Research on cooperation strategy between government and green supply chain based on differential game
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Jingwei Zeng
Abstract
Based on the “three bottom line” and stakeholder theory, the paper considers the relationship and cooperation strategy between the government and the supplier and manufacturer of the green supply chain. By constructing the dynamic differential game model, the paper discusses the differences in the optimal effort level, green degree of product, reputation and the optimal benefit under the three situations of noncooperation, government promotion and collaborative cooperation. The results show that the optimal effort level, green degree of product, reputation and the optimal benefit in collaborative cooperation are obviously higher than the situations of non-cooperation and government promotion, and the cooperation of the three parties can promote the development of green supply chain. Government promotion is better than noncooperation. The government plays an active role in improving the optimal benefit and reputation of green supply chain. Finally, the reliability of the proposed proposition is verified by an example analysis, which provides an important reference for improving the efficiency of green supply chain.
1 Introduction
China’s economic transformation is taking place. It has been recognized that the development of traditional rough type has caused serious environmental pollution and ecological damage [1]. The three bottom line (TBL) theory holds that enterprise development must take into account economic, environmental and social benefits [2]. It provides a basis for the transformation and development of Chinese enterprises. Therefore, in the “13th Five-Year plan”, China put forward the new development concept of innovation, coordination, green, openness and sharing. The formation of green supply chain fits the new development concept. “Actively promoting the green supply chain and strengthening the green supervision of the supply chain” is also clearly required in the guidance of the Chinese government to actively promote the innovation and application of supply chain. However, the green supply chain of China is still in the development stage, and the relevant governance mechanism is not perfect. Therefore, based on stakeholder theory, this paper discusses the cooperation strategy between the government and the green supply chain, and considers the differences between the government and the green supply chain in different game situations, which will help to improve the government’s governance of the green supply chain.
Green Supply Chain Management (GSCM) has become a hot topic in academic circles, and many scholars have done related research [3, 4]. Among them, the green supply chain coordination, consumer willingness, government subsidies have attracted the attention of scholars. With regard to the coordination of green supply chain, Huang et al. [5] use game theory to study the relationship between green supply chain coordination and greenhouse gas emission reduction, and find that product line design, supplier selection, transport mode selection and pricing strategy will affect greenhouse gas emissions. Price and green level sensitivity will affect the coordination of green supply chain, and the cost sharing contract in green supply chain has a positive impact on improving product green level [6]. The development strategy selection of green supply chain is analyzed by differential game model, and it is found that centralized decision can be achieved through contract design to achieve the coordination of green supply chain [7]. On the study of the consumer behavior of the green supply chain, consumers’ green sensitivity plays an important role in the green supply chain of smart phones, and consumers are highly sensitive to the green level of smart phones [8]. Wang et al. [9] find that the cost driven elements and consumer driven elements play an important role in the management practice of the green supply chain, and the role of consumer driven for small companies is more obvious. At the government level, Sheu [10] believes the government’s fiscal intervention will have a significant impact on the strategy choices of members of the green supply chain, and government intervention may have a negative effect on corporate profits and social welfare. Yang and Xiao [11] discuss the relationship between product pricing and green level in green supply chain under government intervention through three game models, it is found that when government intervention increases, the level of green will rise, but government intervention is not always conducive to manufacturers of green supply chains. Sinayi and Rasti-Barzoki [12] construct a two-level model of government and supply chain, analyze the role of government in the green development of supply chain, and point out that different government policies have a significant effect on the profit and environment of supply chain members. Guo et al. [13] compare the impact of different government subsidy policies (subsidies for manufacturers’ green efforts and price subsidies to green products) on the green manufacturing of supply chain, and find that the choice of government subsidy policy depends on the consumer sensitivity to the price of green products. Government subsidies can increase the profits of green supply chain entities and promote the development of green supply chain [14].
To sum up, the studies of green supply chain by scholars have laid a solid foundation for subsequent research, but there are still have some knowledge gaps. First of all, the previous researches usually show the role of government as an impact parameter in consideration of the role of the government, and do not bring the government as a stakeholder into the research framework, which lead to a certain degree of deviation in the analysis of the role of the government. Secondly, the green supply chain should reflect the combination of economic, social and environmental benefits. Most of the previous studies consider the problem of the coordination of green supply chain from the perspective of maximizing the economic profit. Finally, the cooperation of the green supply chain is a dynamic process. The green degree of product and reputation of the supplier and manufacturer will change according to the change of the state variables, and it is a long-term process. The previous researches are mostly based on the static perspective, and the dynamic changes of the characteristics of the green supply chain are not fully considered. Therefore, based on the differential game theory, this paper constructs the dynamic game model between the government and the supplier and manufacturer of the green supply chain, and analyzes the three situations of the noncooperation, the government promotion and the collaborative cooperation. We hope to reveal the specific role of the government in the green supply chain, and then provide suggestions for the development of green supply chain.
