
The efficiency of agro-tourism integration has become an important research object in evaluating agricultural efficiency. However, traditional efficiency evaluation theories and methods assume that all decision-making units are independent of each other and cannot effectively deal with the complex relationship between agriculture and tourism development. Based on the cooperative relationship between agriculture and tourism, this study constructed a data envelopment analysis (DEA) model based on cross-efficiency. It used ten input variables and six output variables from the agricultural and tourism systems to analyze the efficiency of agro-tourism integration in 31 provinces in mainland China from 2010 to 2019. The research results show that the efficiency of agro-tourism integration in China is relatively high and tourism significantly promotes agriculture. Still, the promotion efficiency of agriculture on tourism is low and the integration efficiency of agriculture and tourism in different provinces is significantly different with spatially differentiated features. From the perspective of dynamic trends, hot and sub-hot spots continue to gather in the developed eastern provinces, while cold spots and sub-cold spots mainly gather in the northwest region. Finally, eight indicators were selected to analyze the reasons for forming the spatial differentiation characteristics of China's efficiency of agro-tourism integration.
Citation: Huajin Li, Songbiao Zhang, Yue Deng, Huilin Wang. Research on the efficiency of agro-tourism integration in China: Based on the DEA cross-efficiency model[J]. AIMS Mathematics, 2023, 8(10): 23164-23182. doi: 10.3934/math.20231178
[1] | Yayun Fu, Mengyue Shi . A conservative exponential integrators method for fractional conservative differential equations. AIMS Mathematics, 2023, 8(8): 19067-19082. doi: 10.3934/math.2023973 |
[2] | Yong-Chao Zhang . Least energy solutions to a class of nonlocal Schrödinger equations. AIMS Mathematics, 2024, 9(8): 20763-20772. doi: 10.3934/math.20241009 |
[3] | Tingting Ma, Yuehua He . An efficient linearly-implicit energy-preserving scheme with fast solver for the fractional nonlinear wave equation. AIMS Mathematics, 2023, 8(11): 26574-26589. doi: 10.3934/math.20231358 |
[4] | Karmina K. Ali, Resat Yilmazer . Discrete fractional solutions to the effective mass Schrödinger equation by mean of nabla operator. AIMS Mathematics, 2020, 5(2): 894-903. doi: 10.3934/math.2020061 |
[5] | Erdal Bas, Ramazan Ozarslan . Theory of discrete fractional Sturm–Liouville equations and visual results. AIMS Mathematics, 2019, 4(3): 593-612. doi: 10.3934/math.2019.3.593 |
[6] | Dengfeng Lu, Shuwei Dai . On a class of three coupled fractional Schrödinger systems with general nonlinearities. AIMS Mathematics, 2023, 8(7): 17142-17153. doi: 10.3934/math.2023875 |
[7] | Mubashir Qayyum, Efaza Ahmad, Hijaz Ahmad, Bandar Almohsen . New solutions of time-space fractional coupled Schrödinger systems. AIMS Mathematics, 2023, 8(11): 27033-27051. doi: 10.3934/math.20231383 |
[8] | Xiaojun Zhou, Yue Dai . A spectral collocation method for the coupled system of nonlinear fractional differential equations. AIMS Mathematics, 2022, 7(4): 5670-5689. doi: 10.3934/math.2022314 |
[9] | Zunyuan Hu, Can Li, Shimin Guo . Fast finite difference/Legendre spectral collocation approximations for a tempered time-fractional diffusion equation. AIMS Mathematics, 2024, 9(12): 34647-34673. doi: 10.3934/math.20241650 |
[10] | Xiao-Yu Li, Yu-Lan Wang, Zhi-Yuan Li . Numerical simulation for the fractional-in-space Ginzburg-Landau equation using Fourier spectral method. AIMS Mathematics, 2023, 8(1): 2407-2418. doi: 10.3934/math.2023124 |
The efficiency of agro-tourism integration has become an important research object in evaluating agricultural efficiency. However, traditional efficiency evaluation theories and methods assume that all decision-making units are independent of each other and cannot effectively deal with the complex relationship between agriculture and tourism development. Based on the cooperative relationship between agriculture and tourism, this study constructed a data envelopment analysis (DEA) model based on cross-efficiency. It used ten input variables and six output variables from the agricultural and tourism systems to analyze the efficiency of agro-tourism integration in 31 provinces in mainland China from 2010 to 2019. The research results show that the efficiency of agro-tourism integration in China is relatively high and tourism significantly promotes agriculture. Still, the promotion efficiency of agriculture on tourism is low and the integration efficiency of agriculture and tourism in different provinces is significantly different with spatially differentiated features. From the perspective of dynamic trends, hot and sub-hot spots continue to gather in the developed eastern provinces, while cold spots and sub-cold spots mainly gather in the northwest region. Finally, eight indicators were selected to analyze the reasons for forming the spatial differentiation characteristics of China's efficiency of agro-tourism integration.
Fractional calculus is a popular subject because of having a lot of application areas of theoretical and applied sciences, like engineering, physics, biology, etc. Discrete fractional calculus is more recent area than fractional calculus and it was first defined by Diaz–Osler [1], Miller–Ross [2] and Gray–Zhang [3]. More recently, the theory of discrete fractional calculus have begun to develop rapidly with Goodrich–Peterson [4], Baleanu et al. [5,6], Ahrendt et al. [7], Atici–Eloe [8,9], Anastassiou [10], Abdeljawad et al. [11,12,13,14,15,16], Hein et al. [17] and Cheng et al. [18], Mozyrska [19] and so forth [20,21,22,23,24,25].
Fractional Sturm–Liouville differential operators have been studied by Bas et al. [26,27], Klimek et al.[28], Dehghan et al. [29]. Besides that, Sturm–Liouville differential and difference operators were studied by [30,31,32,33]. In this study, we define DFHA operators and prove the self–adjointness of DFHA operator, some spectral properties of the operator.
More recently, Almeida et al. [34] have studied discrete and continuous fractional Sturm–Liouville operators, Bas–Ozarslan [35] have shown the self–adjointness of discrete fractional Sturm–Liouville operators and proved some spectral properties of the problem.
