• Previous Article
    Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems
  • JIMO Home
  • This Issue
  • Next Article
    Effect of reliability on varying demand and holding cost on inventory system incorporating probabilistic deterioration
doi: 10.3934/jimo.2020155

Preannouncement strategy of platform-type new product for competing platforms: Technical or marketing information

1. 

Center for Behavioral Decision and Control, School of Management and Engineering, Nanjing University, Hankou Road 22, Nanjing, Jiangsu 210093, China

2. 

School of Business, Jiangsu Open University, Jiangdong North Road 399, Nanjing, Jiangsu 210036, China

* Corresponding author: Tiaojun Xiao

Received  March 2020 Revised  July 2020 Published  October 2020

Fund Project: This research is funded by the National Natural Science Foundation of China (No. 71871112) and Jiangsu province's "333 project" training funding project (No. BRA2019040)

What message should be released to consumers and developers is an important part of the preannouncement strategy of platforms' new product. From the perspectives of consumers and developers' information perceptions, we develop a game model of two-sided market, which can better describe the impacts of information preannouncement on consumers, developers, and platforms behavior in a competitive environment. There are two preannouncement strategies: Technical or marketing information. Our studies reveal that (i) when the development capabilities are heterogeneous enough, both platforms release technical information; (ii) both platforms preannounce marketing information when the heterogeneity of development capability is sufficiently small, even if it decreases total social welfare; (iii) the platform lacking competitive advantage is more inclined to adopt a strategy different from the competitive advantage platform, and competitive advantage platform is likely to change the preannouncement strategy constantly; (iv) the heterogeneity of platforms is the prerequisite for the asymmetric equilibrium, even if it may decrease the overall social welfare.

Citation: Ye Jiang, Tiaojun Xiao. Preannouncement strategy of platform-type new product for competing platforms: Technical or marketing information. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020155
References:
[1]

M. AliG. Alireza and Q. Mojtaba, Prediction of air travel demand using a hybrid artificial neural network (ANN) with Bat and Firefly algorithms: A case study, The Journal of Supercomputing, 74 (2018), 5461-5484.   Google Scholar

[2]

M. Armstrong, Competition in two-sided markets, The RAND Journal of Economics, 37 (2006), 668-691.  doi: 10.1007/s00199-006-0114-6.  Google Scholar

[3]

F. M. BassA. KrishnamoorthyA. Prasad and S. P. Sethi, Advertising competition with market expansion for finite horizon firms, Journal of Industrial and Management Optimization, 1 (2005), 1-19.  doi: 10.3934/jimo.2005.1.1.  Google Scholar

[4]

B. L. BayusS. Jain and A. G. Rao, Truth or consequences: An analysis of vporware and new product announcements, Journal of Marketing Research, 38 (2001), 3-13.   Google Scholar

[5]

H. K. Bhargava, Platform technologies and network goods: Insights on product launch and management, Information Technology and Management, 15 (2014), 199-209.   Google Scholar

[6]

R. K. Chellappa and R. Mukherjee, Platform preannouncement strategies: The strategic role of information in two-sided markets competition, Social Science Electronic Publishing, (2015), 55 pp. Google Scholar

[7]

C.-W. Chen and V. Wong, Design and delivery of new product preannouncement messages, Journal of Marketing Theory and Practice, 20 (2014), 203-222.  doi: 10.2753/MTP1069-6679200206.  Google Scholar

[8]

J. P. ChoiE. G. Kristiansen and J. Nahm, Strategic product pre-announcements in markets with network effects, Hitotsubashi Journal of Economics, 60 (2005), 1-20.  doi: 10.2139/ssrn.847986.  Google Scholar

[9]

T. Doganoglu and J. Wright, Multihoming and compatibility, International Journal of Industrial Organization, 24 (2006), 45-67.   Google Scholar

[10]

D. Dranove and N. Gandal, The DVD-vs.-DIVX standard war: Empirical evidence of network effects and preannouncement effects, Journal of Economics & Management Strategy, 12 (2003), 363-386.   Google Scholar

[11]

N. Economides and J. Tag, Network neutrality on the Internet: A two-sided market analysis, Information Economics and Policy, 24 (2012), 91-104.   Google Scholar

[12]

J. Eliashberg and T. S. Robertson, New product preannouncing behavior: A market signaling study, Journal of Marketing Research, 25 (1988), 282-292.   Google Scholar

[13]

M. Gao, Platform pricing in mixed two-sided markets, International Economic Review, 59 (2018), 1103-1129.  doi: 10.1111/iere.12298.  Google Scholar

[14]

H. A. Gerlach, Announcement, entry, and preemption when consumers have switching costs, The RAND Journal of Economics, 35 (2004), 184-202.  doi: 10.2307/1593736.  Google Scholar

[15]

G. AlirezaZ. H. KhademiT. Reza and A. Sadegheih, Multiobjective fuzzy mathematical model for a financially constrained closed-loop supply chain with labor employment, Computational Intelligence, 36 (2020), 4-34.   Google Scholar

[16]

A. Hagiu and H. Halaburda, Information and two-sided platform profits, International Journal of Industrial Organization, 34 (2014), 25-35.   Google Scholar

[17]

J. A. Hoxmeier, Software preannouncements and their impact on customers' perceptions and vendor reputation, Journal of Management Information Systems, 17 (2000), 115-139.   Google Scholar

[18]

