1
![]() |
[24] |
Gbongli K, Xu Y, Amedjonekou KM, et al. (2020) Evaluation and classification of mobile financial services sustainability using structural equation modeling and multiple criteria decision-making methods. Sustainability 12: 1288. https://doi.org/10.3390/su12041288 doi: 10.3390/su12041288
![]() |
[25] |
Goodell JW, Kumar S, Lim WM, et al. (2021) Artificial intelligence and machine learning in finance: Identifying foundations, themes, and research clusters from bibliometric analysis. J Behav Exp Financ 32: 100577. https://doi.org/10.1016/j.jbef.2021.100577 doi: 10.1016/j.jbef.2021.100577
![]() |
[26] |
Gozman D, Willcocks L (2019) The emerging Cloud Dilemma: Balancing innovation with cross-border privacy and outsourcing regulations. J Bus Res 97: 235–256. https://doi.org/10.1016/j.jbusres.2018.06.006 doi: 10.1016/j.jbusres.2018.06.006
![]() |
[27] |
Gruin J, Knaack P (2020) Not just another shadow bank: Chinese authoritarian capitalism and the 'developmental'promise of digital financial innovation. New Polit Econ 25: 370–387. https://doi.org/10.1080/13563467.2018.1562437 doi: 10.1080/13563467.2018.1562437
![]() |
[28] |
Guang-Wen Z, Siddik AB (2023) The effect of Fintech adoption on green finance and environmental performance of banking institutions during the COVID-19 pandemic: the role of green innovation. Environ Sci Pollut Res 30: 25959–25971. https://doi.org/10.1007/s11356-022-23956-z doi: 10.1007/s11356-022-23956-z
![]() |
[29] |
Guo Y, Holland J, Kreander N (2014) An exploration of the value creation process in bank-corporate communications. J Commun manage 18: 254–270. https://doi.org/10.1108/JCOM-10-2012-0079 doi: 10.1108/JCOM-10-2012-0079
![]() |
[30] | Hassan MK, Rabbani MR, Ali MAM (2020) Challenges for the Islamic Finance and banking in post COVID era and the role of Fintech. J Econ Coop Dev 41: 93–116. Available from: https://www.proquest.com/docview/2503186452?pq-origsite = gscholar & fromopenview = true & sourcetype = Scholarly%20Journals. |
[31] | Hassan SM, Rahman Z, Paul J (2022) Consumer ethics: A review and research agenda. Psychol Market 39: 111–130. |
[32] |
He J, Zhang S (2022) How digitalized interactive platforms create new value for customers by integrating B2B and B2C models? An empirical study in China. J Bus Res 142: 694–706. https://doi.org/10.1016/j.jbusres.2022.01.004 doi: 10.1016/j.jbusres.2022.01.004
![]() |
[33] |
Hommel K, Bican PM (2020) Digital entrepreneurship in finance: Fintechs and funding decision criteria. Sustainability 12: 8035. https://doi.org/10.3390/su12198035 doi: 10.3390/su12198035
![]() |
[34] |
Hyun S (2022) Current Status and Challenges of Green Digital Finance in Korea. Green Digital Finance and Sustainable Development Goals, 243–261. https://doi.org/10.1007/978-981-19-2662-4_12 doi: 10.1007/978-981-19-2662-4_12
![]() |
[35] |
Ⅱ WWC, Demrig I (2002) Investment and capitalisation of firms in the USA. Int J Technol Manage 24: 391–418. https://doi.org/10.1504/IJTM.2002.003062 doi: 10.1504/IJTM.2002.003062
![]() |
[36] |
Iman N (2018) Is mobile payment still relevant in the fintech era? Electron Commer Res Appl 30: 72–82. https://doi.org/10.1016/j.elerap.2018.05.009 doi: 10.1016/j.elerap.2018.05.009
![]() |
[37] |
Ji F, Tia A (2022) The effect of blockchain on business intelligence efficiency of banks. Kybernetes 51: 2652–2668. https://doi.org/10.1108/K-10-2020-0668 doi: 10.1108/K-10-2020-0668
![]() |
[38] |
Jibril AB, Kwarteng MA, Botchway RK, et al. (2020) The impact of online identity theft on customers' willingness to engage in e-banking transaction in Ghana: A technology threat avoidance theory. Cogent Bus Manag 7: 1832825. https://doi.org/10.1080/23311975.2020.1832825 doi: 10.1080/23311975.2020.1832825
![]() |
[39] |
