Summary: In this article, we consider the problem of approximating the common solution of monotone inclusions and demicontraction fixed point problems. Firstly, we present Tseng splitting method which incorporates the viscosity technique, new self-adaptive step size and double inertial extrapolations techniques for approximating the solution of the problem in the setting of Hilbert spaces. Unlike some double inertial methods recently studied by many authors, our algorithm does not require computation onto half space, containing the feasible set. The co-coerciveness condition that is often impose on the single-valued operator and the imposition of other stringent assumptions are not required in our method. The suggested method does not require any line search technique. The method uses a new non-monotonic step size which is allowed to increase from iteration to iteration. The step size embeds some relaxation parameters which also improve the rate of convergence of our method. The suggested method only needs one backward computation of the multi-valued operator at each iteration and one forward computation of the single-valued operator; a concept that has not been considered in several splitting methods for strongly convergence in the literature. We prove the strong convergence results of our method under some mild assumptions on the control parameters. We apply our main results to the solutions of several optimization problems. In three numerical experiments, the applicability and efficiency of our new method are compared with some well known methods in the existing literature. Our results improve, unify and generalize several well known results in the literature.