H. Jung, Signaling quality with new product preannouncements: Vaporware and the role of reference quality, Journal of Business Research, 64 (2011), 1251-1258.  doi: 10.1016/j.jbusres.2011.06.032.  Google Scholar

[19]

D. Ke, H. Zhang and Y. Tu, The cannibalization effects of new product preannouncement and launch on the C2C marketplace, 15th workshop on E-business, (2016), 84–94. doi: 10.1007/978-3-319-69644-7_8.  Google Scholar

[20]

T. KlastorinH. Mamani and Y.-P. Zhou, Production and Operations Management, 15th workshop on E-business, 25 (2016), 2051-2064.   Google Scholar

[21]

E. Le Nagard-Assayag and D. Manceau, Modeling the impact of product preannouncements in the context of indirect network externalities, International Journal of Research in Marketing, 18 (2001), 203-219.  doi: 10.1016/S0167-8116(00)00031-8.  Google Scholar

[22]

S. Lee, Factors influencing the social networking service user's value perception and word of mouth decision of corporate post with special reference to the emotional attachment, Information Technology and Management, 17 (2016), 15-27.  doi: 10.1007/s10799-015-0227-3.  Google Scholar

[23]

Y. Lee and G. C. Oconnor, The impact of communication strategy on launching new products: The moderating role of product innovativeness, Journal of Product Innovation Management, 20 (2003), 4-21.  doi: 10.1111/1540-5885.t01-1-201002.  Google Scholar

[24]

B. Lilly and R. Walters, Toward a model of new product preannouncement timing, Journal of Product Innovation Management, 14 (1997), 4-20.  doi: 10.1111/1540-5885.1410004.  Google Scholar

[25]

M. J. Mccabe and C. M. Snyder, Academic journal prices in a digital age: A two-sided market model, Topics in Economic Analysis & Policy, 7 (2007), 1-39.   Google Scholar

[26]

E. Ofek and O. Turut, Vaporware, suddenware, and trueware: New product preannouncements under market uncertainty, Marketing Science, 32 (2013), 342-355.  doi: 10.1287/mksc.1120.0762.  Google Scholar

[27]

J. Rochet and J. Tirole, Platform competition in two-sided markets, Journal of the European Economic Association, 1 (2003), 990-1029.  doi: 10.1162/154247603322493212.  Google Scholar

[28]

J. Rochet and J. Tirole, Two-sided markets: A progress report, The RAND Journal of Economics, 37 (2006), 645-667.  doi: 10.1111/j.1756-2171.2006.tb00036.x.  Google Scholar

[29]

K. Schatzel and R. Calantone, Creating market anticipation: An exploratory examination of the effect of preannouncement behavior on a new product's launch, Journal of the Academy of Marketing Science, 34 (2006), 357-366.  doi: 10.1177/0092070304270737.  Google Scholar

[30]

L. C. SofferS. R. Thiagarajan and B. R. Walther, Earnings preannouncement strategies, Review of Accounting Studies, 5 (2000), 5-26.   Google Scholar

[31]

M. Su and V. R. Rao, New product preannouncement as a signaling strategy: An audience-specific review and analysis, Journal of Product Innovation Management, 27 (2010), 658-672.  doi: 10.1111/j.1540-5885.2010.00743.x.  Google Scholar

[32]

M. Su and V. R. Rao, Timing decisions of new product preannouncement and launch with competition, International Journal of Production Economics, 129 (2011), 51-64.  doi: 10.1016/j.ijpe.2010.09.001.  Google Scholar

[33]

H. Sun and S. Kumar, A manufacturer's new product preannouncement decision and the supplier's response, Production and Operations Management, 29 (2020), 2289-2306.  doi: 10.1111/poms.12900.  Google Scholar

[34]

Y. Trope and N. Liberman, Construal-level theory of psychological distance, Psychological Review, 117.2 (2010), 440-463.  doi: 10.1037/a0018963.  Google Scholar

[35]

M. J. Villena and M. Contreras, Global and local advertising strategies: A dynamic multi-market optimal control model, Journal of Industrial and Management Optimization, 15 (2017), 1017-1048.  doi: 10.3934/jimo.2018084.  Google Scholar

[36]

E. G. Weyl, A price theory of multi-sided platforms, American Economic Review, 100 (2010), 1642-1672.  doi: 10.1257/aer.100.4.1642.  Google Scholar

[37]

H. Zhang and Y. K. Choi, Preannouncement messages: Impetus for electronic word-of-mouth, International Journal of Advertising, 37 (2018), 54-70.  doi: 10.1080/02650487.2017.1391679.  Google Scholar

[38]

J. ZhangQ. Liang and J. Huang, Forward advertising: A competitive analysis of new product preannouncement, Information Economics and Policy, 37 (2016), 3-12.  doi: 10.1016/j.infoecopol.2016.10.004.  Google Scholar

show all references

References:
[1]

M. AliG. Alireza and Q. Mojtaba, Prediction of air travel demand using a hybrid artificial neural network (ANN) with Bat and Firefly algorithms: A case study, The Journal of Supercomputing, 74 (2018), 5461-5484.   Google Scholar

[2]

M. Armstrong, Competition in two-sided markets, The RAND Journal of Economics, 37 (2006), 668-691.  doi: 10.1007/s00199-006-0114-6.  Google Scholar

[3]

