Vol. 10, No. 3, 2017

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Existence of positive solutions for an approximation of stationary mean-field games

Nojood Almayouf, Elena Bachini, Andreia Chapouto, Rita Ferreira, Diogo Gomes, Daniela Jordão, David Evangelista Junior, Avetik Karagulyan, Juan Monasterio, Levon Nurbekyan, Giorgia Pagliar, Marco Piccirilli, Sagar Pratapsi, Mariana Prazeres, João Reis, André Rodrigues, Orlando Romero, Maria Sargsyan, Tommaso Seneci, Chuliang Song, Kengo Terai, Ryota Tomisaki, Hector Velasco-Perez, Vardan Voskanyan and Xianjin Yang

Vol. 10 (2017), No. 3, 473–493
Abstract

Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.

Keywords
mean-field games, low-order regularizations, monotone methods, positive solutions
Mathematical Subject Classification 2010
Primary: 49L25, 91A13, 35J87
Milestones
Received: 12 January 2016
Revised: 21 March 2016
Accepted: 15 April 2016
Published: 14 December 2016

Communicated by Kenneth S. Berenhaut
Authors
Nojood Almayouf
Effat University
Jeddah 23523
Saudi Arabia
Elena Bachini
Department of Mathematics
University of Padova
I-35121 Padova
Italy
Andreia Chapouto
University of Coimbra
5400-000 Chaves
Portugal
Rita Ferreira
CEMSE Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia
Diogo Gomes
CEMSE Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia
Daniela Jordão
University of Coimbra
3001-501 Coimbra
Portugal
David Evangelista Junior
CEMSE Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia
Avetik Karagulyan
Faculty of Mathematics and Mechanics
Yerevan State University
Republic of Armenia, Yerevan, 0025, 1 Alex Manoogian
0025 Yerevan
Armenia
Juan Monasterio
University of Buenos Aires
1414CKS Buenos Aires
Argentina
Levon Nurbekyan
CEMSE Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia
Giorgia Pagliar
University of Verona
I-37045 Legnago
Italy
Marco Piccirilli
University of Rome Tor Vergata
I-35121 Padova
Italy
Sagar Pratapsi
University of Coimbra
4465-040 S. Mamede de Infesta
Portugal
Mariana Prazeres
CEMSE Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia
João Reis
CEMSE Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia
André Rodrigues
University of Coimbra
3040-109 Coimbra
Portugal
Orlando Romero
University of Lisbon
9000-690 Funchal
Portugal
Maria Sargsyan
American University of Armenia
0025 Yerevan
Armenia
Tommaso Seneci
University of Verona
via Brunelleschi 1
I-37178 Verona
Italy
Chuliang Song
MIT
Cambridge, MA 02140
United States
Kengo Terai
Waseda University
Saitama 338-0001
Japan
Ryota Tomisaki
Waseda University
210-10 Nase, Totsuka
Yokohama 245-00051
Japan
Hector Velasco-Perez
National Autonomous University of Mexico
Cerro Copor #16
Campestre Churubusco
02100 Mexico City,
Mexico
Vardan Voskanyan
CEMSE Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia
Xianjin Yang
CEMSE Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia