Stationary Kirchhoff equations and systems with reaction terms

Radu Precup; Andrei Stan; Radu Precup; Andrei Stan

doi:10.3934/math.2022836

AIMS Mathematics

2022, Volume 7, Issue 8: 15258-15281. doi: 10.3934/math.2022836

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Research article

Stationary Kirchhoff equations and systems with reaction terms

Radu Precup ^{1
,
,},
Andrei Stan ²

1.
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania
2.
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania

Received: 19 December 2021 Revised: 07 June 2022 Accepted: 08 June 2022 Published: 17 June 2022
MSC : 35J65, 47J25

In this paper, the operator approach based on the fixed point principles of Banach and Schaefer is used to establish the existence of solutions to stationary Kirchhoff equations with reaction terms. Next, for a coupled system of Kirchhoff equations, it is proved that under suitable assumptions, there exists a unique solution which is a Nash equilibrium with respect to the energy functionals associated to the equations of the system. Both global Nash equilibrium, in the whole space, and local Nash equilibrium, in balls are established. The solution is obtained by using an iterative process based on Ekeland's variational principle and whose development simulates a noncooperative game.
- Kirchhoff equation,
- fixed point principle,
- Nash equilibrium,
- Ekeland variational principle
Citation: Radu Precup, Andrei Stan. Stationary Kirchhoff equations and systems with reaction terms[J]. AIMS Mathematics, 2022, 7(8): 15258-15281. doi: 10.3934/math.2022836

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Abstract

In this paper, the operator approach based on the fixed point principles of Banach and Schaefer is used to establish the existence of solutions to stationary Kirchhoff equations with reaction terms. Next, for a coupled system of Kirchhoff equations, it is proved that under suitable assumptions, there exists a unique solution which is a Nash equilibrium with respect to the energy functionals associated to the equations of the system. Both global Nash equilibrium, in the whole space, and local Nash equilibrium, in balls are established. The solution is obtained by using an iterative process based on Ekeland's variational principle and whose development simulates a noncooperative game.

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