Khan A, Goodell JW, Hassan MK, et al. (2022) A bibliometric review of finance bibliometric papers. Financ Res Lett 47: 102520. https://doi.org/10.1016/j.frl.2021.102520 doi: 10.1016/j.frl.2021.102520
![]() |
[40] |
Khan HU, Sohail M, Nazir S, et al. (2023) Role of authentication factors in Fin-tech mobile transaction security. J Big Data 10: 138. https://doi.org/10.1186/s40537-023-00807-3 doi: 10.1186/s40537-023-00807-3
![]() |
[41] |
Kumar S, Lim WM, Sivarajah U, et al. (2023) Artificial intelligence and blockchain integration in business: trends from a bibliometric-content analysis. Inform Syst Front 25: 871–896. https://doi.org/10.1007/s10796-022-10279-0 doi: 10.1007/s10796-022-10279-0
![]() |
[42] |
Kumari A, Devi NC (2022) The Impact of FinTech and Blockchain Technologies on Banking and Financial Services. Technol Innov Manage Rev 12. https://doi.org/10.22215/timreview/1481 doi: 10.22215/timreview/1481
![]() |
[43] |
Lai X, Yue S, Guo C, et al. (2023) Does FinTech reduce corporate excess leverage? Evidence from China. Econ Anal Policy 77: 281–299. https://doi.org/10.1016/j.eap.2022.11.017 doi: 10.1016/j.eap.2022.11.017
![]() |
[44] |
Lee WS, Sohn SY (2017) Identifying emerging trends of financial business method patents. Sustainability 9: 1670. https://doi.org/10.3390/su9091670 doi: 10.3390/su9091670
![]() |
[45] |
Lekakos G, Vlachos P, Koritos C (2014) Green is good but is usability better? Consumer reactions to environmental initiatives in e-banking services. Ethics Inf Technol 16: 103–117. https://doi.org/10.1007/s10676-014-9337-6 doi: 10.1007/s10676-014-9337-6
![]() |
[46] | Mądra-Sawicka M (2020) Financial management in the big data era. In Management in the Era of Big Data, 71–81, Auerbach Publications. https://doi.org/10.1201/9781003057291-6 |
[47] |
Mejia-Escobar JC, González-Ruiz JD, Duque-Grisales E (2020) Sustainable financial products in the Latin America banking industry: Current status and insights. Sustainability 12: 5648. https://doi.org/10.3390/su12145648 doi: 10.3390/su12145648
![]() |
[48] | Mhlanga D (2023) FinTech for Sustainable Development in Emerging Markets with Case Studies. In FinTech and Artificial Intelligence for Sustainable Development: The Role of Smart Technologies in Achieving Development Goals, 337–363, Springer. https://doi.org/10.1007/978-3-031-37776-1_15 |
[49] |
Mohr I, Fuxman L, Mahmoud AB (2022) A triple-trickle theory for sustainable fashion adoption: the rise of a luxury trend. J Fash Mark Manag Int J 26: 640–660. https://doi.org/10.1108/JFMM-03-2021-0060 doi: 10.1108/JFMM-03-2021-0060
![]() |
[50] |
Muniz Jr AM, Schau HJ (2005) Religiosity in the abandoned Apple Newton brand community. J Consum Res 31: 737–747. https://doi.org/10.1086/426607 doi: 10.1086/426607
![]() |
[51] |
Naruetharadhol P, Ketkaew C, Hongkanchanapong N, et al. (2021) Factors affecting sustainable intention to use mobile banking services. Sage Open 11: 21582440211029925. https://doi.org/10.1177/21582440211029925 doi: 10.1177/21582440211029925
![]() |
[52] |
Nenavath S (2022) Impact of fintech and green finance on environmental quality protection in India: By applying the semi-parametric difference-in-differences (SDID). Renew Energ 193: 913–919. https://doi.org/10.1016/j.renene.2022.05.020 doi: 10.1016/j.renene.2022.05.020
![]() |
[53] |
Nosratabadi S, Pinter G, Mosavi A, et al. (2020) Sustainable banking; evaluation of the European business models. Sustainability 12: 2314. https://doi.org/10.3390/su12062314 doi: 10.3390/su12062314
![]() |
[54] |
Ortas E, Burritt RL, Moneva JM (2013) Socially Responsible Investment and cleaner production in the Asia Pacific: does it pay to be good? J Cleaner Prod 52: 272–280. https://doi.org/10.1016/j.jclepro.2013.02.024 doi: 10.1016/j.jclepro.2013.02.024
![]() |
[55] |
