A double inertial method for non-monotone variational inequality problem

Abstract In this paper, we propose a viscosity-type double inertial algorithm with self-adaptive stepsizes for solving non-monotone variational inequality problems involving uniformly continuous operators in real Hilbert spaces. The proposed method combines viscosity approximation and double inertial extrapolation techniques with a self-adaptive stepsize rule that does not require prior knowledge of the Lipschitz constant. Under suitable conditions, we establish a strong convergence of the generated sequence without assuming monotonicity, Lipschitz continuity, line-search procedures, or additional projection techniques. The obtained results extend several existing inertial and viscosity methods to a broader class of non-monotone and non-Lipschitz variational inequality problems. Numerical examples are presented to illustrate the effectiveness of the proposed algorithm.