F. M. BassA. KrishnamoorthyA. Prasad and S. P. Sethi, Advertising competition with market expansion for finite horizon firms, Journal of Industrial and Management Optimization, 1 (2005), 1-19.  doi: 10.3934/jimo.2005.1.1.  Google Scholar

[4]

B. L. BayusS. Jain and A. G. Rao, Truth or consequences: An analysis of vporware and new product announcements, Journal of Marketing Research, 38 (2001), 3-13.   Google Scholar

[5]

H. K. Bhargava, Platform technologies and network goods: Insights on product launch and management, Information Technology and Management, 15 (2014), 199-209.   Google Scholar

[6]

R. K. Chellappa and R. Mukherjee, Platform preannouncement strategies: The strategic role of information in two-sided markets competition, Social Science Electronic Publishing, (2015), 55 pp. Google Scholar

[7]

C.-W. Chen and V. Wong, Design and delivery of new product preannouncement messages, Journal of Marketing Theory and Practice, 20 (2014), 203-222.  doi: 10.2753/MTP1069-6679200206.  Google Scholar

[8]

J. P. ChoiE. G. Kristiansen and J. Nahm, Strategic product pre-announcements in markets with network effects, Hitotsubashi Journal of Economics, 60 (2005), 1-20.  doi: 10.2139/ssrn.847986.  Google Scholar

[9]

T. Doganoglu and J. Wright, Multihoming and compatibility, International Journal of Industrial Organization, 24 (2006), 45-67.   Google Scholar

[10]

D. Dranove and N. Gandal, The DVD-vs.-DIVX standard war: Empirical evidence of network effects and preannouncement effects, Journal of Economics & Management Strategy, 12 (2003), 363-386.   Google Scholar

[11]

N. Economides and J. Tag, Network neutrality on the Internet: A two-sided market analysis, Information Economics and Policy, 24 (2012), 91-104.   Google Scholar

[12]

J. Eliashberg and T. S. Robertson, New product preannouncing behavior: A market signaling study, Journal of Marketing Research, 25 (1988), 282-292.   Google Scholar

[13]

M. Gao, Platform pricing in mixed two-sided markets, International Economic Review, 59 (2018), 1103-1129.  doi: 10.1111/iere.12298.  Google Scholar

[14]

H. A. Gerlach, Announcement, entry, and preemption when consumers have switching costs, The RAND Journal of Economics, 35 (2004), 184-202.  doi: 10.2307/1593736.  Google Scholar

[15]

G. AlirezaZ. H. KhademiT. Reza and A. Sadegheih, Multiobjective fuzzy mathematical model for a financially constrained closed-loop supply chain with labor employment, Computational Intelligence, 36 (2020), 4-34.   Google Scholar

[16]

A. Hagiu and H. Halaburda, Information and two-sided platform profits, International Journal of Industrial Organization, 34 (2014), 25-35.   Google Scholar

[17]

J. A. Hoxmeier, Software preannouncements and their impact on customers' perceptions and vendor reputation, Journal of Management Information Systems, 17 (2000), 115-139.   Google Scholar

[18]

H. Jung, Signaling quality with new product preannouncements: Vaporware and the role of reference quality, Journal of Business Research, 64 (2011), 1251-1258.  doi: 10.1016/j.jbusres.2011.06.032.  Google Scholar

[19]

D. Ke, H. Zhang and Y. Tu, The cannibalization effects of new product preannouncement and launch on the C2C marketplace, 15th workshop on E-business, (2016), 84–94. doi: 10.1007/978-3-319-69644-7_8.  Google Scholar

[20]

T. KlastorinH. Mamani and Y.-P. Zhou, Production and Operations Management, 15th workshop on E-business, 25 (2016), 2051-2064.   Google Scholar

[21]

E. Le Nagard-Assayag and D. Manceau, Modeling the impact of product preannouncements in the context of indirect network externalities, International Journal of Research in Marketing, 18 (2001), 203-219.  doi: 10.1016/S0167-8116(00)00031-8.  Google Scholar

[22]

S. Lee, Factors influencing the social networking service user's value perception and word of mouth decision of corporate post with special reference to the emotional attachment, Information Technology and Management, 17 (2016), 15-27.  doi: 10.1007/s10799-015-0227-3.  Google Scholar

[23]

Y. Lee and G. C. Oconnor, The impact of communication strategy on launching new products: The moderating role of product innovativeness, Journal of Product Innovation Management, 20 (2003), 4-21.  doi: 10.1111/1540-5885.t01-1-201002.  Google Scholar

[24]

B. Lilly and R. Walters, Toward a model of new product preannouncement timing, Journal of Product Innovation Management, 14 (1997), 4-20.  doi: 10.1111/1540-5885.1410004.  Google Scholar

[25]

M. J. Mccabe and C. M. Snyder, Academic journal prices in a digital age: A two-sided market model, Topics in Economic Analysis & Policy, 7 (2007), 1-39.   Google Scholar

[26]

E. Ofek and O. Turut, Vaporware, suddenware, and trueware: New product preannouncements under market uncertainty, Marketing Science, 32 (2013), 342-355.  doi: 10.1287/mksc.1120.0762.  Google Scholar

[27]

J. Rochet and J. Tirole, Platform competition in two-sided markets, Journal of the European Economic Association, 1 (2003), 990-1029.  doi: 10.1162/154247603322493212.  Google Scholar

[28]

