The variants of projection extragradient algorithms for nonmonotone equilibrium problems in Hilbert spaces

We introduce three novel variants of the projection extragradient algorithm for solving equilibrium problems in real Hilbert spaces, without imposing any generalized monotonicity assumptions on the bifunctions involved. The proposed approach replaces the traditional shrinking projection step with a projection onto the intersection of the feasible set and a suitably chosen half-space, offering a more flexible and effective strategy. To address cases where the bifunctions do not satisfy a Lipschitz-type condition, we incorporate a general linesearch mechanism for determining the step size. When the bifunctions are of Lipschitz-type with known constants, this linesearch is rendered unnecessary; in the absence of such constants, we propose an adaptive step size rule. All three algorithms are rigorously proven to converge strongly to a solution, even without assuming the joint weak continuity of the bifunction-an assumption often required in existing literature. Numerical examples are presented to illustrate the efficiency and practical performance of the proposed methods.