Sturm–Liouville equation having hydrogen atom potential is defined as follows
d2Rdr2+ardRdr−ℓ(ℓ+1)r2R+(E+ar)R=0(0<r<∞). |
In quantum mechanics, the study of the energy levels of the hydrogen atom leads to this equation. Where R is the distance from the mass center to the origin, ℓ is a positive integer, a is real number E is energy constant and r is the distance between the nucleus and the electron.
The hydrogen atom is a two–particle system and it composes of an electron and a proton. Interior motion of two particles around the center of mass corresponds to the movement of a single particle by a reduced mass. The distance between the proton and the electron is identified r and r is given by the orientation of the vector pointing from the proton to the electron. Hydrogen atom equation is defined as Schrödinger equation in spherical coordinates and in consequence of some transformations, this equation is defined as
y′′+(λ−l(l+1)x2+2x−q(x))y=0. |
Spectral theory of hydrogen atom equation is studied by [39,40,41]. Besides that, we can observe that hydrogen atom differential equation has series solution as follows ([39], p.268)
y(x)=a0xl+1{1−k−l−11!(2l+2).2xk+(k−l−1)(k−l−2)2!(2l+2)(2l+3)(2xk)2+…+(−1)n(k−l−1)(k−l−2)…3.2.1(k−1)!(2l+2)(2l+3)…(2l+n)(2xk)n},k=1,2,… | (1.1) |
Recently, Bohner and Cuchta [36,37] studied some special integer order discrete functions, like Laguerre, Hermite, Bessel and especially Cuchta mentioned the difficulty in obtaining series solution of discrete special functions in his dissertation ([38], p.100). In this regard, finding series solution of DFHA equations is an open problem and has some difficulties in the current situation. For this reason, we study to obtain solutions of DFHA eq.s in a different way with representation of solutions.
In this study, we investigate DFHA equation in Riemann–Liouville and Grü nwald–Letnikov sense. The aim of this study is to contribute to the spectral theory of DFHA operator and behaviors of eigenfunctions and also to obtain the solution of DFHA equation.
We investigate DFHA equation in three different ways;
i) (nabla left and right) Riemann–Liouville (R–L)sense,
L1x(t)=∇μa(b∇μx(t))+(l(l+1)t2−2t+q(t))x(t)=λx(t), 0<μ<1, |
ii) (delta left and right) Grünwald–Letnikov (G–L) sense,
L2x(t)=Δμ−(Δμ+x(t))+(l(l+1)t2−2t+q(t))x(t)=λx(t), 0<μ<1, |
iii) (nabla left) Riemann–Liouville (R–L)sense,
L3x(t)=∇μa(∇μax(t))+(l(l+1)t2−2t+q(t))x(t)=λx(t), 0<μ<1. |
Definition 2.1. [42] Falling and rising factorial functions are defined as follows respectively
tα_=Γ(t+1)Γ(t−α+1), | (2.1) |
t¯α=Γ(t+α)Γ(t), | (2.2) |
where Γ is the gamma function, α∈R.
Remark 2.1. Delta and nabla operators hold the following properties
Δtα_=αtα−1_,∇t¯α=αt¯α−1. | (2.3) |
Definition 2.2. [2,8,11] Nabla fractional sum operators are given as below,
(i) The left fractional sum of order μ>0 is defined by
∇−μax(t)=1Γ(μ)t∑s=a+1(t−ρ(s))¯μ−1x(s), t∈Na+1, | (2.4) |
(ii) The right fractional sum of order μ>0 is defined by
b∇−μx(t)=1Γ(μ)b−1∑s=t(s−ρ(t))¯μ−1x(s), t∈ b−1N, | (2.5) |
where ρ(t)=t−1 is called backward jump operators, Na={a,a+1,...}, bN={b,b−1,...}.
Definition 2.3. [12,14] Nabla fractional difference operators are as follows,
(i) The left fractional difference of order μ>0 is defined by
∇μax(t)=∇n∇−(n−μ)ax(t)=∇nΓ(n−μ)t∑s=a+1(t−ρ(s))¯n−μ−1x(s), t∈Na+1, | (2.6) |
(ii) The right fractional difference of order μ>0 is defined by
b∇μx(t)=(−1)n∇n∇−(n−μ)ax(t)=(−1)nΔnΓ(n−μ)b−1∑s=t(s−ρ(t))¯n−μ−1x(s), t∈ b−1N. | (2.7) |
Fractional differences in (2.6−2.7) are called the Riemann–Liouville (R–L) definition of the μ-th order nabla fractional difference.
Definition 2.4. [1,18] Fractional difference operators are given as follows
(i) The delta left fractional difference of order μ, 0<μ≤1, is defined by
Δμ−x(t)=1hμt∑s=0(−1)sμ(μ−1)...(μ−s+1)s!x(t−s), t=1,...,N. | (2.8) |
(ii) The delta right fractional difference of order μ, 0<μ≤1, is defined by
Δμ+x(t)=1hμN−t∑s=0(−1)sμ(μ−1)...(μ−s+1)s!x(t+s), t=0,..,N−1, | (2.9) |
fractional differences in (2.8−2.9) are called the Grünwald–Letnikov (G–L) definition of the μ-th order delta fractional difference.
Definition 2.5 [14] Integration by parts formula for R–L nabla fractional difference operator is defined by, u is defined on bN and v is defined on Na,
b−1∑s=a+1u(s)∇μav(s)=b−1∑s=a+1v(s)b∇μu(s). | (2.10) |
Definition 2.6. [34] Integration by parts formula for G–L delta fractional difference operator is defined by, u, v is defined on {0,1,...,n}, then
n∑s=0u(s)Δμ−v(s)=n∑s=0v(s)Δμ+u(s). | (2.11) |
Definition 2.7. [17] f:Na→R, s∈ℜ, Laplace transform is defined as follows,
La{f}(s)=∞∑k=1(1−s)k−1f(a+k), |
where ℜ=C∖{1} and ℜ is called the set of regressive (complex) functions.
