A novel method for solving split variational inequality and fixed point problems

In this article, we consider the problem of approximating the common minimum-norm solution of split variational inequality problem (when the underlying operators are Lipschiptz continuous and quasimonotone) and fixed point problem (when the underlying operator is demimetric). In solving this problem, we propose a modified subgradient extragradient method which incorporates the original Mann technique and double inertial steps. We prove the strong convergence results of the suggested method with mild conditions on the control parameters. The strong convergence results of our method do not rely on Mann-type and viscosity techniques, unlike several existing methods in the literature. Dependence on the knowledge of the bounded linear operator norm is not required during implementation of our method. We present two numerical examples to test the applicability of our method, which includes double inertial steps and compare the convergence efficiency of our method with some well known methods in the literature with single inertial term, and methods without inertial terms. The results in this article improve, extend and generalize several existing results from the setting of finding solutions of optimization problems in a single solution set to the setting of finding common solutions in two solution sets.