Oseni UA, Adewale AA, Omoola SO (2018) The feasibility of online dispute resolution in the Islamic banking industry in Malaysia: An empirical legal analysis. Int J Law Manag 60: 34–54. https://doi.org/10.1108/IJLMA-06-2016-0057 doi: 10.1108/IJLMA-06-2016-0057
![]() |
[56] |
Paiva BM, Ferreira FA, Carayannis EG, et al. (2021) Strategizing sustainability in the banking industry using fuzzy cognitive maps and system dynamics. Int J Sustain Dev World Ecol 28: 93–108. https://doi.org/10.1080/13504509.2020.1782284 doi: 10.1080/13504509.2020.1782284
![]() |
[57] |
Parmentola A, Petrillo A, Tutore I, et al. (2022) Is blockchain able to enhance environmental sustainability? A systematic review and research agenda from the perspective of Sustainable Development Goals (SDGs). Bus Strateg Environ 31: 194–217. https://doi.org/10.1002/bse.2882 doi: 10.1002/bse.2882
![]() |
[58] |
Paul J, Lim WM, O'Cass A, et al. (2021) Scientific procedures and rationales for systematic literature reviews (SPAR‐4‐SLR). Int J Consum Stud 45: O1–O16. https://doi.org/10.1111/ijcs.12695 doi: 10.1111/ijcs.12695
![]() |
[59] |
Puschmann T, Hoffmann CH, Khmarskyi V (2020) How green FinTech can alleviate the impact of climate change—the case of Switzerland. Sustainability 12: 10691. https://doi.org/10.3390/su122410691 doi: 10.3390/su122410691
![]() |
[60] |
Rahman S, Moral IH, Hassan M, et al. (2022) A systematic review of green finance in the banking industry: perspectives from a developing country. Green Financ 4: 347–363. https://doi.org/10.3934/GF.2022017 doi: 10.3934/GF.2022017
![]() |
[61] |
Ryu HS, Ko KS (2020) Sustainable development of Fintech: Focused on uncertainty and perceived quality issues. Sustainability 12: 7669. https://doi.org/10.3390/su12187669 doi: 10.3390/su12187669
![]() |
[62] |
Sagnier C, Loup-Escande E, Lourdeaux D, et al. (2020) User acceptance of virtual reality: an extended technology acceptance model. Int J Hum–Comput Interact 36: 993–1007. https://doi.org/10.1080/10447318.2019.1708612 doi: 10.1080/10447318.2019.1708612
![]() |
[63] |
Sethi P, Chakrabarti D, Bhattacharjee S (2020) Globalization, financial development and economic growth: Perils on the environmental sustainability of an emerging economy. J Policy Model 42: 520–535. https://doi.org/10.1016/j.jpolmod.2020.01.007 doi: 10.1016/j.jpolmod.2020.01.007
![]() |
[64] |
Singh RK, Mishra R, Gupta S, et al. (2023) Blockchain applications for secured and resilient supply chains: A systematic literature review and future research agenda. Comput Ind Eng 175: 108854. https://doi.org/10.1016/j.cie.2022.108854 doi: 10.1016/j.cie.2022.108854
![]() |
[65] |
Sun Y, Luo B, Wang S, et al. (2021) What you see is meaningful: Does green advertising change the intentions of consumers to purchase eco‐labeled products? Bus Strateg Environ 30: 694–704. https://doi.org/10.1002/bse.2648 doi: 10.1002/bse.2648
![]() |
[66] |
Talom FSG, Tengeh RK (2019) The impact of mobile money on the financial performance of the SMEs in Douala, Cameroon. Sustainability 12: 183. https://doi.org/10.3390/su12010183 doi: 10.3390/su12010183
![]() |
[67] |
Taneja S, Siraj A, Ali L, et al. (2023) Is fintech implementation a strategic step for sustainability in today's changing landscape? An empirical investigation. IEEE T Eng Manage. https://doi.org/10.3390/su12010183 doi: 10.3390/su12010183
![]() |
[68] |
Tara K, Singh S, Kumar R, et al. (2019) Geographical locations of banks as an influencer for green banking adoption. Prabandhan: Indian J Manag 12: 21–35. https://doi.org/10.17010/pijom/2019/v12i1/141425 doi: 10.17010/pijom/2019/v12i1/141425
![]() |
[69] |
Tchamyou VS, Erreygers G, Cassimon D (2019) Inequality, ICT and financial access in Africa. Technol Forecast Soc Change 139: 169–184. https://doi.org/10.1016/j.techfore.2018.11.004 doi: 10.1016/j.techfore.2018.11.004
