Composite Tseng-type extragradient algorithms with adaptive inertial correction strategy for solving bilevel split pseudomonotone VIP under split common fixed-point constraint

This paper investigates the bilevel split pseudomonotone variational inequality problem (BSPVIP) and the split common fixed point problem (SCFPP) involving demimetric mappings in real Hilbert spaces. We propose a novel composite Tseng-type extragradient method that incorporates an adaptive inertial correction term to effectively address the BSPVIP under the SCFPP constraints. Our approach combines an inertial technique with a self-adaptive step size strategy to enhance algorithmic efficiency. The BSPVIP consists of an upper-level variational inequality problem for a strongly monotone operator and a lower-level split variational inequality problem for two pseudomonotone operators. Under mild assumptions, we establish the strong convergence of the proposed algorithm. To demonstrate the practical applicability of the method, we apply it to a BSPVIP under split fixed point problem constraints. A numerical example is provided to illustrate the algorithm’s performance and examine the influence of the involved parameters on its behavior.