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We use the direct variational method, the Ekeland variational principle, themountain pass geometry and Karush-Kuhn-Tucker theorem in order to investigateexistence and multiplicity results for boundary value problems connected withthe discrete p(.)-Laplacian on weighted finite graphs. Several auxiliaryinequalities for the discrete p(.)-Laplacian on finite graphs are also derived.Positive solutions are considered.

Existence and multiplicity results for boundary value problems connected with the discrete p(.)-Laplacian on weighted finite graphs