J. Rochet and J. Tirole, Two-sided markets: A progress report, The RAND Journal of Economics, 37 (2006), 645-667.  doi: 10.1111/j.1756-2171.2006.tb00036.x.  Google Scholar

[29]

K. Schatzel and R. Calantone, Creating market anticipation: An exploratory examination of the effect of preannouncement behavior on a new product's launch, Journal of the Academy of Marketing Science, 34 (2006), 357-366.  doi: 10.1177/0092070304270737.  Google Scholar

[30]

L. C. SofferS. R. Thiagarajan and B. R. Walther, Earnings preannouncement strategies, Review of Accounting Studies, 5 (2000), 5-26.   Google Scholar

[31]

M. Su and V. R. Rao, New product preannouncement as a signaling strategy: An audience-specific review and analysis, Journal of Product Innovation Management, 27 (2010), 658-672.  doi: 10.1111/j.1540-5885.2010.00743.x.  Google Scholar

[32]

M. Su and V. R. Rao, Timing decisions of new product preannouncement and launch with competition, International Journal of Production Economics, 129 (2011), 51-64.  doi: 10.1016/j.ijpe.2010.09.001.  Google Scholar

[33]

H. Sun and S. Kumar, A manufacturer's new product preannouncement decision and the supplier's response, Production and Operations Management, 29 (2020), 2289-2306.  doi: 10.1111/poms.12900.  Google Scholar

[34]

Y. Trope and N. Liberman, Construal-level theory of psychological distance, Psychological Review, 117.2 (2010), 440-463.  doi: 10.1037/a0018963.  Google Scholar

[35]

M. J. Villena and M. Contreras, Global and local advertising strategies: A dynamic multi-market optimal control model, Journal of Industrial and Management Optimization, 15 (2017), 1017-1048.  doi: 10.3934/jimo.2018084.  Google Scholar

[36]

E. G. Weyl, A price theory of multi-sided platforms, American Economic Review, 100 (2010), 1642-1672.  doi: 10.1257/aer.100.4.1642.  Google Scholar

[37]

H. Zhang and Y. K. Choi, Preannouncement messages: Impetus for electronic word-of-mouth, International Journal of Advertising, 37 (2018), 54-70.  doi: 10.1080/02650487.2017.1391679.  Google Scholar

[38]

J. ZhangQ. Liang and J. Huang, Forward advertising: A competitive analysis of new product preannouncement, Information Economics and Policy, 37 (2016), 3-12.  doi: 10.1016/j.infoecopol.2016.10.004.  Google Scholar