![]() |
[70] |
Tripathi R (2023) Framework of green finance to attain sustainable development goals: an empirical evidence from the TCCM approach. Benchmarking. https://doi.org/10.1108/BIJ-05-2023-0311 doi: 10.1108/BIJ-05-2023-0311
![]() |
[71] |
Truby J, Brown R, Dahdal A (2020) Banking on AI: mandating a proactive approach to AI regulation in the financial sector. Law Financ Mark Rev 14: 110–120. https://doi.org/10.1080/17521440.2020.1760454 doi: 10.1080/17521440.2020.1760454
![]() |
[72] |
Tsindeliani IA, Proshunin MM, Sadovskaya TD, et al. (2022) Digital transformation of the banking system in the context of sustainable development. J Money Laund Contro 25: 165–180. https://doi.org/10.1108/JMLC-02-2021-0011 doi: 10.1108/JMLC-02-2021-0011
![]() |
[73] |
Ullah A, Pinglu C, Ullah S, et al. (2023) Impact of intellectual capital efficiency on financial stability in banks: Insights from an emerging economy. Int J Financ Econ 28: 1858–1871. https://doi.org/10.1002/ijfe.2512 doi: 10.1002/ijfe.2512
![]() |
[74] |
Yadav MS (2010) The decline of conceptual articles and implications for knowledge development. J Mark 74: 1–19. https://doi.org/10.1509/jmkg.74.1.1 doi: 10.1509/jmkg.74.1.1
![]() |
[75] |
Yan C, Siddik AB, Yong L, et al. (2022) A two-staged SEM-artificial neural network approach to analyze the impact of FinTech adoption on the sustainability performance of banking firms: The mediating effect of green finance and innovation. Systems 10: 148. https://doi.org/10.3390/systems10050148 doi: 10.3390/systems10050148
![]() |
[76] |
Yang C, Masron TA (2022) Impact of digital finance on energy efficiency in the context of green sustainable development. Sustainability 14: 11250. https://doi.org/10.3390/su14181125 doi: 10.3390/su14181125
![]() |
[77] |
Yigitcanlar T, Cugurullo F (2020) The sustainability of artificial intelligence: An urbanistic viewpoint from the lens of smart and sustainable cities. Sustainability 12: 8548. https://doi.org/10.3390/su1220854 doi: 10.3390/su1220854
![]() |
[78] |
Zhang Y (2023) Impact of green finance and environmental protection on green economic recovery in South Asian economies: mediating role of FinTech. Econ Chang Restruct 56: 2069–2086. https://doi.org/10.1007/s10644-023-09500-0 doi: 10.1007/s10644-023-09500-0
![]() |
[79] | Zhao D, Strotmann A (2015) Analysis and visualization of citation networks. Morgan & Claypool Publishers. |
[80] |
Zhao Q, Tsai PH, Wang JL (2019) Improving financial service innovation strategies for enhancing china's banking industry competitive advantage during the fintech revolution: A Hybrid MCDM model. Sustainability 11: 1419. https://doi.org/10.3390/su11051419 doi: 10.3390/su11051419
![]() |
[81] |
Zuo L, Strauss J, Zuo L (2021) The digitalization transformation of commercial banks and its impact on sustainable efficiency improvements through investment in science and technology. Sustainability 13: 11028. https://doi.org/10.3390/su131911028 doi: 10.3390/su131911028
![]() |
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3. | Manuel De la Sen, Santiago Alonso-Quesada, Asier Ibeas, Aitor J. Garrido, A. E. Matouk, On a Coupled Time-Varying Beverton–Holt Model with Two Habitats Subject to Harvesting, Repopulation, and Mixed Migratory Flows of Populations, 2023, 2023, 1607-887X, 1, 10.1155/2023/6050789 | |
4. | Mariem Mohamed Abdelahi, Mohamed Ahmed Sambe, Elkhomeini Moulay Ely, Existence results for some generalized Sigmoid Beverton-Holt models in time scales, 2023, 10, 2353-0626, 10.1515/msds-2022-0166 | |
5. | Tri Truong, Martin Bohner, Ewa Girejko, Agnieszka B. Malinowska, Ngo Van Hoa, Granular fuzzy calculus on time scales and its applications to fuzzy dynamic equations, 2025, 690, 00200255, 121547, 10.1016/j.ins.2024.121547 |