What does this research mean for the field?

A unified Banach-space framework based on monotone operator theory and low-order regularization successfully establishes the existence of strong solutions for stationary first-order mean-field games, overcoming the limitations of previous Hilbert-space methods. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To develop a unified Banach-space framework using monotone operator theory to prove the existence of strong and weak solutions for stationary first-order mean-field games.

August 22, 2026SIAM Journal on Mathematical AnalysisOpen Access

Solving Mean-Field Games with Monotonicity Methods in Banach Spaces

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Authors

RFRita FerreiraAdministração Regional de Saúde de Lisboa e Vale do Tejo DGDiogo A. GomesKing Abdullah University of Science and Technology MUMelih UcerKing Abdullah University of Science and Technology

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Implication

Theoretical analysis demonstrates existence of strong solutions for stationary mean-field games in Banach spaces, highlighting a direct path for numerical algorithms.

Key Points

To develop a unified Banach-space framework using monotone operator theory to prove the existence of strong and weak solutions for stationary first-order mean-field games.
Formulated the coupled mean-field game system as a variational inequality within Banach spaces rather than Hilbert spaces.
Restored operator coercivity using a low-order p-Laplacian regularization, deriving uniform a priori estimates and passing to the limit via Minty's method.
Applied infimal convolution to regularize Hamiltonians satisfying only minimal growth hypotheses.
Established the existence of strong solutions for mean-field game models with standard power-growth and singular congestion using low-order regularization.
Demonstrated the existence of weak solutions under minimal Hamiltonian growth hypotheses via infimal convolution.
Unified prior existence results into a framework that avoids high-order smoothing and directly accommodates modern numerical algorithms.

Cite This Study

Ferreira et al. (2026) studied this question.

synapsesocial.com/papers/6a895e6bca7ade938187c87c https://doi.org/10.1137/25m1784521

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Implication

Theoretical analysis demonstrates existence of strong solutions for stationary mean-field games in Banach spaces, highlighting a direct path for numerical algorithms.

Solving Mean-Field Games with Monotonicity Methods in Banach Spaces

Discussion

Member takes

Implication

Key Points

Cite This Study

Solving Mean-Field Games with Monotonicity Methods in Banach Spaces

Discussion

Member takes

Implication

Key Points

Cite This Study

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