Figure 1.  Game structure of the PNPP
Figure 2.  (a) SPNE change with $ t_{e} $ when $ t = 15 $ and $ f_{e} = 1 $ and (b)SPNE change with $ f_{e} $ when $ t = 1 $ and $ t_{e} = 15 $
Figure 3.  (a) SPNE distribution when $ \beta_{b} = 1.01 $ and (b)SPNE distribution when $ \beta_{b} = 5.01 $
Figure 4.  (a) Consumer price changes with $ \delta $ and (b)Developer price changes with $ \delta $
Figure 5.  (a) SPNE changes with $ f_{e} $ when $ \delta = 0.2 $ and (b)SPNE changes with $ f_{e} $ when $ \delta = 1.2 $
Figure 6.  SPNE changes with $ \delta $
Figure 7.  Total social welfare changes with $ \delta $
Table 1.  Equilibrium results under $ (M, M) $
Decisions
Consumers' price $ p_{bi}^{MM}=t+t_{e}+\beta_{b}(1-g)/(4g)-\beta_{b}(2v_{d} +\beta_{b}+3\beta_{d})/(8gc_{d}) $
Developers' price $ p_{di}^{MM}=[2v_{d}-2c_{d}(1-g)+\beta_{b}-\beta_{d}]/4 $
Consumers' number $ n_{bi}^{MM}=1/2 $
Developers' number $ n_{di}^{MM}=[2v_{d}-2c_{d}(1-g)+\beta_{b}+\beta_{d}]/(8gc_{d}) $
Equilibrium profits/total utilities
Platforms' profit $ \Pi_{i}^{MM}=(t+t_{e})/2+(4V_{1}^2-B_{0}-2\beta_{b}\beta_{d})/(32gc_{d})-c_{M} $
Consumers' total utility $ BS_{i}^{MM}=[4v_{b}-5(t+t_{e})]/8+[B_{0}+2(\beta_{b}+\beta_{d})V_{1}]/(16gc_{d}) $
Developers' total utility $ DS_{i}^{MM}=(2V_{1}+\beta_{b}+\beta_{d})^2/(64gc_{d}) $
Decisions
Consumers' price $ p_{bi}^{MM}=t+t_{e}+\beta_{b}(1-g)/(4g)-\beta_{b}(2v_{d} +\beta_{b}+3\beta_{d})/(8gc_{d}) $
Developers' price $ p_{di}^{MM}=[2v_{d}-2c_{d}(1-g)+\beta_{b}-\beta_{d}]/4 $
Consumers' number $ n_{bi}^{MM}=1/2 $
Developers' number $ n_{di}^{MM}=[2v_{d}-2c_{d}(1-g)+\beta_{b}+\beta_{d}]/(8gc_{d}) $
Equilibrium profits/total utilities
Platforms' profit $ \Pi_{i}^{MM}=(t+t_{e})/2+(4V_{1}^2-B_{0}-2\beta_{b}\beta_{d})/(32gc_{d})-c_{M} $
Consumers' total utility $ BS_{i}^{MM}=[4v_{b}-5(t+t_{e})]/8+[B_{0}+2(\beta_{b}+\beta_{d})V_{1}]/(16gc_{d}) $
Developers' total utility $ DS_{i}^{MM}=(2V_{1}+\beta_{b}+\beta_{d})^2/(64gc_{d}) $
Table 2.  Nash equilibrium under $ (T, T) $
Decisions
Consumers' price $ p_{bi}^{TT}=t+\beta_{b}(1-g)/(4g)-\beta_{b}[2(v_{d}+f_{e}) +\beta_{b}+3\beta_{d}]/(8gc_{d}) $
Developers' price $ p_{di}^{TT}=(2v_{d}-2c_{d}+2c_{d}g+2f_{e}+\beta_{b}-\beta_{d})/4 $
Consumers' number $ n_{bi}^{TT}=1/2 $
Developers' number $ n_{di}^{TT}=[2v_{d}+2f_{e}+\beta_{b}+\beta_{d}-2c_{d}(1-g)]/(8gc_{d}) $
Equilibrium profits/total utilities
Platforms' profit $ \Pi_{i}^{TT}=t/2+[4(V_{1}+f_{e})^2-B_{0}-2\beta_{b}\beta_{d}]/(32gc_{d})-c_{T} $
Consumers' total utility $ BS_{i}^{TT}=(4v_{b}-5t)/8+[B_{0}+2(\beta_{b}+\beta_{d})(V_{1}+f_{e})]/(16gc_{d}) $
Developers' total utility $ DS_{i}^{TT}=[2(V_{1}+f_{e})+\beta_{b}+\beta_{d}]^2/(64gc_{d}) $
Decisions
Consumers' price $ p_{bi}^{TT}=t+\beta_{b}(1-g)/(4g)-\beta_{b}[2(v_{d}+f_{e}) +\beta_{b}+3\beta_{d}]/(8gc_{d}) $
Developers' price $ p_{di}^{TT}=(2v_{d}-2c_{d}+2c_{d}g+2f_{e}+\beta_{b}-\beta_{d})/4 $
Consumers' number $ n_{bi}^{TT}=1/2 $
Developers' number $ n_{di}^{TT}=[2v_{d}+2f_{e}+\beta_{b}+\beta_{d}-2c_{d}(1-g)]/(8gc_{d}) $
Equilibrium profits/total utilities
Platforms' profit $ \Pi_{i}^{TT}=t/2+[4(V_{1}+f_{e})^2-B_{0}-2\beta_{b}\beta_{d}]/(32gc_{d})-c_{T} $
Consumers' total utility $ BS_{i}^{TT}=(4v_{b}-5t)/8+[B_{0}+2(\beta_{b}+\beta_{d})(V_{1}+f_{e})]/(16gc_{d}) $
Developers' total utility $ DS_{i}^{TT}=[2(V_{1}+f_{e})+\beta_{b}+\beta_{d}]^2/(64gc_{d}) $
Table 3.  Nash equilibrium under $ (M, T) $
Decisions
Consumers' price $p_{bi}^{TT}=t+\beta_{b}(1-g)/(4g)-\beta_{b}[2(v_{d}+f_{e}) +\beta_{b}+3\beta_{d}]/(8gc_{d})$
Developers' price $p_{di}^{TT}=(2v_{d}-2c_{d}+2c_{d}g+2f_{e}+\beta_{b}-\beta_{d})/4$
Consumers' number $n_{bi}^{TT}=1/2$