Definition 2.8. [17] Let f,g:Na→R, all t∈Na+1, convolution of f and g is defined as follows
(f∗g)(t)=t∑s=a+1f(t−ρ(s)+a)g(s), |
where ρ(s) is the backward jump function defined in [42] as
ρ(s)=s−1. |
Theorem 2.1. [17] f,g:Na→R, convolution theorem is expressed as follows,
La{f∗g}(s)=La{f}La{g}(s). |
Lemma 2.1. [17] f:Na→R, the following property is valid,
La+1{f}(s)=11−sLa{f}(s)−11−sf(a+1). |
Theorem 2.2. [17] f:Na→R, 0<μ<1, Laplace transform of nabla fractional difference
La+1{∇μaf}(s)=sμLa+1{f}(s)−1−sμ1−sf(a+1),t∈Na+1. |
Definition 2.9. [17] For |p|<1, α>0, β∈R and t∈Na, Mittag–Leffler function is defined by
Ep,α,β(t,a)=∞∑k=0pk(t−a)¯αk+βΓ(αk+β+1). |
Theorem 2.3. [17] For |p|<1, α>0, β∈R, |1−s|<1 and |s|α>p, Laplace transform of Mittag–Leffler function is as follows,
La+1{Ep,α,β(.,a)}(s)=sα−β−1sα−p. |
Let us consider equations in three different forms;
i) L1 DFHA operator L1 is defined in (nabla left and right) R–L sense,
L1x(t)=∇μa(p(t)b∇μx(t))+(l(l+1)t2−2t+q(t))x(t)=λx(t), 0<μ<1, | (3.1) |
where l is a positive integer or zero, q(t)+2t−l(l+1)t2 are named potential function., λ is the spectral parameter, t∈[a+1,b−1], x(t)∈l2[a+1,b−1], a>0.
ii) L2 DFHA operator L2 is defined in (delta left and right) G–L sense,
L2x(t)=Δμ−(p(t)Δμ+x(t))+(l(l+1)t2−2t+q(t))x(t)=λx(t), 0<μ<1, | (3.2) |
where p,q,l,λ is as defined above, t∈[1,n], x(t)∈l2[0,n].
iii) L3 DFHA operator L3 is defined in (nabla left) R–L sense,
L3x(t)=∇μa(∇μax(t))+(l(l+1)t2−2t+q(t))x(t)=λx(t), 0<μ<1, | (3.3) |
p,q,l,λ is as defined above, t∈[a+1,b−1], a>0.
Theorem 3.1. DFHA operator L1 is self–adjoint.
Proof.
u(t)L1v(t)=u(t)∇μa(p(t)b∇μv(t))+u(t)(l(l+1)t2−2t+q(t))v(t), | (3.4) |
v(t)L1u(t)=v(t)∇μa(p(t)b∇μu(t))+v(t)(l(l+1)t2−2t+q(t))u(t). | (3.5) |
Subtracting (16−17) from each other
u(t)L1v(t)−v(t)L1u(t)=u(t)∇μa(p(t)b∇μv(t))−v(t)∇μa(p(t)b∇μu(t)) |
and applying definite sum operator to both side of the last equality, we have
b−1∑s=a+1(u(s)L1v(s)−v(s)L1u(s))=b−1∑s=a+1u(s)∇μa(p(s)b∇μv(s))−b−1∑s=a+1v(s)∇μa(p(s)b∇μu(s)). | (3.6) |
Applying the integration by parts formula (2.10) to right hand side of (18), we have
b−1∑s=a+1(u(s)L1v(s)−v(s)L1u(s))=b−1∑s=a+1p(s)b∇μv(s)b∇μu(s)−b−1∑s=a+1p(s)b∇μu(s)b∇μv(s)=0, |
⟨L1u,v⟩=⟨u,L1v⟩. |
The proof completes.
Theorem 3.2. Eigenfunctions, corresponding to distinct eigenvalues, of the equation (3.2) are orthogonal.
Proof. Assume that λα and λβ are two different eigenvalues corresponds to eigenfunctions u(n) and v(n) respectively for the equation (3.1),
∇μa(p(t)b∇μu(t))+(l(l+1)t2−2t+q(t))u(t)−λαu(t)=0,∇μa(p(t)b∇μv(t))+(l(l+1)t2−2t+q(t))v(t)−λβv(t)=0, |
Multiplying last two equations to v(n) and u(n) respectively, subtracting from each other and applying sum operator, since the self–adjointness of the operator L1, we get
(λα−λβ)b−1∑s=a+1r(s)u(s)v(s)=0, |
since λα≠λβ,
b−1∑s=a+1r(s)u(s)v(s)=0,⟨u(t),v(t)⟩=0, |
and the proof completes.
Theorem 3.3. All eigenvalues of the equation (3.1) are real.
Proof. Assume λ=α+iβ, since the self–adjointness of the operator L1, we have
⟨L1u,u⟩=⟨u,L1u⟩,⟨λu,u⟩=⟨u,λu⟩, |
(λ−¯λ)⟨u,u⟩=0 |
Since ⟨u,u⟩r≠0,
λ=¯λ |
and hence β=0. So, the proof is completed.
Self–adjointness of L2 DFHA operator G–L sense, reality of eigenvalues and orthogonality of eigenfunctions of the equation 3.2 can be proven in a similar way to the Theorem 3.1–3.2–3.3 by means of Definition 2.5.
Theorem 3.4.
L3x(t)=∇μa(∇μax(t))+(l(l+1)t2−2t+q(t))x(t)=λx(t),0<μ<1, | (3.7) |
x(a+1)=c1,∇μax(a+1)=c2, | (3.8) |
where p(t)>0, r(t)>0, q(t) is defined and real valued, λ is the spectral parameter. The sum representation of solution of the problem (3.7)−(3.8) is given as follows,
x(t)=c1((1+l(l+1)(a+1)2−2a+1+q(a+1))Eλ,2μ,μ−1(t,a)−λEλ,2μ,2μ−1(t,a))+c2(Eλ,2μ,2μ−1(t,a)−Eλ,2μ,μ−1(t,a))−t∑s=a+1Eλ,2μ,2μ−1(t−ρ(s)+a)(l(l+1)s2−2s+q(s))x(s). | (3.9) |
Proof. Taking Laplace transform of the equation (3.7) by Theorem 2.2 and take (l(l+1)t2−2t+q(t))x(t)=g(t),
La+1{∇μa(∇μax)}(s)+La+1{g}(s)=λLa+1{x}(s),=sμLa+1{∇μax}(s)−1−sμ1−sc2=λLa+1{x}(s)−La+1{g}(s),=sμ(sμLa+1{x}(s)−1−sμ1−sc1)−1−sμ1−sc2=λLa+1{x}(s)−La+1{g}(s), |
=La+1{x}(s)=1−sμ1−s1s2μ−λ(sμc1+c2)−1s2μ−λLa+1{g}(s). |
Using Lemma 2.1, we have
La{x}(s)=c1(sμ−λs2μ−λ)−1−ss2μ−λ(11−sLa{g}(s)−11−sg(a+1))+c2(1−sμs2μ−λ). | (3.10) |
Now, taking inverse Laplace transform of the equation (3.10) and applying convolution theorem, then we have the representation of solution of the problem (3.7)−(3.8), |λ|<1, |1−s|<1 and |s|α>λ from Theorem 2.3., i.e.
