Geodesic
Existence and uniqueness of common solutions of strict Stampacchia and Minty variational inequalities with non-monotone operators in Banach spaces
Minimax theory and its applications, Tome 8 (2023) no. 2
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We study the existence of common solutions of the Stampacchia and Minty variational inequalities associated to non-monotone operators in Banach spaces, as a consequence of a general saddle-point theorem. We prove, in particular, that if (X,∥ · ∥) is a Banach space, whose norm has suitable convexity and differentiability properties, Bρ := {x ∈ X : ∥x∥ ≤ ρ}, and Φ : Bρ → X∗ is a C1 function with Lipschitzian derivative, with Φ(0)= 0, then for each r > 0 small enough, there exists a unique x∗ ∈ Br, with ∥x∥ = r, such that max{〈Φ(x∗),x∗ −x〉,〈Φ(x),x∗ −x〉} < 0 for all x ∈Br\{x∗}. Our results extend to the setting of Banach spaces some results previously obtained by B. Ricceri in the setting of Hilbert spaces.
Mots-clés : Saddle point, minimax theorem, Banach space, modulus of convexity, C1 function, Stampacchia and Minty variational inequalities, ball, non-monotone operators 
@article{MTA_2023_8_2_a6,
     author = {Filippo Cammaroto and Paolo Cubiotti},
     title = {Existence and uniqueness of common solutions of strict {Stampacchia} and {Minty} variational inequalities with non-monotone operators in {Banach} spaces},
     journal = {Minimax theory and its applications},
     year = {2023},
     volume = {8},
     number = {2},
     zbl = {1550.47034},
     url = {http://geodesic.mathdoc.fr/item/MTA_2023_8_2_a6/}
}
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Filippo Cammaroto; Paolo Cubiotti. Existence and uniqueness of common solutions of strict Stampacchia and Minty variational inequalities with non-monotone operators in Banach spaces. Minimax theory and its applications, Tome 8 (2023) no. 2. http://geodesic.mathdoc.fr/item/MTA_2023_8_2_a6/