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Blum and Oettli in their seminal paper studied monotone bifunctions and proved the existence of an equilibrium point. In this work, we extend theirmain result by replacing monotone bifunction with a more general bifunction and prove the existence of an equilibrium point.

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Abstract. Let X be a separable Banach space and M be a subspace of X. A bounded Linear operator T on X is subspace balanced convex-cyclic for a subspace M, if there exists a vector x∈X such that the intersection of balanced convex hull of orb(T,x) with M is dense in M. We give an example of subspace balanced convex-cyclic operator that is not balanced convex-cyclic. Also we give an improvement of the Kitailike criterion for subspace balanced convex-cyclicity and bring on with the Hahn-Banach characterization for subspace balanced convex-cyclicity.