This paper is arranged as follows. Section 2 presents the problem description and research assumptions. In Section 3, three differential game models are formulated according to the three situations. Section 4 presents a comparison of the different situations. Simulations and example analysis are conducted in Section 5, and conclusions are presented in Section 6.
2 Problem description and research assumptions
A simple system is used to describe the relationship between the government and the supplier and manufacturer of the green supply chain. In the green supply chain, the supplier and the manufacturer constitute a two-level supply chain. The main body of the supply chain needs to improve and update the raw materials and equipment, so as to better satisfy the relevant conditions of green innovation. Supplier provide raw materials to manufacturer, and manufacturer provide green products to consumers through production, assembly and sales [7]. Due to the growing awareness of consumers’ environmental protection, a preference for green products is formed [15]. The consumers play an important role in the game because their environmental awareness has a signihcant influence on market demand. The role of government in green supply chain is to provide a series of policies and subsidies to supplier and manufacturer. The government’s green supervision will also promote the green production of the supply chain, further reduce energy consumption and environmental pollution. We assume that government behavior will encourage green supply chain to carry out green innovation and enhance the green degree of product. It is conducive to the unification of economic, environmental and social benefits [15].
In green supply chain, supplier and manufacturer will make efforts to enhance the green degree of product and reputation. Let NS(t) denotes the effort level of supplier at time t, and let NM(t) denotes the effort level of manufacturer at time t. According to the assumption of cost in previous studies [17, 18], we assume that the cost of upgrading the green degree of product and reputation is a convex function of effort. Therefore, the cost of supplier and manufacturer of green supply chain can be written as
It is proposed that the promotion of product green degree is a time-varied dynamic process. It depends on the efforts of the supplier, manufacturer and government toward the promotion of product green degree. The efforts of green supply chain and government during the production process contribute to the promotion of product green degree, which evolves according to the Nerlove and Arrow model (1962). When there is no effort, the green degree of product will decay because the related equipment will depreciate with time. In addition, the green degree of the supplier’s product can affect the green degree of the manufacturer’s product. Let DS(t) and DM(t) respectively denote the product green degree of supplier and manufacturer at time t. The differential equation of the product green degree of supplier can be expressed as
For the manufacturer, the green degree of the product supplied by the supplier will affect the green degree of the manufacturer’s product to a certain extent. According to formula (1), the differential equation of green degree of manufacturer’s product can be expressed as
where α1 and α2 are respectively the influence coefficient of supplier and manufacturer’s effort level on product green degree, β1 and β2 are respectively the influence coefficient of government effort level on product green degree of green supply chain. η1 and η2 are respectively the attenuation coefficient of product green degree for supplier and manufacturer. μ is the promotion coefficient of supplier product to the green degree of manufacturer product. In the initial state, DS(0) = d1 ≥ 0, DM(0) = d2 ≥ 0.
Considering that green products are environmentally friendly and can be recognized by consumers, the promotion of reputation will also benefit the green supply chain and the whole system. Based on this, the consideration of reputation effect will further improve the research of green supply chain. Reputation also depends on the efforts of green supply chains and government [19]. Similar to the product green degree, reputation also decays over time without effort. Let QS(t) and QM(t) respectively denote the reputation of supplier and manufacturer at time t. The differential equation for the reputation of supplier and manufacturer can be written as
where δ1 and δ2 are respectively the influence coefficient of supplier and manufacturer effort level on reputation, ϕ1 and ϕ2 are respectively the influence coefficient of government effort level on reputation of green supply chain. γ1 and γ2 are respectively the attenuation coefficient of reputation for supplier and manufacturer. In the initial state, QS (0) = q1 ≥ 0, QM (0) = q2 ≥ 0.
Let πS(t) and πM(t) denote the benefits (including economic benefit, environmental benefit and social benefit) of supplier and manufacturer at time t. As we all know, the benefits come from the efforts of manufacturer, supplier and government, the green dgree of products and reputation. According to literature [15], the benefit function of supplier and manufacturer can be expressed as
where p1 and p2 are the influence coefficient of supplier and manufacturer effort level on benefit, k1 and k2 are the influence coefficient of government effort level on benefit of green supply chain. l1 and l2 are the influence coefficient of product green degree on benefit, b1 and b2 are the influence coefficient of reputation on benefit.
3 Differential game equilibrium analysis
Cooperation between government and green supply chain is a long-term dynamic process, and the effects of the supplier, manufacturer and government can be inter-temporal. Supplier, manufacturer and government are far sighted, they seek benefit maximization in the long run. It is useful to introduce the dynamic framework into the research on cooperation between government and green supply chain. Therefore, we use the differential game to analyze this problem and discuss the differences under the three situations of non-cooperation, government promotion and collaborative cooperation.
3.1 Situation of non-cooperation
In the situation of non-cooperation, both supplier and manufacturer will independently decide their level of effort to maximize their own benefit. The only regulatory mechanism in this situation is based on the market. Namely, the government will supervise green supply chain and provide corresponding efforts. However, the government does not subsidize the cost of product development of supplier and manufacturer of green supply chain. The benefit of green supply chain will be allocated between the government and the green supply chain. Let θi(0 ≤ θi ≤ 1, i = 1, 2) respectively denote the benefit coefficient of the government from supplier and manufacturer, let 1 – θ1 denotes the benefit coefficient of the supplier, let 1 – θ2 denotes the benefit coefficient of the manufacturer. λS , λM and λG are the cost coefficients of the supplier, manufacturer and government at time t, respectively. We assume that the discount rate r is the same. In this situation, the objective functions (net benefit) of the supplier, manufacturer and government be expressed as
For convenience in writing and understanding, time t is omitted in the next analysis [20].
Proposition 1
In the situation of non-cooperation, the feedback non-cooperative game Nash equilibria are
Proof
According to the optimal control theory and the sucient conditions of the Nash equilibrium strategy, the three parties of the supplier, manufacturer of the green supply chain and government have benet optimal value functions. VS(DS, DM, QS, QM), VM(DS, DM, QS, QM) and VG(DS, DM, QS, QM) respectively denote the benet optimal value functions of the supplier, manufacturer and government. For all Di ≥ 0 and QS ≥ 0, i ∈ {S, M}, the benefit optimal value functions satisfy the following Hamilton-Jacobi-Bellman equation
For solving formula (14), (15) and (17), using extreme conditions and searching for the optimal value of NS, NM, NG1 and NG2 by setting the all partial derivative equal to zero, we can get
Substituting the results of (17), (18), (19) and (20) into (14), (15) and (17), we can obtain
Through the above formulas and the optimal control theory, we can see that the linear optimal function of the DS, DM, QS, QM is the solution of the HJB equation. Based on the predecessor’s literature [16, 21], we can get
where σi, qi, gi(i = 1, 2, 3, 4) and ωi(i = 1, 2, 3) are the constants to be solved. We can have
Substituting the result of (24) and (25) into (21), (22) and (23), we can get
Using the DS, DM, QS, QM≥ 0 to (26), (27) and (28), parameter values of the optimal value function can be expressed as follows
Substituting the results of (29), (30), (31) and (32) into (24), we can obtain
In this case, the benefit of the whole system is
Solving the partial derivative of (33), (34) and (35), and substituting the results into (17), (18), (19) and (20), we can get the optimal eort level of supplier and manufacturer and government as the proposition 1. In the situation of non-cooperation, according to formula (1) and (3), we use the general solution method of dierential equations, and can get the supplier’s product green degree and reputation as
Similar to supplier, according to formulas (2) and (4), the manufacturer’s product green degree and reputation are
From proposition 1, we can find that when supplier, manufacturer and government only consider maximizing their own benefits, the optimal eort level is related to some parameters. For the supplier and manufacturer of the green supply chain, pi, αi, li, δi, bi(i = 1, 2) are positively related to the optimal level of eort, ηi, γi(i = 1, 2), γS, γM, r are negatively related to the optimal level of eort, ki(i = 1, 2) are not related to the optimal level of eort, it shows that government efforts have no impact on the optimal level of eort of supplier and manufacturer.
For the government, θi, ki, βi, li, φi, bi(i = 1, 2) and μ are positively related to the optimal level of eort, ηi, γi(i = 1, 2), γG and r are negatively related to the optimal level of eort. Government needs to integrate various factors to formulate corresponding incentive policies and regulatory mechanisms.
3.2 Situation of government promotion
The government plays an important role in promoting green supply chain cooperation. Cost subsidy is a common measure for the government to promote the development of green supply chain. Therefore, in the situation of government promotion, we focus on the cost sharing of the government to the green supply chain. We assume that the cost sharing coefficient for the supplier and manufacturer are ε1 and ε2. The government is a leader, the first to make the cost sharing coefficient, the supplier and manufacturer in the green supply chain as the followers, and then decide how much eort should be made. So, the situation of the Stackelberg master-slave game is formed. In this situation, the objective functions (net benefit) of the supplier, manufacturer and government be expressed as
Proposition 2
In the situation of government promotion, the feedback Stackelberg master-slave equilibria are (for the proof, see the Appendix)
Similar to proposition 1, in the situation of government promotion, we can get the supplier’s product green degree and reputation as
The manufacturer’s product green degree and reputation are
From proposition 2, in the case of government promotion, we find that the efforts of supplier and manufacturer in green supply chain are positively related to the government’s benefit distribution coefficient. The more benefit the government gains, the higher the efforts ofthe supplier and manufacturer, which reects the characteristics of the three parties driven by the government. The government is a leader, the first to make the cost sharing coefficient, the supplier and manufacturer in the green supply chain as the followers, if the government expects to get more benefit, it will enhance the subsidy level and promote the eciency of green supply chain. When the benefit distribution coefficient is too small, the government will not provide a cost subsidy strategy to the green supply chain. The other inuence coefficients are the same as the situation of non-cooperation.