Developers' number $n_{di}^{TT}=[2v_{d}+2f_{e}+\beta_{b}+\beta_{d}-2c_{d}(1-g)]/(8gc_{d})$
Equilibrium profits/total utilities
Platforms' profit $\Pi_{i}^{TT}=t/2+[4(V_{1}+f_{e})^2-B_{0}-2\beta_{b}\beta_{d}]/(32gc_{d})-c_{T}$
Consumers' total utility $BS_{i}^{TT}=(4v_{b}-5t)/8+[B_{0}+2(\beta_{b}+\beta_{d})(V_{1}+f_{e})]/(16gc_{d})$
Developers' total utility $DS_{i}^{TT}=[2(V_{1}+f_{e})+\beta_{b}+\beta_{d}]^2/(64gc_{d})$
Decisions
Consumers' price $p_{bi}^{TT}=t+\beta_{b}(1-g)/(4g)-\beta_{b}[2(v_{d}+f_{e}) +\beta_{b}+3\beta_{d}]/(8gc_{d})$
Developers' price $p_{di}^{TT}=(2v_{d}-2c_{d}+2c_{d}g+2f_{e}+\beta_{b}-\beta_{d})/4$
Consumers' number $n_{bi}^{TT}=1/2$
Developers' number $n_{di}^{TT}=[2v_{d}+2f_{e}+\beta_{b}+\beta_{d}-2c_{d}(1-g)]/(8gc_{d})$
Equilibrium profits/total utilities
Platforms' profit $\Pi_{i}^{TT}=t/2+[4(V_{1}+f_{e})^2-B_{0}-2\beta_{b}\beta_{d}]/(32gc_{d})-c_{T}$
Consumers' total utility $BS_{i}^{TT}=(4v_{b}-5t)/8+[B_{0}+2(\beta_{b}+\beta_{d})(V_{1}+f_{e})]/(16gc_{d})$
Developers' total utility $DS_{i}^{TT}=[2(V_{1}+f_{e})+\beta_{b}+\beta_{d}]^2/(64gc_{d})$
Table 4.  Factors of effect asymmetric strategy to be a SPNE
$ t=1 $ $ c_{d}=1 $
$ c_{d} $ $ \hat{f}_{e1} $ $ \hat{f}_{e2} $ $ \hat{f}_{e2}-\hat{f}_{e1} $ $ t $ $ \hat{f}_{e1} $ $ \hat{f}_{e2} $ $ \hat{f}_{e2}-\hat{f}_{e1} $
1 0.31 0.61 0.30 1 0.31 0.61 0.30
3 0.93 1.72 0.79 3 0.34 0.58 0.24
5 1.52 2.74 1.22 5 0.36 0.56 0.20
7 2.08 3.69 1.61 7 0.37 0.54 0.17
9 2.62 4.59 1.96 9 0.38 0.53 0.15
$ t=1 $ $ c_{d}=1 $
$ c_{d} $ $ \hat{f}_{e1} $ $ \hat{f}_{e2} $ $ \hat{f}_{e2}-\hat{f}_{e1} $ $ t $ $ \hat{f}_{e1} $ $ \hat{f}_{e2} $ $ \hat{f}_{e2}-\hat{f}_{e1} $
1 0.31 0.61 0.30 1 0.31 0.61 0.30
3 0.93 1.72 0.79 3 0.34 0.58 0.24
5 1.52 2.74 1.22 5 0.36 0.56 0.20
7 2.08 3.69 1.61 7 0.37 0.54 0.17
9 2.62 4.59 1.96 9 0.38 0.53 0.15
Table 5.  Nash equilibrium of non-identical platforms
$ (M, M) $ $ (T, T) $
$p_{bi, E}^{MM}=p_{bi}^{MM}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{1})-E_{3}/(12gc_{d})]$ $p_{bi, E}^{TT}=p_{bi}^{TT}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{2})-E_{3}/(12gc_{d})]$
$p_{di, E}^{MM}=p_{di}^{MM}+F(i)E_{2}/W_{1}+(v_{di}-v_{d})/2$ $p_{di, E}^{TT}=p_{di}^{TT}+F(i)E_{2}/(4W_{2})+(v_{di}-v_{d})/2$
$n_{bi, E}^{MM}=n_{bi}^{MM}+F(i)E_{3}/W_{1}$ $n_{bi, E}^{TT}=n_{bi}^{TT}+F(i)E_{3}/W_{2}$
$n_{di, E}^{MM}=n_{di}^{MM}+F(i)E_{4}/W_{1}+(v_{di}-v_{d})/(4gc_{d})$ $n_{di, E}^{TT}=n_{di}^{TT}+F(i)E_{4}/W_{2}+(v_{di}-v_{d})/(4gc_{d})$
$\Pi_{i, E}^{MM}=\Pi_{i}^{MM}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}-\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$} $\Pi_{i, E}^{TT}=\Pi_{i}^{TT}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}+\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$}
$(M, T)$
$p_{bi, E}^{MT}=p_{bi}^{MT}-\beta_{b}(v_{di}-v_{d})/(4gc_{d})-F(i)E_{3}(B_{2}-2W_{3})/(24gc_{d}W_{3})$
$p_{di, E}^{MT}=p_{di}^{MT}+F(i)E_{2}/(4W_{3})+(v_{di}-v_{d})/2$
$n_{bi, E}^{MT}=n_{bi}^{MT}+F(i)E_{3}/(2W_{3})$
$n_{di, E}^{MT}=n_{di}^{MT}+F(i)E_{4}/W_{3}+(v_{di}-v_{d})/(4gc_{d})$
$\Pi_{1, E}^{MT}=\Pi_{1}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}+\delta-2\phi)\delta]/(96gc_{d})$ $-(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})-(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$
$\Pi_{2, E}^{MT}=\Pi_{2}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}-\delta-2\phi+2f_{e})\delta]/(96gc_{d})$ $+(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$
$(T, M)$
$\Pi_{1, E}^{TM}=\Pi_{1, E}^{MT}+[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[8\delta(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2]/(24gc_{d}W_{3})$}}
$\Pi_{2, E}^{TM}=\Pi_{2, E}^{MT}-[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[(\beta_{b}-\beta_{d})^2-8\delta(\beta_{b}+\beta_{d})]/(24gc_{d}W_{3})$
$ (M, M) $ $ (T, T) $
$p_{bi, E}^{MM}=p_{bi}^{MM}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{1})-E_{3}/(12gc_{d})]$ $p_{bi, E}^{TT}=p_{bi}^{TT}-3\beta_{b}(v_{di}-v_{d})/(12gc_{d})$ $-F(i)[(\beta_{b}+2\beta_{d})E_{2}/(24gc_{d}W_{2})-E_{3}/(12gc_{d})]$
$p_{di, E}^{MM}=p_{di}^{MM}+F(i)E_{2}/W_{1}+(v_{di}-v_{d})/2$ $p_{di, E}^{TT}=p_{di}^{TT}+F(i)E_{2}/(4W_{2})+(v_{di}-v_{d})/2$
$n_{bi, E}^{MM}=n_{bi}^{MM}+F(i)E_{3}/W_{1}$ $n_{bi, E}^{TT}=n_{bi}^{TT}+F(i)E_{3}/W_{2}$
$n_{di, E}^{MM}=n_{di}^{MM}+F(i)E_{4}/W_{1}+(v_{di}-v_{d})/(4gc_{d})$ $n_{di, E}^{TT}=n_{di}^{TT}+F(i)E_{4}/W_{2}+(v_{di}-v_{d})/(4gc_{d})$
$\Pi_{i, E}^{MM}=\Pi_{i}^{MM}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}-\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$} $\Pi_{i, E}^{TT}=\Pi_{i}^{TT}+E_{2}^2/(96gc_{d}W_{1}^2)+E_{3}^2/(24gc_{d}W_{1})$ $+F(i)E_{3}(\beta_{b}+\beta_{d})^2/(48gc_{d}W_{1})+E_{3}/(12gc_{d})$ $+[(v_{di}+v_{d})-2c_{d}+2c_{d}g](v_{di}-v_{d})/(8gc_{d})$}
$(M, T)$
$p_{bi, E}^{MT}=p_{bi}^{MT}-\beta_{b}(v_{di}-v_{d})/(4gc_{d})-F(i)E_{3}(B_{2}-2W_{3})/(24gc_{d}W_{3})$
$p_{di, E}^{MT}=p_{di}^{MT}+F(i)E_{2}/(4W_{3})+(v_{di}-v_{d})/2$
$n_{bi, E}^{MT}=n_{bi}^{MT}+F(i)E_{3}/(2W_{3})$
$n_{di, E}^{MT}=n_{di}^{MT}+F(i)E_{4}/W_{3}+(v_{di}-v_{d})/(4gc_{d})$
$\Pi_{1, E}^{MT}=\Pi_{1}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}+\delta-2\phi)\delta]/(96gc_{d})$ $-(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})-(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$
$\Pi_{2, E}^{MT}=\Pi_{2}^{MT}+(\beta_{b}-\beta_{d})^2E_{3}(E_{3}-2R_{2})/(96gc_{d}W_{3}^2)+[8E_{3}+12(2v_{d}-\delta-2\phi+2f_{e})\delta]/(96gc_{d})$ $+(\beta_{b}+\beta_{d})\delta[8gc_{d}t_{e}+4f_{e}(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2-2E_{3}]/(48gc_{d}W_{3})$
$(T, M)$
$\Pi_{1, E}^{TM}=\Pi_{1, E}^{MT}+[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[8\delta(\beta_{b}+\beta_{d})+(\beta_{b}-\beta_{d})^2]/(24gc_{d}W_{3})$}}
$\Pi_{2, E}^{TM}=\Pi_{2, E}^{MT}-[3f_{e}^2+8gc_{d}t_{e}+f_{e}(6\delta+6V_1+4\beta_{b}+4\beta_{d})]/(24gc_{d})+(\beta_{b}-\beta_{d})R_{2}E_{4}/(3W_{3}^2)$ $+R_{2}[(\beta_{b}-\beta_{d})^2-8\delta(\beta_{b}+\beta_{d})]/(24gc_{d}W_{3})$
Table 1A.  Mathematical abbreviation and threshold
$V_{1}=v_{d}-c_{d}(1-g)$ $B_{0}=\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d}$
$B_{1}=\beta_{b}^2+\beta_{d}^2+6\beta_{b}\beta_{d}$ $\phi=c_{d}(1-g)$
$R_{1}=\beta_{b}^2-2\beta_{d}^2+\beta_{b}\beta_{d}$ $R_{2}=f_{e}(\beta_{b}+\beta_{d})+2gc_{d}t_{e}$
$R_{3}=24gc_{d}(c_{T}-c_{M})-6f_{e}[v_{d}-c_{d}(1-g)]$ $+2gc_{d}t_{e}-f_{e}(3f_{e}+2\beta_{b}+2\beta_{d})$ $R_{4}=f_{e}(\beta_{b}+\beta_{d})+5gc_{d}t_{e}$
$R_{5}=6(\beta_{b}+\beta_{d})(2v_{d}-2c_{d}+2c_{d}g+f_{e})$ $+48gc_{d}v_{b}+2\beta_{b}^2+8\beta_{b}\beta_{d}+\beta_{d}^2$ $R_{6}=2v_{d}-2c_{d}(1-g)+f_{e}+\beta_{b}+\beta_{d}$
$R_{7}=f_{e}(\beta_{b}+\beta_{d})+8gc_{d}t_{e}$ $E_{2}=2\delta(\beta_{b}^2-\beta_{d}^2)$
$E_{3}=2\delta(\beta_{b}+\beta_{d})$ $E_{4}=2\delta(\beta_{b}+\beta_{d})^2/(8gc_{d})$
$E_{5}=2v_{d}+\beta_{b}+\beta_{d}$ $W_{1}=12gc_{d}(t+t_{e})-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$
$W_{2}=12gc_{d}t-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$ $W_{3}=6gc_{d}(2t+t_{e})-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$
$H_{1}=[6f_{e}V_{1}+3f_{e}^2+2f_{e}(\beta_{b}+\beta_{d})$ $-2gc_{d}t_{e}]/(24gc_{d})$} $H_{2}=R_{2}[4gc_{d}t_{e}-(\beta_{b}-\beta_{d})^2$ $+2f_{e}(\beta_{b}+\beta_{d})]/(48gc_{d}W_{3})$}
$H_{3}=R_{2}[4gc_{d}t_{e}+(\beta_{b}-\beta_{d})^2$ $+2f_{e}(\beta_{b}+\beta_{d})]/(48gc_{d}W_{3})$} $H_{4}=R_{2}^2(\beta_{b}-\beta_{d})^2/(96gc_{d}W_{3}^2)$