L−1a{sμs2μ−λ}=Eλ,2μ,μ−1(t,a),L−1a{1s2μ−λ}=Eλ,2μ,2μ−1(t,a), |
L−1a{1s2μ−λLa{q(s)x(s)}}=t∑s=a+1Eλ,2μ,2μ−1(t−ρ(s)+a)q(s)x(s). |
Consequently, we have sum representation of solution for DFHA problem 3.7–3.8
x(t)=c1((1+l(l+1)(a+1)2−2a+1+q(a+1))Eλ,2μ,μ−1(t,a)−λEλ,2μ,2μ−1(t,a))+c2(Eλ,2μ,2μ−1(t,a)−Eλ,2μ,μ−1(t,a))−t∑s=a+1Eλ,2μ,2μ−1(t−ρ(s)+a)(l(l+1)s2−2s+q(s))x(s). |
Presume that c1=1,c2=0,a=0 in the representation of solution (3.9) and hence we may observe the behaviors of solutions in following figures (Figures 1–7) and tables (Tables 1–3);
x(t) | μ=0.3 | μ=0.35 | μ=0.4 | μ=0.45 | μ=0.5 |
x(1) | 1 | 1 | 1 | 1 | 1 |
x(2) | 0.612 | 0.714 | 1.123 | 0.918 | 1.020 |
x(3) | 0.700 | 0.900 | 1.515 | 1.370 | 1.641 |
x(5) | 0.881 | 1.336 | 2.402 | 2.747 | 3.773 |
x(7) | 1.009 | 1.740 | 3.352 | 4.566 | 7.031 |
x(9) | 1.099 | 2.100 | 4.332 | 6.749 | 11.461 |
x(12) | 1.190 | 2.570 | 5.745 | 10.623 | 20.450 |
x(15) | 1.249 | 2.975 | 6.739 | 15.149 | 32.472 |
x(16) | 1.264 | 3.098 | 7.235 | 16.793 | 37.198 |
x(18) | 1.289 | 3.330 | 8.233 | 20.279 | 47.789 |
x(20) | 1.309 | 3.544 | 9.229 | 24.021 | 59.967 |
x(t) | q(t)=1 | q(t)=t | q(t)=√t |
x(1) | 1 | 1 | 1 |
x(2) | 7.37∗10−17 | 4.41∗10−17 | 5.77∗10−17 |
x(3) | −0.131 | −0.057 | −0.088 |
x(5) | −0.123 | −0.018 | −0.049 |
x(7) | −0.080 | −0.006 | −0.021 |
x(9) | −0.050 | −0.003 | −0.011 |
x(12) | −0.028 | −0.001 | −0.005 |
x(15) | −0.017 | −0.0008 | −0.003 |
x(16) | −0.015 | −0.0006 | −0.0006 |
x(18) | −0.012 | −0.0005 | −0.002 |
x(20) | −0.010 | −0.0003 | −0.001 |
x(t) | λ=0.1 | λ=0.11 | λ=0.12 |
x(1) | 1 | 1 | 1 |
x(2) | 1 | 1.025 | 1.052 |
x(3) | 1.668 | 1.751 | 1.841 |
x(5) | 3.876 | 4.216 | 4.595 |
x(7) | 7.243 | 8.107 | 9.095 |
x(9) | 11.941 | 13.707 | 12.130 |
x(12) | 22.045 | 26.197 | 25.237 |
x(15) | 36.831 | 45.198 | 46.330 |
x(16) | 43.042 | 53.369 | 55.687 |
x(18) | 57.766 | 73.092 | 78.795 |
x(20) | 76.055 | 98.154 | 127.306 |
We have analyzed DFHA equation in Riemann–Liouville and Grü nwald–Letnikov sense. Self–adjointness of the DFHA operator is presented and also, we have proved some significant spectral properties for instance, orthogonality of distinct eigenfunctions, reality of eigenvalues. Moreover, we give sum representation of the solutions for DFHA problem and find the solutions of the problem. We have carried out simulation analysis with graphics and tables. The aim of this paper is to contribute to the theory of hydrogen atom fractional difference operator.
We observe the behaviors of solutions by changing the order of the derivative μ in Figure 1 and Figure 5, by changing the potential function q(t) in Figure 2, we compare solutions under different λ eigenvalues in Figure 3, and Figure 7, also we observe the solutions by changing μ with a specific eigenvalue in Figure 4 and by changing l values in Figure 6.
We have shown the solutions by changing the order of the derivative μ in Table 1, by changing the potential function q(t) and λ eigenvalues in Table 2, Table 3.
The authors would like to thank the editor and anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.
The authors declare no conflict of interest.