3.3 Situation of collaborative cooperation
In the situation of collaborative cooperation, the government and the supplier and manufacturer of the green supply chain will choose their optimal eort levels based on maximization of their total benefit. Therefore, product green degree and reputation can be further improved through cooperation between the government and the supplier and manufacturer of the green supply chain.
Proposition 3
In the situation of collaborative cooperation, the feedback cooperative game equilibria are (for the proof, see the Appendix)
Similar to proposition 1, in the situation of collaborative cooperation, we can get the supplier’s product green degree and reputation as
The manufacturer’s product green degree and reputation are
From proposition 3, in the case of collaborative cooperation, we find that the benefit allocation coefficient no longer aects the decisions of supplier, manufacturer and government on the optimal eort level. The reason is that supplier, manufacturer and government work together to maximize the benefit of the whole system. It is no longer binding on the benefit obtained by each other. Therefore, benefit allocation coefficient does not aect the optimal eort level. The other inuence coefficients are the same as the situation of non-cooperation and government promotion.
4 Comparison between situations
According to the analysis of the three situations, we get dierent optimal effort level, optimal benefit, product green degree and reputation. So, we develop four propositions for the dierent game equilibriums in the aspects of dierent optimal eort level, optimal benefit, product green degree and reputation.
Proposition 4
If the benefit distribution coefficient from the government satisfies the condition
Proof
From formula (10), (44) and (54), we can get
According to the
From formula (11), (45) and (55), we can get
According to the
From formula (13), (47) and (57), we can get
and
So, the cost sharing coefficient of the government are
In proposition 4, we can find that the optimal effort of supplier and manufacturer has been improved in the situation of government promotion, and this promotion has a significant correlation with the government’s cost subsidy coefficient, but the optimal effort level of government has not changed. The reason is that the government will work out the level of effort based on the related factors of its own benefit, which is the same as that in the non-cooperative situation, so the optimal effort level of the government has not changed. In the situation of collaborative cooperation, government, supplier and manufacturer will both make higher levels of effort to improve the product green degree and reputation.
Proposition 5
If the benefit distribution coefficient from the government satises the condition
In proposition 5, we can see that government cost subsidies play an important role in promoting the product green degree of green supply chain. Both supplier and manufacturer’s product green degree has been increased. Collaborative cooperation is optimal in three situations.
Proposition 6
If the benefit distribution coefficient from the government satises the condition
In proposition 6, we can see that government cost subsidies play an important role in promoting the reputation of green supply chain. Both supplier and manufacturer’s reputation has been increased. Collaborative cooperation is optimal in three situations.
Proposition 7
If the benefit distribution coefficient from the government satises the condition
In proposition 7 we can see that the benefits of government, supplier, manufacturer and the whole system have been increased in the situations of government promotion and collaborative cooperation. Government cost subsidies play an important role in promoting the reputation of green supply chain.
5 Simulation and example analysis
To better understand the long-term relationship between the equilibriums of three situations, simulation and example analysis of the models are conducted in this section. According to the assignment method of relevant literature [22, 23], through the assignment of relevant parameters, three situations can be compared. The specific parameter assignment is shown in Table 1.
parameter assignment.
| λS | λM | λG | α1 | α2 | β1 | β2 | η1 | η2 | μ | δ1 |
| 2 | 3 | 1 | 3 | 4 | 1 | 1 | 0.3 | 0.4 | 0.2 | 2 |
| δ2 | φ1 | φ2 | γ1 | γ2 | d1 | d2 | q1 | q2 | p1 | p2 |
| 2 | 3 | 3 | 0.2 | 0.25 | 0 | 0 | 0 | 0 | 2 | 3 |
| k1 | k2 | l1 | l2 | b1 | b2 | r | θ1 | θ2 | ||
| 1 | 2 | 2 | 3 | 2 | 3 | 0.1 | 0.4 | 0.5 | ||
We calculate the optimal effort level of the government and the supplier and manufacturer of the green supply chain in three situations, as shown in Table 2.