Threshold
$\hat{t}^{MM}=\beta_{b}\beta_{d}/(2gc_{d})-t_{e}$ $\hat{t}^{MT}=\beta_{b}\beta_{d}/(2gc_{d})-t_{e}/2$
$\hat{t}^{TT}=\beta_{b}\beta_{d}/(2gc_{d})$ $\hat{t}_{e7}=-(1+g)f_{e}(2\phi+f_{e}+E_{5})/(gc_{d})$
$\hat{f}_{e7}=\{2\phi-2v_{d}-\beta_{b}-\beta_{d}+\sqrt{(1+g)[(1+g)(E_{5}-2\phi)^2-4gc_{d}t_{e}]}/(1+g)\}/2$
$V_{1}=v_{d}-c_{d}(1-g)$ $B_{0}=\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d}$
$B_{1}=\beta_{b}^2+\beta_{d}^2+6\beta_{b}\beta_{d}$ $\phi=c_{d}(1-g)$
$R_{1}=\beta_{b}^2-2\beta_{d}^2+\beta_{b}\beta_{d}$ $R_{2}=f_{e}(\beta_{b}+\beta_{d})+2gc_{d}t_{e}$
$R_{3}=24gc_{d}(c_{T}-c_{M})-6f_{e}[v_{d}-c_{d}(1-g)]$ $+2gc_{d}t_{e}-f_{e}(3f_{e}+2\beta_{b}+2\beta_{d})$ $R_{4}=f_{e}(\beta_{b}+\beta_{d})+5gc_{d}t_{e}$
$R_{5}=6(\beta_{b}+\beta_{d})(2v_{d}-2c_{d}+2c_{d}g+f_{e})$ $+48gc_{d}v_{b}+2\beta_{b}^2+8\beta_{b}\beta_{d}+\beta_{d}^2$ $R_{6}=2v_{d}-2c_{d}(1-g)+f_{e}+\beta_{b}+\beta_{d}$
$R_{7}=f_{e}(\beta_{b}+\beta_{d})+8gc_{d}t_{e}$ $E_{2}=2\delta(\beta_{b}^2-\beta_{d}^2)$
$E_{3}=2\delta(\beta_{b}+\beta_{d})$ $E_{4}=2\delta(\beta_{b}+\beta_{d})^2/(8gc_{d})$
$E_{5}=2v_{d}+\beta_{b}+\beta_{d}$ $W_{1}=12gc_{d}(t+t_{e})-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$
$W_{2}=12gc_{d}t-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$ $W_{3}=6gc_{d}(2t+t_{e})-(\beta_{b}^2+\beta_{d}^2+4\beta_{b}\beta_{d})$
$H_{1}=[6f_{e}V_{1}+3f_{e}^2+2f_{e}(\beta_{b}+\beta_{d})$ $-2gc_{d}t_{e}]/(24gc_{d})$} $H_{2}=R_{2}[4gc_{d}t_{e}-(\beta_{b}-\beta_{d})^2$ $+2f_{e}(\beta_{b}+\beta_{d})]/(48gc_{d}W_{3})$}
$H_{3}=R_{2}[4gc_{d}t_{e}+(\beta_{b}-\beta_{d})^2$ $+2f_{e}(\beta_{b}+\beta_{d})]/(48gc_{d}W_{3})$} $H_{4}=R_{2}^2(\beta_{b}-\beta_{d})^2/(96gc_{d}W_{3}^2)$
Threshold
$\hat{t}^{MM}=\beta_{b}\beta_{d}/(2gc_{d})-t_{e}$ $\hat{t}^{MT}=\beta_{b}\beta_{d}/(2gc_{d})-t_{e}/2$
$\hat{t}^{TT}=\beta_{b}\beta_{d}/(2gc_{d})$ $\hat{t}_{e7}=-(1+g)f_{e}(2\phi+f_{e}+E_{5})/(gc_{d})$
$\hat{f}_{e7}=\{2\phi-2v_{d}-\beta_{b}-\beta_{d}+\sqrt{(1+g)[(1+g)(E_{5}-2\phi)^2-4gc_{d}t_{e}]}/(1+g)\}/2$
[1]

Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182

[2]

David Cantala, Juan Sebastián Pereyra. Endogenous budget constraints in the assignment game. Journal of Dynamics & Games, 2015, 2 (3&4) : 207-225. doi: 10.3934/jdg.2015002

[3]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409

[4]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

[5]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451

[6]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[7]

Xiaoyi Zhou, Tong Ye, Tony T. Lee. Designing and analysis of a Wi-Fi data offloading strategy catering for the preference of mobile users. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021038

[8]

Akio Matsumot, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021069

[9]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349

[10]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247

[11]

John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021004

[12]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[13]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881

[14]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[15]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933

[16]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021025

[17]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

[18]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[19]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[20]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020

2019 Impact Factor: 1.366

Article outline

Figures and Tables

[Back to Top]