[1] |
A. Fleischer, A. Tchetchik, Does rural tourism benefit from agriculture?, Tourism Manage., 26 (2005), 493–501. https://doi.org/10.1016/j.tourman.2003.10.003 doi: 10.1016/j.tourman.2003.10.003
![]() |
[2] | C. Quintas-Soriano, A. J. Castro, H. Castro, M. García-Llorente, Impacts of land use change on ecosystem services and implications for human well-being in Spanish drylands, Land Use Pol., 54 (2016), 534–548. https://doi.org/1016/j.landusepol.2016.03.011 |
[3] |
Y. Liu, X. Cheng, Does agro-ecological efficiency contribute to poverty alleviation? An empirical study based on panel data regression, Environ. Sci. Pollut. Res., 29 (2022), 51892–51908. https://doi.org/10.1007/s11356-022-19408-3 doi: 10.1007/s11356-022-19408-3
![]() |
[4] |
J. Yang, R. X. Yang, M. H. Chen, C. Su, Y. Zhi, J. Xi, Effects of rural revitalization on rural tourism, J. Hosp. Tour. Manag., 47 (2021), 35–45. https://doi.org/10.1016/j.jhtm.2021.02.008 doi: 10.1016/j.jhtm.2021.02.008
![]() |
[5] |
G. R. Jiang, How does agro-tourism integration influence the rebound effect of China's agricultural eco-efficiency? An economic development perspective, Front. Environ. Sci., 10 (2022), 136–149. https://doi.org/10.3389/fenvs.2022.921103 doi: 10.3389/fenvs.2022.921103
![]() |
[6] |
E. Panyik, C. Costa, T. Ratz, Implementing integrated rural tourism: An event-based approach, Tour. Manag., 32 (2011), 1352–1363. https://doi.org/10.1016/j.tourman.2011.01.009 doi: 10.1016/j.tourman.2011.01.009
![]() |
[7] |
G. H. Liu, Z. F. Liu, H. F. Hu, G. Wu, L. Dai, The impact of tourism on agriculture in Lugu Lake region, Int. J. Sustain. Dev. World Ecol., 15 (2008), 3–9. https://doi.org/10.1080/13504500809469762 doi: 10.1080/13504500809469762
![]() |
[8] | E. Di-Clemente, J. M. Hernandez-Mogollon, T. Lopez-Guzman, Culinary tourism as an effective strategy for a profitable cooperation between agriculture and tourism, Soc. Sci-Basel, 9 (2020), 15. https://doi.org/10.3390/socsci9030025 |
[9] |
A. A. Alola, U. V. Alola, Agricultural land usage and tourism impact on renewable energy consumption among coastline mediterranean countries, Energy Environ., 29 (2018), 1438–1454. https://doi.org/10.1177/0958305x18779577 doi: 10.1177/0958305x18779577
![]() |
[10] |
X. T. Chi, H. Han, Emerging rural tourism in China's current tourism industry and tourist behaviors: The case of Anji County, J. Travel Tour. Mark., 38 (2021), 58–74. https://doi.org/10.1080/10548408.2020.1862026 doi: 10.1080/10548408.2020.1862026
![]() |
[11] |
G. C. M. Laeis, R. A. Scheyvens, C. Morris, Cuisine: A new concept for analysing tourism-agriculture linkages?, J. Tour. Cult. Chang., 18 (2020), 643–658. https://doi.org/10.1080/14766825.2019.1624763 doi: 10.1080/14766825.2019.1624763
![]() |
[12] | J. Kim, S. Kim, C. Ju, H. Son, Unmanned aerial vehicles in agriculture: A review of perspective of platform, control, and applications, IEEE Access, 7 (2019), 105100–105115. https://doi.org/10.1109/access.2019.2932119 |
[13] | J. S. Zhu, A. Siriphon, D. Airey, J. Mei-lan-Anatolia, Chinese tourism diplomacy: A Chinese-style modernity review, Anatolia, 33 (2022), 550–563. https://doi.org/1080/13032917.2021.1978515 |
[14] |
L. Sun. S. Jing, L. Moucheng, M. Qingwen, Agricultural production under rural tourism on the Qinghai-Tibet plateau: From the perspective of smallholder farmers, Land Use Policy, 103 (2021), 1053–1063. https://doi.org/10.1016/j.landusepol.2021.105329 doi: 10.1016/j.landusepol.2021.105329
![]() |
[15] | P. Y. Chen, X. Kong, Tourism-led commodification of place and rural transformation development: A case study of Xixinan village, Huangshan, China, Land, 10 (2021), 156–167. https://doi.org/10.3390/land10070694 |
[16] |
Z. S. Yang, J. M. Cai, R. Sliuzas, Agro-tourism enterprises as a form of multi-functional urban agriculture for peri-urban development in China, Habitat Int., 34 (2010), 374–385. https://doi.org/10.1016/j.habitatint.2009.11.002 doi: 10.1016/j.habitatint.2009.11.002
![]() |
[17] | Y. Liu, W. Zhang, Z. Zhang, Q. Xu, W. Li, Risk factor detection and landslide susceptibility mapping using geo-detector and random forest models: The 2018 Hokkaido eastern Iburi earthquake, Remote Sens., 13 (2021). https://doi.org/17.10.3390/rs13061157 |
[18] |
Z. X. Zhou, M. T. Li, Spatial-temporal change in urban agricultural land use efficiency from the perspective of agricultural multi-functionality: A case study of the Xi'an metropolitan zone, J. Geogr. Sci., 27 (2017), 1499–1520. https://doi.org/10.1007/s11442-017-1449-6 doi: 10.1007/s11442-017-1449-6
![]() |
[19] | J. Pueyo-Ros, The role of tourism in the ecosystem services framework, Land, 7 (2018), 13. https://doi.org/10.3390/land7030111 |
[20] |