Green supply chain and government effort level under different situations.
| NS | NM | NG1 | NG2 | |
|---|---|---|---|---|
| non cooperation | 9.100 | 7.357 | 10.550 | 16.857 |
| government promotion | 10.842 | 11.036 | 10.550 | 16.857 |
| collaborative cooperation | 15.617 | 14.714 | 26.300 | 33.714 |
| size relation |
In the three situations, we can find that the level of effort of supplier, manufacturer and government is the lowest in the non-cooperative differential game. The government’s cost subsidy strategy enhances the level of effort of supplier and manufacturer in green supply chain. The level of effort supplier, manufacturer and government have greatly improved in the situation of collaborative cooperation. This verifies the truth of proposition 4. Based on the results of Table 2, we can further calculate the product green degree, reputation and benefit, as shown in Table 3.
In order to make more intuitive comparison, we use Origin to draw out the trend of product green degree and reputation changing with time, as shown in Figures 1 and 2. As time goes on, the green degree of product and reputation of the supplier and manufacturer are growing rapidly in the initial period, then the speed decreases, and finally tends to be smooth. In the case of collaborative cooperation, supplier and manufacturer in green supply chain have the highest level of effort, and government promotion is higher than non-cooperation. This verifies propositions 5 and 6.
Change trend of supplier’s product green degree and reputation.
Change trend of manufacturer’s product green degree and reputation.
Product green degree, reputation and benefit in different situations.
| Situation | Product green degree, reputation and benefit |
|---|---|
| non cooperation | |
| government promotion | |
| collaborative cooperation | |
We compare the benefits of supplier, manufacturer, government and the whole system in different situations, as shown in Figure 3. When the government adopts the cost subsidy strategy, the benefit of the supplier and manufacturer of the green supply chain will be greatly improved than the non-cooperation situation. It shows that the government plays an important role in promoting the development of green supply chain. A relatively suitable strategy can effectively improve the benefits of the green supply chain and the government. Figure 3 (d) shows the whole system benefits of government, supplier and manufacturer in different situations. It can be found that the whole system benefit reaches the maximum in the situation of collaborative cooperation, followed by the situation of government promotion, and the lowest one is the non-cooperation situation. Figure 3 verifies the related conclusions of proposition 7.
Change trend in the benefits of supplier, manufacturer and government.
6 Conclusion
In this paper, we have shown a differential game model under the three situations of non-cooperation, government promotion and cooperative cooperation. We consider the relationship and cooperation strategy between the government and the supplier and manufacturer of the green supply chain, and discuss the differences in the optimal effort level, green degree of product, reputation and the optimal benefit of each subject. By comparing and analyzing of equilibrium results, we have known that the optimal effort level, product green degree, reputation and optimal benefit are obviously higher in the situation of collaborative cooperation than that under non-cooperation and government promotion. The government’s cost subsidy strategy enhances the level of effort of supplier and manufacturer in green supply chain. The cost coefficient, the natural attenuation coefficient of product green degree, the discount rate and the negative influence factors of reputation are negatively correlated with the optimal level of effort. The influence coefficient of the level of effort, the influence coefficient of product green degree on the benefit, the influence coefficient of reputation on the benefit, the influence coefficient of the degree of effort on the green degree of the product, and the influence coefficient of the effort on the reputation are positively related to the optimal level of effort.
Acknowledgments
We are very grateful to Yin Shi and Hou Jie for their help in paper writing. Special thanks to Chen Yuanhong from Fujian Normal University for his help in typesetting. This research was supported by the National Fund Key Project on Social Science of China (14AGL004) and the Fundamental Research Funds for the Central Universities (HEUCFW170901).
Appendix
Proof of Proposition 2
In order to obtain the Stackelberg equilibrium, there exist the benefit optimal value function of the green supply chain, VS(DS, DM, QS, QM) and VM(DS, DM, QS, QM), which are continuous differentiable functions. First, we use backward induction to solve optimal control problem. For all Di ≥ 0 and QS ≥ 0, i ∈ {S, M}, the benefit optimal value functions satisfy the following Hamilton-Jacobi-Bellman equation
For solving formula (A1) and (A2), using extreme conditions and searching for the optimal value of NS and NM by setting the all partial derivative equal to zero, we can get
Second, the benefit optimal value function, VG(DS, DM, QS, QM) satisfies the following Hamilton-Jacobi-Bellman equation
Substituting the results of (A3) and (A4) into (54), we can obtain
Substituting the results of (A3), (A4), (A6), (A7), (A8) and (A9) into (A1), (A2) and (A5), we can obtain
Through the above formulas and the optimal control theory, we can see that the linear optimal function of the DS, DM, QS, QM is the solution of the HJB equation. We can assume
where σi, qi, gi (i = 1, 2, 3, 4) and ωi (i = 1, 2, 3) are the constants to be solved. We can have
Substituting the result of (A13) and (A14) into (A10), (A11) and (A12), we can get
Using the DS, DM, QS, QM = 0 to (A15), (A16) and (A17), parameter values of the optimal value function can be expressed as follows
Substituting the results of (A18), (A19), (A20) and (A21) into (A13), we can obtain
In this case, the benefit of the whole system is
Solving the partial derivative of (A22), (A23) and (A24), and substituting the results into (A3), (A4), (A6), (A7), (A8) and (A9), we can get the Stackelberg master-slave equilibria of supplier and manufacturer and government as the proposition 2.
