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It is well known that the projection method is not convergent for monotoneequilibrium problems. Recently Sosa \textit{et al.} in \cite{SS2011} proposed aprojection algorithm ensuring convergence for paramonotone equilibriumproblems. In this paper we modify this algorithm to obtain a splittingconvergent one for the case when the bifunction is the sum of the two ones. Ateach iteration, two strongly convex subprograms are required to solveseparately, one for each component bifunction. We show that the algorithm isconvergent for paramonotone bifunction without any Lipschitz type condition aswell as H\"older continuity of the involved bifunctions. Furthermore, we showthat the ergodic sequence defined by the algorithm's iterates converges to asolution without paramonotonicity property. We use the proposed algorithm tosolve a jointly constrained Cournot-Nash model. The computational results showthat this algorithm is efficient for the model with a restart strategy.

Splitting gradient algorithms for solving monotone equilibrium problems