L. Scherer, S. Pfister, Modelling spatially explicit impacts from phosphorus emissions in agriculture, Int. J. Life Cycle Assess, 20 (2015), 785–795. https://doi.org/10.1007/s11367-015-0880-0 doi: 10.1007/s11367-015-0880-0
![]() |
[21] |
J. J. Wang, F. M. Zhou, A. L. Xie, J. Shi, Impacts of the integral development of agriculture and tourism on agricultural eco-efficiency: A case study of two river basins in China, Environ. Dev. Sustain., 34 (2022), 1–30. https://doi.org/10.1007/s10668-022-02781-x doi: 10.1007/s10668-022-02781-x
![]() |
[22] |
D. Light, S. Richards, P. Lvanova, Exploring "gothic tourism": A new form of urban tourism?, Int. J. Tour. Cities, 7 (2021), 224–236. https://doi.org/10.1108/ijtc-01-2020-0012 doi: 10.1108/ijtc-01-2020-0012
![]() |
[23] |
Y. Li, Q. Cui, Analyzing the role of competition and cooperation in airline environmental efficiency through two dynamic environmental cross-efficiency models, Int. J. Sustain. Transp., 15 (2021), 850–864. https://doi.org/10.1080/15568318.2020.1821415 doi: 10.1080/15568318.2020.1821415
![]() |
[24] |
C. N. Wang, H. P. Nguyen, C. W. Chang, Environmental efficiency evaluation in the top Asian economies: An application of DEA, Mathematics, 9 (2021), 19. https://doi.org/10.3390/math9080889 doi: 10.3390/math9080889
![]() |
[25] |
D. Liu, Q. X. Chen, A regret cross-efficiency ranking method considering consensus consistency, Expert. Syst. Appl., 208 (2022), 16. https://doi.org/10.1016/j.eswa.2022.118192 doi: 10.1016/j.eswa.2022.118192
![]() |
[26] | M. Akram, S. M. U. Shah, M. M. A. Al-Shamiri, Extended DEA method for solving multi-objective transportation problem with Fermatean fuzzy sets, AIMS Math., 8 (2023), 924–961. https://doi.org/10.3934/math.2023045 |
[27] | M. Akbar, A. Alireza, K. Sohrab, M. Vaez-Ghasemi, Closest reference point on the strong efficient frontier in data envelopment analysis, AIMS Math., 5 (2020), 811–827. https://doi.org/10.3934/math.2020055 |
[28] |
F. Yang, C. C. Yang, L. A. Liang, S. Du, New approach to determine common weights in DEA efficiency evaluation model, J. Syst. Eng. Electron., 21 (2010), 609–615. https://doi.org/10.3969/j.issn.1004-4132.2010.04.013 doi: 10.3969/j.issn.1004-4132.2010.04.013
![]() |
[29] |
Y. Huang, Y. M. Wang, J. Lin, Two-stage fuzzy cross-efficiency aggregation model using fuzzy information retrieval method, Int. J. Fuzzy Syst., 21 (2019), 2650–2666. https://doi.org/10.1007/s40815-019-00733-8 doi: 10.1007/s40815-019-00733-8
![]() |
[30] |
Y. Huang, Y. M. Wang, Pareto approach for DEA cross efficiency evaluation based on interval programming, J. Intell. Fuzzy Syst., 33 (2017), 2375–2389. https://doi.org/10.3233/jifs-17514 doi: 10.3233/jifs-17514
![]() |
[31] |
F. Najafzade, S. Khoshfetrat, S. Saeidi, A cross-efficiency model using bargaining theory in a two-stage network DEA, Comput. Appl. Math., 41 (2022), 23. https://doi.org/10.1007/s40314-022-02027-6 doi: 10.1007/s40314-022-02027-6
![]() |
[32] |
L. Chen, Y. M. Wang, Data envelopment analysis cross-efficiency model in fuzzy environments, J. Intell. Fuzzy Syst., 30 (2016), 2601–2609. https://doi.org/10.3233/ifs-151878 doi: 10.3233/ifs-151878
![]() |
[33] |
X. Shi, Environmental efficiency evaluation of Chinese industry systems by using non-cooperative two-stage DEA model, Math. Probl. Eng., 2019 (2019), https://doi.org/10.10.1155/2019/9208367 doi: 10.10.1155/2019/9208367
![]() |
[34] | L. Angulo-Meza, M. Gonzalez-Araya, A. Iriarte, R. Rebolledo-Leiva, J. Carlos Soares de Mello, A multiobjective DEA model to assess the eco-efficiency of agricultural practices within the CF plus DEA method, Comput. Electron. Agric., 161 (2019), 151–161. https://doi.org/10.1016/j.compag.2018.05.037 |
[35] |
L. Wen, H. P. Li, Estimation of agricultural energy efficiency in five provinces: Based on data envelopment analysis and malmquist index model, Energy Sources. Part A-Recovery Util. Environ. Eff., 44 (2022), 2900–2913. https://doi.org/10.1080/15567036.2019.1651798 doi: 10.1080/15567036.2019.1651798
![]() |
[36] |
L. Liang, J. Wu, W. D. Cook, J. Zhu, Alternative secondary goals in DEA cross-efficiency evaluation, Int. J. Prod. Econ., 113 (2008), 1025–1030. https://doi.org/10.1016/j.ijpe.2007.12.006 doi: 10.1016/j.ijpe.2007.12.006
![]() |
[37] |
Y. M. Wang, K. S. Chin, S. Wang, DEA models for minimizing weight disparity in cross-efficiency evaluation, J. Oper. Res. Soc., 63 (2012), 1079–1088. https://doi.org/10.1057/jors.2011.116 doi: 10.1057/jors.2011.116
![]() |
[38] |
L. Li, W. Dai, J. Chu, Y. Wang, DEA altruism and exclusiveness cross-efficiency evaluation models, Sci. Iran., 27 (2020), 2499–2513. https://doi.org/10.24200/sci.2020.21994 doi: 10.24200/sci.2020.21994
![]() |
[39] |