Proof of Proposition 3
In order to obtain the cooperative equilibrium, there exists a benefit optimal value function of VT(DS, DM, QS, QM, which is a continuous differentiable function. For all DS, DM, QS, QM = 0, the benefit optimal value function satisfies the following Hamilton-Jacobi-Bellman equation
For solving formula (A27), using extreme conditions and searching for the optimal value of NS, NM, NG 1 and NG 2 by setting the partial derivative equal to zero, we can get
Substituting the results of (A28), (A29), (A30) and (A31) into (A27), we can obtain
Through the above formulas and the optimal control theory, we can see that the linear optimal function of the DS, DM, QS, QM is the solution of the HJB equation. We can get
Substituting the results of (A33) and (A34) into (A32), we can get
Using the DS, DM, QS, QM ≥ 0 to (A35), parameter values of the optimal value function can be expressed as follows
Substituting the results of (A36) and (A37) into (A35), we can get
Solving the partial derivative of (A38), and substituting the results into (A28), (A29), (A30) and (A31), we can get the cooperative game equilibria of supplier and manufacturer and government as the proposition 3.
Proof of Proposition 5
From formula (37), (50) and (58), we can get
According to the
From formula (39), (52) and (60), we can get
According to the
Proof of Proposition 6
From formula (38), (51) and (59), we can get
According to the
According to the
Proof of Proposition 7
From formula (33) and (A22), we can get
From formula (34) and (A23), we can get
From formula (35) and (A24), we can get
According to the
Based on the above analysis, we can find
From the formula (A25) and (A38), we can get
According to the
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© 2019 Zeng and Li, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 Public License.
Artikel in diesem Heft
- Regular Articles
- On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one
- Centralizers of automorphisms permuting free generators
- Extreme points and support points of conformal mappings
- Arithmetical properties of double Möbius-Bernoulli numbers
- The product of quasi-ideal refined generalised quasi-adequate transversals
- Characterizations of the Solution Sets of Generalized Convex Fuzzy Optimization Problem
- Augmented, free and tensor generalized digroups
- Time-dependent attractor of wave equations with nonlinear damping and linear memory
- A new smoothing method for solving nonlinear complementarity problems
- Almost periodic solution of a discrete competitive system with delays and feedback controls
- On a problem of Hasse and Ramachandra
- Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge
- A note on the formulas for the Drazin inverse of the sum of two matrices
- Completeness theorem for probability models with finitely many valued measure
- Periodic solution for ϕ-Laplacian neutral differential equation
- Asymptotic orbital shadowing property for diffeomorphisms
- Modular equations of a continued fraction of order six
- Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas
- Stability Problems and Analytical Integration for the Clebsch’s System
- Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures
- On split Lie color triple systems
- Triangular Surface Patch Based on Bivariate Meyer-König-Zeller Operator
- Generators for maximal subgroups of Conway group Co1
- Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model
- Characterizations of Convex spaces and Anti-matroids via Derived Operators
- On Partitions and Arf Semigroups
- Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions
- A concise proof to the spectral and nuclear norm bounds through tensor partitions
- A categorical approach to abstract convex spaces and interval spaces
- Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations
- Parity results for broken 11-diamond partitions
- A new fourth power mean of two-term exponential sums
- The new operations on complete ideals
- Soft covering based rough graphs and corresponding decision making
- Complete convergence for arrays of ratios of order statistics
- Sufficient and necessary conditions of convergence for ρ͠ mixing random variables
- Attractors of dynamical systems in locally compact spaces
- Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains
- Statistical approximation properties of λ-Bernstein operators based on q-integers
- An investigation of fractional Bagley-Torvik equation
- Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer
- On the hybrid power mean of two kind different trigonometric sums
- Embedding of Supplementary Results in Strong EMT Valuations and Strength
- On Diophantine approximation by unlike powers of primes
- A General Version of the Nullstellensatz for Arbitrary Fields
- A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces
- Random Polygons and Estimations of π
- The optimal pebbling of spindle graphs
- MBJ-neutrosophic ideals of BCK/BCI-algebras
- A note on the structure of a finite group G having a subgroup H maximal in 〈H, Hg〉
- A fuzzy multi-objective linear programming with interval-typed triangular fuzzy numbers
- Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization
- Stability property of the prey free equilibrium point
- Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem
- Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders
- Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching
- Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts
- Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application
- Nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems
- Empirical likelihood confidence regions of the parameters in a partially single-index varying-coefficient model
- Algebraic loop structures on algebra comultiplications
- Two weight estimates for a class of (p, q) type sublinear operators and their commutators
- Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays
- 2-closures of primitive permutation groups of holomorph type
- Monotonicity properties and inequalities related to generalized Grötzsch ring functions
- Variation inequalities related to Schrödinger operators on weighted Morrey spaces
- Research on cooperation strategy between government and green supply chain based on differential game
- Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting
- *-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds
- Some improved bounds on two energy-like invariants of some derived graphs
- Pricing under dynamic risk measures
- Finite groups with star-free noncyclic graphs
- A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies
- S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain
- On Diophantine equations involving Lucas sequences
- A new way to represent functions as series
- Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system
- Some remarks on a pair of seemingly unrelated regression models
- Lyapunov stable homoclinic classes for smooth vector fields
- Stabilizers in EQ-algebras
- The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles
- Spectrum perturbations of compact operators in a Banach space
- The non-commuting graph of a non-central hypergroup
- Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation
- Positive solutions of the discrete Dirichlet problem involving the mean curvature operator
- Dislocated quasi cone b-metric space over Banach algebra and contraction principles with application to functional equations
- On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis
- Differential polynomials of L-functions with truncated shared values
- Exclusion sets in the S-type eigenvalue localization sets for tensors
- Continuous linear operators on Orlicz-Bochner spaces
- Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity
- Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces
- A quantitative obstruction to collapsing surfaces
- Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species
- Coexistence for a kind of stochastic three-species competitive models
- Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
- On the two-term exponential sums and character sums of polynomials
- F-biharmonic maps into general Riemannian manifolds
- Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn
- Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains
- Power graphs and exchange property for resolving sets
- On nearly Hurewicz spaces
- Least eigenvalue of the connected graphs whose complements are cacti
- Determinants of two kinds of matrices whose elements involve sine functions
- A characterization of translational hulls of a strongly right type B semigroup
- Common fixed point results for two families of multivalued A–dominated contractive mappings on closed ball with applications
- Lp estimates for maximal functions along surfaces of revolution on product spaces
- Path-induced closure operators on graphs for defining digital Jordan surfaces
- Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras
- Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system
- Injective hulls of many-sorted ordered algebras
- Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3
- Global properties of virus dynamics with B-cell impairment
- The monotonicity of ratios involving arc tangent function with applications
- A family of Cantorvals
- An asymptotic property of branching-type overloaded polling networks
- Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales
- Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion
- L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras
- L-topological-convex spaces generated by L-convex bases
- An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior
- New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices
- Hankel determinant of order three for familiar subsets of analytic functions related with sine function
- On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5
- Results on existence for generalized nD Navier-Stokes equations
- Regular Banach space net and abstract-valued Orlicz space of range-varying type
- Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means
- On a new convergence in topological spaces
- On a fixed point theorem with application to functional equations
- Coupled system of a fractional order differential equations with weighted initial conditions
- Rough quotient in topological rough sets
- Split Hausdorff internal topologies on posets
- A preconditioned AOR iterative scheme for systems of linear equations with L-matrics
- New handy and accurate approximation for the Gaussian integrals with applications to science and engineering
- Special Issue on Graph Theory (GWGT 2019)
- The general position problem and strong resolving graphs
- Connected domination game played on Cartesian products
- On minimum algebraic connectivity of graphs whose complements are bicyclic
- A novel method to construct NSSD molecular graphs
Artikel in diesem Heft
- Regular Articles
- On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one
- Centralizers of automorphisms permuting free generators
- Extreme points and support points of conformal mappings
- Arithmetical properties of double Möbius-Bernoulli numbers
- The product of quasi-ideal refined generalised quasi-adequate transversals
- Characterizations of the Solution Sets of Generalized Convex Fuzzy Optimization Problem
- Augmented, free and tensor generalized digroups
- Time-dependent attractor of wave equations with nonlinear damping and linear memory
- A new smoothing method for solving nonlinear complementarity problems
- Almost periodic solution of a discrete competitive system with delays and feedback controls
- On a problem of Hasse and Ramachandra
- Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge
- A note on the formulas for the Drazin inverse of the sum of two matrices
- Completeness theorem for probability models with finitely many valued measure
- Periodic solution for ϕ-Laplacian neutral differential equation
- Asymptotic orbital shadowing property for diffeomorphisms
- Modular equations of a continued fraction of order six
- Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas
- Stability Problems and Analytical Integration for the Clebsch’s System
- Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures
- On split Lie color triple systems
- Triangular Surface Patch Based on Bivariate Meyer-König-Zeller Operator
- Generators for maximal subgroups of Conway group Co1
- Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model
- Characterizations of Convex spaces and Anti-matroids via Derived Operators
- On Partitions and Arf Semigroups
- Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions
- A concise proof to the spectral and nuclear norm bounds through tensor partitions
- A categorical approach to abstract convex spaces and interval spaces
- Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations
- Parity results for broken 11-diamond partitions
- A new fourth power mean of two-term exponential sums
- The new operations on complete ideals
- Soft covering based rough graphs and corresponding decision making
- Complete convergence for arrays of ratios of order statistics
- Sufficient and necessary conditions of convergence for ρ͠ mixing random variables
- Attractors of dynamical systems in locally compact spaces
- Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains
- Statistical approximation properties of λ-Bernstein operators based on q-integers
- An investigation of fractional Bagley-Torvik equation
- Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer
- On the hybrid power mean of two kind different trigonometric sums
- Embedding of Supplementary Results in Strong EMT Valuations and Strength
- On Diophantine approximation by unlike powers of primes
- A General Version of the Nullstellensatz for Arbitrary Fields
- A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces
- Random Polygons and Estimations of π
- The optimal pebbling of spindle graphs
- MBJ-neutrosophic ideals of BCK/BCI-algebras
- A note on the structure of a finite group G having a subgroup H maximal in 〈H, Hg〉
- A fuzzy multi-objective linear programming with interval-typed triangular fuzzy numbers
- Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization
- Stability property of the prey free equilibrium point
- Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem
- Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders
- Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching
- Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts
- Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application
- Nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems
- Empirical likelihood confidence regions of the parameters in a partially single-index varying-coefficient model
- Algebraic loop structures on algebra comultiplications
- Two weight estimates for a class of (p, q) type sublinear operators and their commutators
- Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays
- 2-closures of primitive permutation groups of holomorph type
- Monotonicity properties and inequalities related to generalized Grötzsch ring functions
- Variation inequalities related to Schrödinger operators on weighted Morrey spaces
- Research on cooperation strategy between government and green supply chain based on differential game
- Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting
- *-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds
- Some improved bounds on two energy-like invariants of some derived graphs
- Pricing under dynamic risk measures
- Finite groups with star-free noncyclic graphs
- A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies
- S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain
- On Diophantine equations involving Lucas sequences
- A new way to represent functions as series
- Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system
- Some remarks on a pair of seemingly unrelated regression models
- Lyapunov stable homoclinic classes for smooth vector fields
- Stabilizers in EQ-algebras
- The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles
- Spectrum perturbations of compact operators in a Banach space
- The non-commuting graph of a non-central hypergroup
- Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation
- Positive solutions of the discrete Dirichlet problem involving the mean curvature operator
- Dislocated quasi cone b-metric space over Banach algebra and contraction principles with application to functional equations
- On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis
- Differential polynomials of L-functions with truncated shared values
- Exclusion sets in the S-type eigenvalue localization sets for tensors
- Continuous linear operators on Orlicz-Bochner spaces
- Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity
- Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces
- A quantitative obstruction to collapsing surfaces
- Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species
- Coexistence for a kind of stochastic three-species competitive models
- Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
- On the two-term exponential sums and character sums of polynomials
- F-biharmonic maps into general Riemannian manifolds
- Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn
- Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains
- Power graphs and exchange property for resolving sets
- On nearly Hurewicz spaces
- Least eigenvalue of the connected graphs whose complements are cacti
- Determinants of two kinds of matrices whose elements involve sine functions
- A characterization of translational hulls of a strongly right type B semigroup
- Common fixed point results for two families of multivalued A–dominated contractive mappings on closed ball with applications
- Lp estimates for maximal functions along surfaces of revolution on product spaces
- Path-induced closure operators on graphs for defining digital Jordan surfaces
- Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras
- Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system
- Injective hulls of many-sorted ordered algebras
- Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3
- Global properties of virus dynamics with B-cell impairment
- The monotonicity of ratios involving arc tangent function with applications
- A family of Cantorvals
- An asymptotic property of branching-type overloaded polling networks
- Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales
- Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion
- L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras
- L-topological-convex spaces generated by L-convex bases
- An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior
- New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices
- Hankel determinant of order three for familiar subsets of analytic functions related with sine function
- On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5
- Results on existence for generalized nD Navier-Stokes equations
- Regular Banach space net and abstract-valued Orlicz space of range-varying type
- Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means
- On a new convergence in topological spaces
- On a fixed point theorem with application to functional equations
- Coupled system of a fractional order differential equations with weighted initial conditions
- Rough quotient in topological rough sets
- Split Hausdorff internal topologies on posets
- A preconditioned AOR iterative scheme for systems of linear equations with L-matrics
- New handy and accurate approximation for the Gaussian integrals with applications to science and engineering
- Special Issue on Graph Theory (GWGT 2019)
- The general position problem and strong resolving graphs
- Connected domination game played on Cartesian products
- On minimum algebraic connectivity of graphs whose complements are bicyclic
- A novel method to construct NSSD molecular graphs