G. L. Yang, J. B. Yang, W. B. Liu, X. Li, Cross-efficiency aggregation in DEA models using the evidential-reasoning approach, Eur. J. Oper. Res., 231 (2013), 393–404. https://doi.org/10.1016/j.ejor.2013.05.017 doi: 10.1016/j.ejor.2013.05.017
![]() |
[40] |
Q. Wang, K. K. Wei, Y. Zhang, Comprehensive cross-efficiency methods with common weight restrictions in data envelopment analysis, Asia Pac. J. Oper. Res., 37 (2020), 21. https://doi.org/10.1142/s0217595920500190 doi: 10.1142/s0217595920500190
![]() |
[41] |
C. M. Rogerson, Tourism-agriculture linkages in rural South Africa: Evidence from the accommodation sector, J. Sustain. Tour., 20 (2012), 477–495. https://doi.org/10.1080/09669582.2011.617825 doi: 10.1080/09669582.2011.617825
![]() |
[42] |
Y. Z. Song, J. F. Wang, Y. Ge, C. Xu, An optimal parameters-based geographical detector model enhances geographic characteristics of explanatory variables for spatial heterogeneity analysis: Cases with different types of spatial data, GISci. Remote Sens., 57 (2020), 593–610. https://doi.org/10.1080/15481603.2020.1760434 doi: 10.1080/15481603.2020.1760434
![]() |
[43] | G. Guo, K. Li, D. Zhang, M. Lei, Quantitative source apportionment and associated driving factor identification for soil potential toxicity elements via combining receptor models, SOM, and geo-detector method, Sci. Total Environ., 830 (2022), 12. https://doi.org/10.1016/j.scitotenv.2022.154721 |
[44] |
H. H. Orkcu, V. S. Ozsoy, M. Orkcu, H. Bal, A neutral cross efficiency approach for basic two stage production systems, Expert. Syst. Appl., 125 (2019), 333–344. https://doi.org/10.1016/j.eswa.2019.01.067 doi: 10.1016/j.eswa.2019.01.067
![]() |
[45] | M. C. Zewdie, M. Moretti, D. B. Tenessa, Z. A. Ayele, J. Nyssen, E. A. Tsegaye, et al., Agricultural technical efficiency of smallholder farmers in ethiopia: A stochastic frontier approach, Land, 10 (2021), 246. https://doi.org/10.3390/land10030246 |
[46] |
G. Cheng, J. W. Han, P. C. Zhou, L. Guo, Multi-class geospatial object detection and geographic image classification based on collection of part detectors, ISPRS J. Photogramm. Remote. Sens., 98 (2014), 119–132. https://doi.org/1016/j.isprsjprs.2014.10.002 doi: 1016/j.isprsjprs.2014.10.002
![]() |
[47] | X. Xu, J. Li, D. Wu, X. Zhang, The intellectual capital efficiency and corporate sustainable growth nexus: Comparison from agriculture, tourism and renewable energy sector, Environ. Dev. Sustain., 23 (2021), 16038–16056. https://doi.org/10.1007/s10668-021-01319-x |
[48] | E. Slabbert, E. Du Plessis, O. Digun-Aweto, Impacts of tourism in predicting residents' opinions and interest in tourism activities, J. Tour. Cult. Chang., 19 (2021), 819–837. https://doi.org/10.1080/14766825.2020.1803891 |
[49] | X. Yang, L. Shu, J. N. Chen, M. A. Ferrag, J. Wu, E. Nurellari, et al., A survey on smart agriculture: development modes, technologies, and security and privacy challenges, IIEEE-CAA J. Automatica Sin., 8 (2021), 273–302. https://doi.org/10.1109/jas.2020.1003536 |
[50] | Y. Zhu, C. J. Huo, The impact of agricultural production efficiency on agricultural carbon emissions in China, Energies, 15 (2022), 22. Math-9075-english.pdf |
[51] |
B. K. T. Edjah, J. P. Wu, J. J. Tian, Research on the comparative advantage and complementarity of China-Ghana agricultural product trade, Sustainability, 14 (2022), 14. https://doi.org/10.3390/su142013136 doi: 10.3390/su142013136
![]() |
[52] |
Y. Cui, Y. Han, N. Lv, H. Zhu, Characteristics of local officials and high quality development of chinese agriculture-evidence from chinese governors andsecretaries of the provincial party committee, Appl. Math. Nonlinear. Sci., 5 (2020), 543–556. https://doi.org/10.2478/amns.2020.2.00019 doi: 10.2478/amns.2020.2.00019
![]() |
[53] |
M. Wysokinski, J. Domagala, A. Gromada, M. Golonko, P. Trębska, Economic and energy efficiency of agriculture, Agric. Econ., 66 (2020), 355–364. https://doi.org/10.17221/170/2020-agricecon doi: 10.17221/170/2020-agricecon
![]() |
[54] |
W. T. Pan, M. E. Zhuang, Y. Y. Zhou, J. J. Yang, Research on sustainable development and efficiency of China's e-agriculture based on a data envelopment analysis-Malmquist model, Technol. Forecast. Soc. Chang., 162 (2021), 6. https://doi.org/10.1016/j.techfore.2020.120298 doi: 10.1016/j.techfore.2020.120298
![]() |
[55] |
B. Duan, C. Arcodia, E. Ma, A. Hsiao, Understanding wine tourism in China using an integrated product-level and experience economy framework, Asia Pac. J. Tour. Res., 23 (2018), 949–960. https://doi.org/10.1080/10941665.2018.1512506 doi: 10.1080/10941665.2018.1512506
![]() |
[56] |
F. B. Shang, W. J. Zhu, Planning of ecological agricultural tourist attractions based on the concept of circular economy, Acta Agric. Scand. Sect. B-Soil. Plant Sci., 72 (2022), 538–552. https://doi.org/10.1080/09064710.2021.2021278 doi: 10.1080/09064710.2021.2021278
![]() |
[57] |
S. H. Liu, B. Gao, Econometric analysis of leisure agriculture and rural tourism scenery based on spatial data analysis, Comput. Intell. Neurosci., 2022 (2022), 8. https://doi.org/10.1155/2022/2286803 doi: 10.1155/2022/2286803
![]() |
[58] | X. Gong, W. C. Zhu, S. Liu, The strategy of eco-agriculture economic development along the coast based on improving the rural eco-tourism environment, J. Coast Res., (2020), 652–655. https://doi.org/10.2112/jcr-si104-112.1 |
[59] |
S. Corak, S. B. Zivoder, Z. Marusic, Opportunities for tourism recovery and development during and after COVID-19: Views of tourism scholars versus tourism practitioners, Tourism, 68 (2020), 434–449. https://doi.org/10.37741/t.68.4.5 doi: 10.37741/t.68.4.5
![]() |
[60] |
F. Higgins-Desbiolles, B. C. Bigby, A. Doering, Socialising tourism after COVID-19: reclaiming tourism as a social force?, J. Tour. Futures, 8 (2022), 208–219. https://doi.org/10.1108/jtf-03-2021-0058 doi: 10.1108/jtf-03-2021-0058
![]() |
[61] | J. H. Jung, M. Maeda, A. J. Chang, M. Bhandari, A. Ashapure, J. Landivar-Bowles, The potential of remote sensing and artificial intelligence as tools to improve the resilience of agriculture production systems, Curr. Opin. Biotechnol., 70 (2021), 15–22. https://doi.org/10.1016/j.copbio.2020.09.003 |
1. | Erdal Bas, Funda Metin Turk, Ramazan Ozarslan, Ahu Ercan, Spectral data of conformable Sturm–Liouville direct problems, 2021, 11, 1664-2368, 10.1007/s13324-020-00428-6 | |
2. | Tom Cuchta, Dallas Freeman, Discrete Polylogarithm Functions, 2023, 84, 1338-9750, 19, 10.2478/tmmp-2023-0012 | |
3. | B. Shiri, Y. Guang, D. Baleanu, Inverse problems for discrete Hermite nabla difference equation, 2025, 33, 2769-0911, 10.1080/27690911.2024.2431000 | |
4. | Muhammad Sulthan Zacky, Heru Sukamto, Lila Yuwana, Agus Purwanto, Eny Latifah, The performance of space-fractional quantum carnot engine, 2025, 100, 0031-8949, 025306, 10.1088/1402-4896/ada9de |
x(t) | μ=0.3 | μ=0.35 | μ=0.4 | μ=0.45 | μ=0.5 |
x(1) | 1 | 1 | 1 | 1 | 1 |
x(2) | 0.612 | 0.714 | 1.123 | 0.918 | 1.020 |
x(3) | 0.700 | 0.900 | 1.515 | 1.370 | 1.641 |
x(5) | 0.881 | 1.336 | 2.402 | 2.747 | 3.773 |
x(7) | 1.009 | 1.740 | 3.352 | 4.566 | 7.031 |
x(9) | 1.099 | 2.100 | 4.332 | 6.749 | 11.461 |
x(12) | 1.190 | 2.570 | 5.745 | 10.623 | 20.450 |
x(15) | 1.249 | 2.975 | 6.739 | 15.149 | 32.472 |
x(16) | 1.264 | 3.098 | 7.235 | 16.793 | 37.198 |
x(18) | 1.289 | 3.330 | 8.233 | 20.279 | 47.789 |
x(20) | 1.309 | 3.544 | 9.229 | 24.021 | 59.967 |
x(t) | q(t)=1 | q(t)=t | q(t)=√t |
x(1) | 1 | 1 | 1 |
x(2) | 7.37∗10−17 | 4.41∗10−17 | 5.77∗10−17 |
x(3) | −0.131 | −0.057 | −0.088 |
x(5) | −0.123 | −0.018 | −0.049 |
x(7) | −0.080 | −0.006 | −0.021 |
x(9) | −0.050 | −0.003 | −0.011 |
x(12) | −0.028 | −0.001 | −0.005 |
x(15) | −0.017 | −0.0008 | −0.003 |
x(16) | −0.015 | −0.0006 | −0.0006 |
x(18) | −0.012 | −0.0005 | −0.002 |
x(20) | −0.010 | −0.0003 | −0.001 |
x(t) | λ=0.1 | λ=0.11 | λ=0.12 |
x(1) | 1 | 1 | 1 |
x(2) | 1 | 1.025 | 1.052 |
x(3) | 1.668 | 1.751 | 1.841 |
x(5) | 3.876 | 4.216 | 4.595 |
x(7) | 7.243 | 8.107 | 9.095 |
x(9) | 11.941 | 13.707 | 12.130 |
x(12) | 22.045 | 26.197 | 25.237 |
x(15) | 36.831 | 45.198 | 46.330 |
x(16) | 43.042 | 53.369 | 55.687 |
x(18) | 57.766 | 73.092 | 78.795 |
x(20) | 76.055 | 98.154 | 127.306 |
x(t) | μ=0.3 | μ=0.35 | μ=0.4 | μ=0.45 | μ=0.5 |
x(1) | 1 | 1 | 1 | 1 | 1 |
x(2) | 0.612 | 0.714 | 1.123 | 0.918 | 1.020 |
x(3) | 0.700 | 0.900 | 1.515 | 1.370 | 1.641 |
x(5) | 0.881 | 1.336 | 2.402 | 2.747 | 3.773 |
x(7) | 1.009 | 1.740 | 3.352 | 4.566 | 7.031 |
x(9) | 1.099 | 2.100 | 4.332 | 6.749 | 11.461 |
x(12) | 1.190 | 2.570 | 5.745 | 10.623 | 20.450 |
x(15) | 1.249 | 2.975 | 6.739 | 15.149 | 32.472 |
x(16) | 1.264 | 3.098 | 7.235 | 16.793 | 37.198 |
x(18) | 1.289 | 3.330 | 8.233 | 20.279 | 47.789 |
x(20) | 1.309 | 3.544 | 9.229 | 24.021 | 59.967 |
x(t) | q(t)=1 | q(t)=t | q(t)=√t |
x(1) | 1 | 1 | 1 |
x(2) | 7.37∗10−17 | 4.41∗10−17 | 5.77∗10−17 |
x(3) | −0.131 | −0.057 | −0.088 |
x(5) | −0.123 | −0.018 | −0.049 |
x(7) | −0.080 | −0.006 | −0.021 |
x(9) | −0.050 | −0.003 | −0.011 |
x(12) | −0.028 | −0.001 | −0.005 |
x(15) | −0.017 | −0.0008 | −0.003 |
x(16) | −0.015 | −0.0006 | −0.0006 |
x(18) | −0.012 | −0.0005 | −0.002 |
x(20) | −0.010 | −0.0003 | −0.001 |
x(t) | λ=0.1 | λ=0.11 | λ=0.12 |
x(1) | 1 | 1 | 1 |
x(2) | 1 | 1.025 | 1.052 |
x(3) | 1.668 | 1.751 | 1.841 |
x(5) | 3.876 | 4.216 | 4.595 |
x(7) | 7.243 | 8.107 | 9.095 |
x(9) | 11.941 | 13.707 | 12.130 |
x(12) | 22.045 | 26.197 | 25.237 |
x(15) | 36.831 | 45.198 | 46.330 |
x(16) | 43.042 | 53.369 | 55.687 |
x(18) | 57.766 | 73.092 | 78.795 |
x(20) | 76.055 | 98.154 | 127.306 |